Content-Type: multipart/mixed; boundary="-------------0405070533445" This is a multi-part message in MIME format. ---------------0405070533445 Content-Type: text/plain; name="04-148.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="04-148.keywords" center manifold, KAM theory, homoclinic orbits, Poincar\'e-Melnikov method ---------------0405070533445 Content-Type: application/postscript; name="Tori3.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Tori3.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.86e Copyright 2001 Radical Eye Software %%Title: Tori3.dvi %%Pages: 27 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips Tori3 %DVIPSParameters: dpi=600, compressed %DVIPSSource: TeX output 2004.05.07:1211 %%BeginProcSet: texc.pro %! /TeXDict 300 dict def TeXDict begin/N{def}def/B{bind def}N/S{exch}N/X{S N}B/A{dup}B/TR{translate}N/isls false N/vsize 11 72 mul 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Fo(;)g(\021)t Ft(\))p Fo(;)44 b(F)2295 1947 y Fu(2)2362 1932 y Ft(=)28 b Fo(f)2514 1947 y Fu(2)2553 1932 y Ft(\()p Fo(\030)2634 1947 y Fu(1)2673 1932 y Fo(;)17 b(\030)2760 1947 y Fu(2)2799 1932 y Fo(;)g(\021)t Ft(\))p Fo(;)-118 2096 y Ft(with)35 b Fo(h;)17 b(f)255 2111 y Fu(1)294 2096 y Fo(;)g(f)386 2111 y Fu(2)461 2096 y Ft(smo)s(oth)34 b(and)i Fo(\030)1044 2111 y Fu(1)1115 2096 y Ft(=)d(\()p Fo(x)1317 2060 y Fu(2)1317 2121 y(1)1381 2096 y Ft(+)24 b Fo(y)1533 2060 y Fu(2)1529 2121 y(1)1571 2096 y Ft(\))p Fo(=)p Ft(2,)36 b Fo(\030)1813 2111 y Fu(2)1884 2096 y Ft(=)d(\()p Fo(x)2086 2060 y Fu(2)2086 2121 y(2)2150 2096 y Ft(+)24 b Fo(y)2302 2060 y Fu(2)2298 2121 y(2)2340 2096 y Ft(\))p Fo(=)p Ft(2,)36 b Fo(\021)g Ft(=)c Fo(x)2786 2111 y Fu(3)2826 2096 y Fo(y)2874 2111 y Fu(3)2913 2096 y Ft(.)52 b(This)36 b(result)f(for)g(the)-118 2216 y(analytic)c(case)j (w)m(as)f(pro)m(v)m(ed)h(b)m(y)g(V)-8 b(ey)33 b(in)f([16)o(])h(and)g(w) m(as)g(extended)i(to)d(the)h Fo(C)2746 2180 y Fn(1)2821 2216 y Ft(-case)g(b)m(y)g(Eliasson)f([5].)28 2337 y(The)j(quoted)g (result)e(allo)m(ws)g(us)h(to)g(describ)s(e)g(completely)f(the)h (structure)h(of)f Fo(X)3017 2352 y Fm(H)3075 2361 y Fl(0)3147 2337 y Ft(lo)s(cally)d(near)j Fo(O)s Ft(.)-118 2457 y(The)27 b(represen)m(tation)g(obtained)e(p)s(ermits)g(to)h(write)g(do)m(wn)h (the)f(in)m(tegrals)f(in)g(the)i(follo)m(wing)c(form)i(where,)-118 2578 y(without)32 b(loss)g(of)g(generalit)m(y)-8 b(,)32 b(w)m(e)i(can)f(supp)s(ose)h(that)e(all)e(in)m(tegrals)i(v)-5 b(anish)32 b(at)g Fo(O)1208 2765 y(H)1289 2780 y Fu(0)1356 2765 y Ft(=)c Fo(!)1521 2780 y Fu(1)1560 2765 y Fo(\030)1603 2780 y Fu(1)1664 2765 y Ft(+)22 b Fo(!)1823 2780 y Fu(2)1862 2765 y Fo(\030)1905 2780 y Fu(2)1966 2765 y Ft(+)g Fo(\025\021)k Ft(+)c Fi(\001)17 b(\001)g(\001)e Fo(;)1208 3006 y(F)1271 3021 y Fu(1)1339 3006 y Ft(=)27 b Fo(\027)1490 3021 y Fu(1)1530 3006 y Fo(\030)1573 3021 y Fu(1)1634 3006 y Ft(+)22 b Fo(\027)1780 3021 y Fu(2)1820 3006 y Fo(\030)1863 3021 y Fu(2)1924 3006 y Ft(+)g Fo(\026\021)j Ft(+)d Fi(\001)17 b(\001)g(\001)e Fo(;)1208 3247 y(F)1271 3262 y Fu(2)1339 3247 y Ft(=)27 b Fo(\024)1498 3262 y Fu(1)1538 3247 y Fo(\030)1581 3262 y Fu(1)1642 3247 y Ft(+)22 b Fo(\024)1796 3262 y Fu(2)1835 3247 y Fo(\030)1878 3262 y Fu(2)1940 3247 y Ft(+)g Fo(\033)t(\021)j Ft(+)d Fi(\001)17 b(\001)g(\001)e Fo(;)3655 3007 y Ft(\(1\))-118 3442 y(here)34 b(dots)h(mean)e(terms)h (of)f(higher)g(order)h(in)f(the)h(v)-5 b(ariables)33 b(\()p Fo(\030)2280 3457 y Fu(1)2319 3442 y Fo(;)17 b(\030)2406 3457 y Fu(2)2444 3442 y Fo(;)g(\021)t Ft(\).)47 b(In)34 b(this)f(form)g(the)h(simplicit)m(y)-118 3562 y(condition)d(for)h(the)h (p)s(oin)m(t)f Fo(O)j Ft(is)d(equiv)-5 b(alen)m(t)32 b(to)g(the)h(condition)1339 3871 y(\001)28 b(=)1552 3672 y Fh(\014)1552 3722 y(\014)1552 3771 y(\014)1552 3821 y(\014)1552 3871 y(\014)1552 3921 y(\014)1552 3971 y(\014)1621 3750 y Fo(!)1682 3765 y Fu(1)1804 3750 y Fo(!)1865 3765 y Fu(2)1989 3750 y Fo(\025)1621 3870 y(\027)1669 3885 y Fu(1)1810 3870 y Fo(\027)1858 3885 y Fu(2)1987 3870 y Fo(\026)1621 3991 y(\024)1677 4006 y Fu(1)1806 3991 y Fo(\024)1862 4006 y Fu(2)1987 3991 y Fo(\033)2088 3672 y Fh(\014)2088 3722 y(\014)2088 3771 y(\014)2088 3821 y(\014)2088 3871 y(\014)2088 3921 y(\014)2088 3971 y(\014)2143 3871 y Fi(6)p Ft(=)g(0)p Fo(:)-118 4180 y Ft(Belo)m(w,)37 b(without)f(loss)g(of)g(generalit)m(y)-8 b(,)36 b(w)m(e)i(regard)e Fo(\025)g Ft(to)g(b)s(e)g(p)s(ositiv)m(e)g(and,)h(for)f (de\014niteness,)j(supp)s(ose)-118 4300 y Fo(!)-57 4315 y Fu(1)18 4300 y Fo(>)e Ft(0.)58 b(In)38 b(V)-8 b(ey-Eliasson)38 b(co)s(ordinates)f(all)f(lo)s(cal)f(in)m(v)-5 b(arian)m(t)36 b(manifolds)g(through)h Fo(O)j Ft(can)e(b)s(e)g(easily)-118 4421 y(c)m(haracterized.)43 b(In)27 b(particular,)g(the)h(lo)s(cal)d (one-dimensional)g(stable)i(manifold)d Fo(W)2981 4384 y Fm(s)3045 4421 y Ft(is)j(the)h(axis)f Fo(x)3549 4436 y Fu(3)3589 4421 y Ft(,)h(the)-118 4541 y(lo)s(cal)22 b(one-dimensional)f(unstable)j(manifold)d Fo(W)1694 4505 y Fm(u)1763 4541 y Ft(is)i(the)i(axis)e Fo(y)2249 4556 y Fu(3)2288 4541 y Ft(,)j(the)e(lo)s(cal)e(4-dimensional)f(symplectic) -118 4661 y(cen)m(ter)37 b(manifold)32 b Fo(W)687 4625 y Fm(c)756 4661 y Ft(is)j(the)h(disk)f Fo(x)1289 4676 y Fu(3)1361 4661 y Ft(=)e Fo(y)1518 4676 y Fu(3)1589 4661 y Ft(=)f(0,)k(the)f(lo)s(cal)e(5-dimensional)f(cen)m(ter-stable)k (manifold)-118 4782 y Fo(W)-12 4746 y Fm(cs)87 4782 y Ft(is)c Fo(y)233 4797 y Fu(3)300 4782 y Ft(=)c(0)k(and)h(the)g(cen)m (ter-unstable)g(manifold)d Fo(W)2033 4746 y Fm(cu)2141 4782 y Ft(is)i Fo(x)2294 4797 y Fu(3)2361 4782 y Ft(=)c(0.)28 4902 y(Our)33 b(next)h(assumption)e(concerns)i(the)g(orbit)d(b)s(eha)m (vior)i(on)g Fo(W)2409 4866 y Fm(c)2443 4902 y Ft(.)44 b(The)34 b(restriction)e(of)g Fo(X)3374 4917 y Fm(H)3432 4926 y Fl(0)3503 4902 y Ft(on)h Fo(W)3745 4866 y Fm(c)-118 5023 y Ft(giv)m(es)42 b(a)g(2)g(d.o.f)g(Hamiltonian)c(v)m(ector)43 b(\014eld)f(with)g(an)g(elliptic)d(singular)h(p)s(oin)m(t)h(at)h Fo(O)s Ft(.)71 b(This)42 b(v)m(ector)-118 5143 y(\014eld)28 b(is)g(in)m(tegrable,)g(and)h(the)g(restriction)e(of)h(an)m(y)h(of)f Fo(F)1928 5158 y Fu(1)1968 5143 y Fo(;)17 b(F)2075 5158 y Fu(2)2143 5143 y Ft(giv)m(es)28 b(an)h(additional)c(in)m(tegral.)41 b(Moreo)m(v)m(er,)-118 5263 y(since)34 b Fo(O)i Ft(is)e(simple,)f(it)f (will)g(also)h(b)s(e)h(simple)f(for)g(the)h(v)m(ector)h(\014eld)f(on)g Fo(W)2656 5227 y Fm(c)2690 5263 y Ft(,)g(since)h(if)e(\001)d Fi(6)p Ft(=)g(0,)k(then)g(one)-118 5384 y(of)i(the)g(determinan)m(ts)g (of)g(the)h(second)g(order)g(comp)s(osed)f(from)f(the)i(\014rst)f(and)h (second)g(columns)f(in)f(\001)-118 5504 y(do)s(es)e(not)f(v)-5 b(anish.)1806 5753 y(2)p eop %%Page: 3 3 3 2 bop -118 407 a Fk(Assumption)36 b(2)49 b Fj(The)35 b(r)-5 b(estriction)35 b(of)h Fo(X)1476 422 y Fm(H)1534 431 y Fl(0)1608 407 y Fj(on)f Fo(W)1854 371 y Fm(c)1924 407 y Fj(has)g Fo(O)j Fj(as)d(a)g(singular)g(p)-5 b(oint)35 b(of)g(gener)-5 b(al)35 b(el)5 b(liptic)-118 527 y(typ)-5 b(e)35 b(\(that)g(is,)g(nonde)-5 b(gener)g(ate)33 b(or)i(iso)-5 b(ener)g(getic)g(al)5 b(ly)34 b(nonde)-5 b(gener)g(ate\).)28 751 y Ft(Let)46 b Fo(H)297 766 y Fu(0)336 751 y Fi(j)364 780 y Fm(W)441 761 y Fg(c)509 751 y Ft(=)33 b Fo(!)679 766 y Fu(1)718 751 y Fo(\030)761 766 y Fu(1)824 751 y Ft(+)24 b Fo(!)985 766 y Fu(2)1024 751 y Fo(\030)1067 766 y Fu(2)1130 751 y Ft(+)g Fo(A\030)1351 715 y Fu(2)1346 776 y(1)1414 751 y Ft(+)g(2)p Fo(B)5 b(\030)1685 766 y Fu(1)1724 751 y Fo(\030)1767 766 y Fu(2)1831 751 y Ft(+)24 b Fo(C)7 b(\030)2056 715 y Fu(2)2051 776 y(2)2119 751 y Ft(+)24 b Fi(\001)17 b(\001)g(\001)33 b Ft(b)s(e)j(the)g (restriction)e(of)i Fo(X)3342 766 y Fm(H)3400 775 y Fl(0)3474 751 y Ft(on)f Fo(W)3718 715 y Fm(c)3752 751 y Ft(.)-118 872 y(Recall)44 b(that)h(the)h(condition)e(for)h(an)h(elliptic)c (singular)i(p)s(oin)m(t)h(to)g(b)s(e)h(of)f(the)h(general)f(elliptic)e (t)m(yp)s(e)-118 992 y(requires)34 b(that)e(some)h(quan)m(tities)g (calculated)e(in)i(terms)f(of)h(the)g(fourth)g(order)g(in)f(the)h (normal)e(form)h(do)-118 1112 y(not)g(v)-5 b(anish.)44 b(F)-8 b(or)31 b(our)i(co)s(ordinates)f(it)f(giv)m(es)i(either)-2 1292 y Fh(\014)-2 1342 y(\014)-2 1391 y(\014)-2 1441 y(\014)-2 1491 y(\014)70 1380 y Fo(A)87 b(B)67 1501 y(B)j(C)350 1292 y Fh(\014)350 1342 y(\014)350 1391 y(\014)350 1441 y(\014)350 1491 y(\014)406 1441 y Fi(6)p Ft(=)27 b(0)32 b(\(nondegenerate\))i(or)1447 1242 y Fh(\014)1447 1292 y(\014)1447 1342 y(\014)1447 1391 y(\014)1447 1441 y(\014)1447 1491 y(\014)1447 1541 y(\014)1530 1320 y Fo(A)107 b(B)99 b(!)1944 1335 y Fu(1)1527 1440 y Fo(B)110 b(C)102 b(!)1944 1455 y Fu(2)1516 1561 y Fo(!)1577 1576 y Fu(1)1699 1561 y Fo(!)1760 1576 y Fu(2)1908 1561 y Ft(0)2024 1242 y Fh(\014)2024 1292 y(\014)2024 1342 y(\014)2024 1391 y(\014)2024 1441 y(\014)2024 1491 y(\014)2024 1541 y(\014)2080 1441 y Fi(6)p Ft(=)27 b(0)32 b(\(iso)s(energetically)f(nondegenerate\))q Fo(:)-118 1770 y Ft(Th)m(us)j(w)m(e)g(see)g(that)f(if)e(the)i (restriction)f(of)h Fo(X)1560 1785 y Fm(H)1618 1794 y Fl(0)1689 1770 y Ft(on)g Fo(W)1931 1734 y Fm(s)2000 1770 y Ft(giv)m(es)g(a)g(p)s(oin)m(t)e(of)i(general)f(elliptic)e(t)m(yp)s (e,)k(then)-118 1891 y(p)s(oin)m(t)e(this)g(p)s(ersists)h(for)f(a)g(p)s (erturb)s(ed)i(v)m(ector)f(\014eld)g(on)f(its)g(lo)s(cal)e(cen)m(ter)k (manifold.)28 2011 y(This)h(ensures)i(the)e(existence)i(of)d(in)m(v)-5 b(arian)m(t)34 b(2-dimensional)d(KAM)36 b(tori)d(under)j(the)f(p)s (erturbation)-118 2131 y(of)d Fo(H)74 2146 y Fu(0)146 2131 y Ft(on)g(the)h(related)f(p)s(ersisten)m(t)h(cen)m(ter)h (submanifold)c([6].)44 b(They)34 b(b)s(ecome)e(h)m(yp)s(erb)s(olic)g (tori)f(for)h(the)-118 2252 y(whole)g(3)h(d.o.f.)43 b(Hamiltonian.)28 2372 y(Our)j(third)e(assumption)h(\(see)i(b)s(elo)m(w\))e(is)g(usually) g(ful\014lled)e(for)i(an)h(in)m(tegrable)e(system)j(with)e(a)-118 2493 y(\(unique\))e(singular)f(p)s(oin)m(t)g(of)g(the)i(t)m(yp)s(e)g (under)g(consideration.)74 b(It)43 b(concerns)h(the)g(b)s(eha)m(vior)f (of)f(the)-118 2613 y(global)35 b(stable)h(and)i(unstable)f(manifolds.) 54 b(W)-8 b(e)38 b(assume)g(that)e(they)i(coalesce)g(forming)d(a)i (homo)s(clinic)-118 2733 y(lo)s(op.)66 b(Indeed,)44 b(lo)s(cally)38 b(near)j Fo(O)s Ft(,)h(the)g(set)f Fo(W)1644 2697 y Fm(u)1729 2733 y Ft(consists)h(of)e(three)h(orbits:)59 b Fo(O)43 b Ft(and)e(t)m(w)m(o)g(semi-orbits)-118 2854 y(whic)m(h)32 b(tend)f(to)g Fo(O)i Ft(when)g Fo(t)28 b Fi(!)f(\0001)p Ft(.)43 b(Supp)s(ose)32 b(that)f(the)h(con)m(tin)m(uation)e(in)g Fo(t)i Ft(for)e(p)s(ositiv)m(e)h(time)f(of)g(one)-118 2974 y(of)39 b(these)j(orbits)d(sta)m(ys)i(in)e(a)h(compact)g(domain)e (of)h(the)i(phase)f(space)i Fo(M)10 b Ft(.)66 b(Then,)43 b(it)c(can)h(b)s(e)g(sho)m(wn)-118 3094 y([9,)31 b(8])f(that,)i(under)f (some)g(mild)e(assumptions)h(on)h(the)h(b)s(eha)m(vior)e(of)h(the)g(in) m(tegrable)f(v)m(ector)i(\014eld)f Fo(X)3656 3109 y Fm(H)3714 3118 y Fl(0)3752 3094 y Ft(,)-118 3215 y(this)i(orbit)g(has)h(to)f (return)h(in)m(to)f(a)g(neigh)m(b)s(orho)s(o)s(d)g(of)g Fo(O)j Ft(and)d(then)i(should)e(tend)h(to)g Fo(O)h Ft(when)g Fo(t)30 b Fi(!)f(1)p Ft(,)-118 3335 y(forming)35 b(a)h(homo)s(clinic)e (orbit)i(to)g Fo(O)j Ft(\(its)d(closure)h(is)g(frequen)m(tly)h(called)d (a)i(homo)s(clinic)d(lo)s(op)h(\000\).)56 b(W)-8 b(e)-118 3456 y(assume)33 b(that)f(this)h(is)f(the)h(case,)g(namely)-118 3680 y Fk(Assumption)j(3)49 b Fo(X)689 3695 y Fm(H)747 3704 y Fl(0)821 3680 y Fj(has)34 b(a)h(homo)-5 b(clinic)33 b(lo)-5 b(op)34 b Ft(\000)h Fj(to)g Fo(O)s Fj(.)-118 3904 y Fk(Remark)i(1)49 b Fj(The)29 b(same)g(as)g(in)g(assumption)g(3)h (is)f(true)h(for)g(the)g(se)-5 b(c)g(ond)28 b(semi-orbit)h(in)h Fo(W)3259 3867 y Fm(u)3303 3904 y Fj(,)h(so)e(usual)5 b(ly)-118 4024 y(in)30 b(an)g(inte)-5 b(gr)g(able)29 b(system)i(without)f(discr)-5 b(ete)30 b(symmetries)g(and)g(with)g(a)g (c)-5 b(omp)g(act)30 b(Hamiltonian)f(level)h(of)-118 4144 y(the)g(p)-5 b(oint)29 b Fo(O)s Fj(,)h(two)g(outgoing)g (semi-orbits)f(mer)-5 b(ge)29 b(with)h(two)f(ingoing)g(semi-orbits)g (forming)g(the)h(\\\014gur)-5 b(e)-118 4265 y(eight")34 b(set.)28 4489 y Ft(Let)29 b(us)g(summarize)e(the)i(con)m(tributions)e (of)h(this)g(pap)s(er,)i(whic)m(h)f(are)f(concerned)i(with)e(the)h(b)s (eha)m(vior)-118 4609 y(of)37 b(the)h(p)s(erturb)s(ed)g(Hamiltonian)c Fo(H)44 b Ft(=)36 b Fo(H)1515 4624 y Fu(0)1580 4609 y Ft(+)26 b Fo("H)1809 4624 y Fu(1)1885 4609 y Ft(on)38 b(a)f(neigh)m(b)s(orho)s(o)s(d)f(of)i(one)f(lo)s(op)g(in)f(the)j(phase) -118 4729 y(space.)28 4850 y(W)-8 b(e)38 b(start,)g(in)f(Section)g(2,)h (b)m(y)g(studying)f(the)h(b)s(eha)m(vior)f(of)g(the)g(in)m(tegrable)f (Hamilonian)e Fo(H)3519 4865 y Fu(0)3595 4850 y Ft(near)-118 4970 y(the)i(lo)s(op)e(\000.)53 b(T)-8 b(o)36 b(this)f(end,)i(w)m(e)g (c)m(ho)s(ose)g(four)e(lo)s(cal)f(sections)i(to)f(orbits)g(in)g Fo(W)2858 4934 y Fm(s)2930 4970 y Ft(and)h Fo(W)3229 4934 y Fm(u)3274 4970 y Ft(,)g(de\014ned)i(b)m(y)-118 5091 y(the)33 b(equalities)979 5211 y Fo(N)1067 5170 y Fm(s)1057 5236 y Fn(\006)1144 5211 y Ft(=)28 b Fi(f)p Fo(x)1353 5226 y Fu(3)1420 5211 y Ft(=)g Fi(\006)p Fo(d)p Fi(g)p Fo(;)211 b(N)2028 5170 y Fm(u)2018 5236 y Fn(\006)2105 5211 y Ft(=)28 b Fi(f)p Fo(y)2307 5226 y Fu(3)2373 5211 y Ft(=)g Fi(\006)p Fo(d)p Fi(g)p Fo(;)973 b Ft(\(2\))-118 5384 y(resp)s(ectiv)m(ely)-8 b(,)30 b(and)f(w)m(e)g(describ)s(e)g(the)g (map)f(from)f Fo(N)1818 5347 y Fm(s)1808 5408 y Fn(\006)1895 5384 y Ft(to)h Fo(N)2098 5347 y Fm(u)2088 5408 y Fn(\006)2176 5384 y Ft(\(lo)s(cal)e(map\))i(de\014ned)h(b)m(y)h(the)f(\015o)m(w.)42 b(After)-118 5504 y(that)c(w)m(e)i(discuss)g(the)e(form)g(of)g(the)h (map)f(from)f Fo(N)1830 5468 y Fm(u)1820 5529 y Fn(\006)1917 5504 y Ft(to)i Fo(N)2131 5468 y Fm(s)2121 5529 y Fn(\006)2218 5504 y Ft(de\014ned)h(b)m(y)g(the)f(\015o)m(w)g(along)e(the)i(global) 1806 5753 y(3)p eop %%Page: 4 4 4 3 bop -118 407 a Ft(piece)40 b(of)e(the)i(homo)s(clinic)c(orbit)i (\(global)f(map\).)63 b(Their)40 b(comp)s(osition)c(giv)m(es)k(us)g (the)g(P)m(oincar)m(\023)-46 b(e)39 b(map)-118 527 y(near)33 b(the)g(homo)s(clinic)c(orbit.)28 648 y(In)37 b(Section)g(3,)h(w)m(e)g (consider)f(the)h(lo)s(cal)c(map)j(for)f(the)h(p)s(erturb)s(ed)h (Hamiltonian.)53 b(With)36 b(the)i(help)-118 768 y(of)h(the)i(cen)m (ter)g(manifold)c(theory)j(and)g(KAM)h(theory)-8 b(,)42 b(w)m(e)f(in)m(tro)s(duce)f(new)g(symplectic)g(co)s(ordinates)-118 888 y(suc)m(h)29 b(that)f(the)g(p)s(ersisten)m(t)h(h)m(yp)s(erb)s(olic) f(in)m(v)-5 b(arian)m(t)26 b(tori)h(\(on)g(the)i(p)s(erturb)s(ed)f(cen) m(ter)i(manifold)25 b(near)j(the)-118 1009 y(equilibrium\),)h(as)k(w)m (ell)f(as)h(their)f(stable)g(and)h(unstable)g(manifolds,)d(b)s(ecome)j (\015at.)28 1129 y(W)-8 b(e)28 b(consider)g(in)f(Section)g(4)g(the)h(p) s(erturb)s(ed)g(global)e(map)g(and)i(study)h(the)e(existence)j(of)d (homo)s(clinic)-118 1249 y(in)m(tersections)34 b(b)s(et)m(w)m(een)i (the)e(stable)g(and)g(unstable)g(manifolds)d(asso)s(ciated)j(to)f(p)s (ersisten)m(t)i(h)m(yp)s(erb)s(olic)-118 1370 y(tori.)66 b(If)40 b(the)h(p)s(erturbation)e(is)h(suc)m(h)i(that)e(the)h(homo)s (clinic)c(lo)s(op)i(is)h(preserv)m(ed,)45 b(w)m(e)c(sho)m(w)h(that,)g (for)-118 1490 y(a)h(wide)g(set)h(of)f(tori,)i(the)e(manifolds)e(in)m (tersect)j(along)e(8,)k(12)d(or)g(16)g(transv)m(erse)i(homo)s(clinic)40 b(orbits)-118 1611 y(\(Theorem)46 b(1\).)83 b(On)46 b(the)h(other)f (hand,)j(in)d(a)f(more)g(generic)h(situation)f(in)g(whic)m(h)h(the)h (lo)s(op)d(is)i(not)-118 1731 y(preserv)m(ed,)35 b(the)e(manifolds)d (in)m(tersect)k(along)d(4)h(transv)m(erse)j(homo)s(clinic)29 b(orbits)j(\(Theorem)h(2\).)28 1851 y(In)e(Section)f(5)g(w)m(e)i(use)f (the)g(P)m(oincar)m(\023)-46 b(e{Melnik)m(o)m(v)31 b(metho)s(d)e(and)i (pro)m(v)m(e)h(that)e(a)g(\014rst)h(appro)m(ximation)-118 1972 y(for)41 b(the)h(splitting)d(distance)j(b)s(et)m(w)m(een)i(the)d (in)m(v)-5 b(arian)m(t)41 b(manifolds)e(can)j(b)s(e)f(expressed)k(in)40 b(terms)i(of)f(a)-118 2092 y(Melnik)m(o)m(v)g(p)s(oten)m(tial)e (\(Theorem)j(3\).)68 b(Then,)44 b(the)d(transv)m(erse)i(homo)s(clinic) 38 b(orbits)j(can)g(b)s(e)g(detected)-118 2213 y(as)34 b(nondegenerate)i(critical)c(p)s(oin)m(ts)h(of)h(the)h(Melnik)m(o)m(v)f (p)s(oten)m(tial,)f(and)h(this)g(allo)m(ws)f(us)i(to)f(obtain)f(the) -118 2333 y(results)e(on)f(the)i(existence)g(of)e(suc)m(h)i(orbits)e (in)g(a)g(more)g(precise)h(form)m(ulation)d(\(Theorems)k(4)e(and)h (5\).)42 b(In)-118 2453 y(concrete)26 b(examples,)h(the)f(n)m(um)m(b)s (er)f(of)g(homo)s(clinic)d(orbits)i(can)i(b)s(e)f(found)g(b)m(y)i (computing)c(the)j(Melnik)m(o)m(v)-118 2574 y(p)s(oten)m(tial)31 b(explicitly)-8 b(.)28 2694 y(It)32 b(has)g(to)g(b)s(e)g(p)s(oin)m(ted) f(out)h(that,)f(for)h(the)g(v)-5 b(alidit)m(y)29 b(of)j(the)g(P)m (oincar)m(\023)-46 b(e{Melnik)m(o)m(v)32 b(metho)s(d,)f(a)h(small)-118 2814 y(neigh)m(b)s(orho)s(o)s(d)39 b(of)h(the)h(lo)s(op)e(has)i(to)g(b) s(e)f(excluded.)69 b(T)-8 b(o)41 b(o)m(v)m(ercome)g(this,)i(in)c (Section)i(6)f(w)m(e)i(dev)m(elop)-118 2935 y(an)48 b(alternativ)m(e)e (Melnik)m(o)m(v)j(appro)m(ximation)c(in)i(order)h(to)f(study)i(the)g (splitting)c(of)i(the)i(lo)s(op)d(itself)-118 3055 y(\(Theorem)33 b(6\).)-118 3344 y Ff(1.1)135 b(Some)45 b(examples)-118 3529 y Ft(W)-8 b(e)30 b(end)h(this)e(in)m(tro)s(duction)f(with)i(some)f (concrete)i(examples)f(of)f(in)m(tegrable)g(Hamiltonians)e(for)i(whic)m (h)-118 3649 y(the)h(previous)f(assumptions)h(are)f(ful\014lled.)40 b(In)30 b(symplectic)f(\(global\))e(co)s(ordinates)i(\()p Fo(x)3092 3664 y Fu(1)3131 3649 y Fo(;)17 b(y)3223 3664 y Fu(1)3262 3649 y Fo(;)g(x)3361 3664 y Fu(2)3401 3649 y Fo(;)g(y)3493 3664 y Fu(2)3531 3649 y Fo(;)g(p;)g(q)t Ft(\),)-118 3770 y(consider)788 4024 y Fo(H)869 4039 y Fu(0)936 4024 y Ft(=)28 b Fo(h)1096 4039 y Fu(0)1135 4024 y Ft(\()p Fo(\030)1216 4039 y Fu(1)1255 4024 y Fo(;)17 b(\030)1342 4039 y Fu(2)1381 4024 y Ft(\))22 b(+)1549 3956 y Fo(p)1598 3920 y Fu(2)p 1549 4001 89 4 v 1569 4092 a Ft(2)1669 4024 y(+)g Fo(V)g Ft(\()p Fo(q)t Ft(\))p Fo(;)788 4209 y(h)844 4224 y Fu(0)884 4209 y Ft(\()p Fo(\030)965 4224 y Fu(1)1004 4209 y Fo(;)17 b(\030)1091 4224 y Fu(2)1129 4209 y Ft(\))28 b(=)g Fo(!)1360 4224 y Fu(1)1399 4209 y Fo(\030)1442 4224 y Fu(1)1503 4209 y Ft(+)22 b Fo(!)1662 4224 y Fu(2)1701 4209 y Fo(\030)1744 4224 y Fu(2)1805 4209 y Ft(+)g Fo(A\030)2024 4168 y Fu(2)2019 4234 y(1)2085 4209 y Ft(+)g(2)p Fo(B)5 b(\030)2354 4224 y Fu(1)2393 4209 y Fo(\030)2436 4224 y Fu(2)2498 4209 y Ft(+)22 b Fo(C)7 b(\030)2721 4168 y Fu(2)2716 4234 y(2)2781 4209 y Ft(+)22 b Fi(\001)17 b(\001)g(\001)e Fo(;)-118 4429 y Ft(with)40 b Fo(A)p Ft(,)i Fo(B)5 b Ft(,)43 b Fo(C)k Ft(as)40 b(required)h(in)f(assumption)f(2,)j(and)f(a)f (giv)m(en)g(p)s(oten)m(tial)f Fo(V)22 b Ft(\()p Fo(q)t Ft(\).)66 b(This)40 b(p)s(oten)m(tial)f(is)-118 4550 y(assumed)d(to)g(ha)m(v)m(e)h(a)e(nondegenerate)i(maxim)m(um)c(at)j Fo(q)h Ft(=)32 b(0;)37 b(sa)m(y)g Fo(V)22 b Ft(\(0\))32 b(=)h Fo(V)2837 4514 y Fn(0)2860 4550 y Ft(\(0\))g(=)f(0,)37 b Fo(V)3317 4514 y Fn(00)3360 4550 y Ft(\(0\))32 b(=)h Fi(\000)p Ft(1.)-118 4670 y(Next,)52 b(w)m(e)c(pro)m(vide)g(some)f (examples;)55 b(the)48 b(asso)s(ciated)f(homo)s(clinic)e(lo)s(ops)h (are)i(parameterized)f(b)m(y)-118 4790 y(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)33 b Ft(_)-43 b Fo(q)335 4805 y Fu(0)372 4790 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(q)570 4805 y Fu(0)610 4790 y Ft(\()p Fo(t)p Ft(\)\))32 b(with)h Fo(q)1057 4805 y Fu(0)1096 4790 y Ft(\()p Fo(t)p Ft(\))g(as)g(giv)m(en)f(b)s(elo)m(w.)1 4994 y(1.)49 b(P)m(endulum:)43 b Fo(V)22 b Ft(\()p Fo(q)t Ft(\))27 b(=)h(cos)17 b Fo(q)26 b Fi(\000)d Ft(1,)32 b Fo(q)g Fi(2)c Fe(T)p Ft(;)k Fo(q)1712 5009 y Fu(0)1751 4994 y Ft(\()p Fo(t)p Ft(\))c(=)g Fi(\006)p Ft(4)17 b(arctan)f Fo(e)2469 4958 y Fm(t)2499 4994 y Ft(.)1 5197 y(2.)49 b(Du\016ng:)43 b Fo(V)22 b Ft(\()p Fo(q)t Ft(\))27 b(=)h Fi(\000)p Fo(q)966 5161 y Fu(2)1005 5197 y Fo(=)p Ft(2)22 b(+)g Fo(q)1270 5161 y Fu(4)1309 5197 y Fo(=)p Ft(4,)33 b Fo(q)e Fi(2)d Fe(R)p Ft(;)39 b Fo(q)1798 5212 y Fu(0)1837 5197 y Ft(\()p Fo(t)p Ft(\))28 b(=)g Fi(\006)2157 5115 y(p)p 2240 5115 49 4 v 82 x Ft(2)17 b(sec)m(h)h Fo(t)p Ft(.)1 5401 y(3.)49 b(Cubic:)44 b Fo(V)21 b Ft(\()p Fo(q)t Ft(\))27 b(=)h Fi(\000)p Fo(q)902 5365 y Fu(2)942 5401 y Fo(=)p Ft(2)21 b(+)h Fo(q)1206 5365 y Fu(3)1246 5401 y Fo(=)p Ft(3,)32 b Fo(q)f Fi(2)d Fe(R)p Ft(;)39 b Fo(q)1734 5416 y Fu(0)1774 5401 y Ft(\()p Fo(t)p Ft(\))27 b(=)2026 5361 y Fu(3)p 2026 5377 36 4 v 2026 5435 a(2)2088 5401 y Ft(sec)m(h)2265 5358 y Fu(2)2335 5361 y Fm(t)p 2331 5377 V 2331 5435 a Fu(2)2376 5401 y Ft(.)1806 5753 y(4)p eop %%Page: 5 5 5 4 bop -118 407 a Ft(W)-8 b(e)35 b(p)s(oin)m(t)f(out)g(that)g(in)g (examples)h(1)f(and)h(2)f(there)i(are)e(t)m(w)m(o)h(symmetric)f(homo)s (clinic)e(lo)s(ops;)i(instead,)-118 527 y(in)39 b(example)h(3)f(there)i (is)e(only)h(one)g(lo)s(op)e(\(the)j(second)g(semi-orbit)d(is)h(not)h (con)m(tained)g(in)f(a)h(compact)-118 648 y(set\).)28 768 y(In)c(those)g(examples,)g(the)f(symplectic)g(\(lo)s(cal\))e(co)s (ordinates)i(leading)f(to)h(the)g(represen)m(tations)i(\(1\))-118 888 y(are)23 b(giv)m(en)g(b)m(y)g(the)h(w)m(ell-kno)m(wn)f (transformation)d(to)j(Birkho\013)f(normal)f(form)g(for)h(a)h (1-degree-of-freedom)-118 1009 y(Hamiltonian)29 b(near)k(a)f(h)m(yp)s (erb)s(olic)g(equilibrium)d(p)s(oin)m(t)j([12].)43 b(First,)32 b(the)h(linear)e(c)m(hange)1150 1115 y Fh( )1257 1200 y Fo(p)1258 1320 y(q)1348 1115 y Fh(!)1441 1261 y Ft(=)1596 1194 y(1)p 1555 1238 132 4 v 1555 1256 a Fi(p)p 1638 1256 49 4 v 82 x Ft(2)1713 1115 y Fh( )1859 1200 y Ft(1)121 b(1)1820 1320 y Fi(\000)p Ft(1)83 b(1)2120 1115 y Fh(!)16 b( )2315 1200 y Ft(~)-55 b Fo(x)2364 1215 y Fu(3)2320 1320 y Ft(~)f Fo(y)2361 1335 y Fu(3)2446 1115 y Fh(!)-118 1518 y Ft(tak)m(es,)34 b(in)e(the)h(three)g(cases,)h(the)f(term)f Fo(p)1412 1481 y Fu(2)1451 1518 y Fo(=)p Ft(2)22 b(+)g Fo(V)g Ft(\()p Fo(q)t Ft(\))32 b(in)g Fo(H)2098 1533 y Fu(0)2169 1518 y Ft(in)m(to)1207 1870 y(~)-55 b Fo(x)1256 1885 y Fu(3)1302 1870 y Ft(~)g Fo(y)1344 1885 y Fu(3)1405 1870 y Ft(+)1503 1620 y Fh(8)1503 1695 y(>)1503 1720 y(>)1503 1745 y(>)1503 1770 y(<)1503 1919 y(>)1503 1944 y(>)1503 1969 y(>)1503 1994 y(:)1646 1669 y Fu(1)p 1628 1685 71 4 v 1628 1743 a(96)1709 1708 y Ft(\()7 b(~)-56 b Fo(y)1795 1723 y Fu(3)1856 1708 y Fi(\000)28 b Ft(~)-55 b Fo(x)2010 1723 y Fu(3)2050 1708 y Ft(\))2088 1672 y Fu(4)2150 1708 y Fi(\000)22 b(\001)17 b(\001)g(\001)e Fo(;)1646 1823 y Fu(1)p 1628 1839 V 1628 1896 a(16)1709 1862 y Ft(\()7 b(~)-56 b Fo(y)1795 1877 y Fu(3)1856 1862 y Fi(\000)28 b Ft(~)-55 b Fo(x)2010 1877 y Fu(3)2050 1862 y Ft(\))2088 1826 y Fu(4)2128 1862 y Fo(;)1675 1976 y Fu(1)p 1628 1992 130 4 v 1628 2057 a(6)1663 2002 y Fn(p)p 1722 2002 36 3 v 55 x Fu(2)1767 2016 y Ft(\()7 b(~)-56 b Fo(y)1853 2031 y Fu(3)1914 2016 y Fi(\000)29 b Ft(~)-55 b Fo(x)2069 2031 y Fu(3)2109 2016 y Ft(\))2147 1979 y Fu(3)2186 2016 y Fo(:)-118 2221 y Ft(Then,)35 b(a)f(symplectic)f(c)m(hange)i(\()6 b(~)-55 b Fo(x)1146 2236 y Fu(3)1186 2221 y Fo(;)23 b Ft(~)-55 b Fo(y)1278 2236 y Fu(3)1316 2221 y Ft(\))30 b(=)f(\()p Fo(x)1582 2236 y Fu(3)1622 2221 y Fo(;)17 b(y)1714 2236 y Fu(3)1753 2221 y Ft(\))23 b(+)g Fi(O)1992 2236 y Fu(2)2031 2221 y Ft(\()p Fo(x)2124 2236 y Fu(3)2164 2221 y Fo(;)17 b(y)2256 2236 y Fu(3)2295 2221 y Ft(\))34 b(with)f(a)h(con)m(v)m(ergen)m(t)h (expansion)g(in)e(a)-118 2342 y(neigh)m(b)s(orho)s(o)s(d)j(leads)h(to)g (a)g(function)g(of)g Fo(\021)i Ft(=)c Fo(x)1714 2357 y Fu(3)1754 2342 y Fo(y)1802 2357 y Fu(3)1841 2342 y Ft(.)58 b(Note)37 b(that)g Fo(\025)f Ft(=)f(1)i(in)g(the)h(three)g (examples.)57 b(In)-118 2462 y(all)33 b(three)i(examples)g(w)m(e)h(ma)m (y)e(tak)m(e)i(the)f(\014rst)h(in)m(tegrals)d Fo(F)2099 2477 y Fu(1)2170 2462 y Ft(=)f Fo(\030)2321 2477 y Fu(1)2360 2462 y Ft(,)j Fo(F)2485 2477 y Fu(2)2556 2462 y Ft(=)d Fo(\030)2707 2477 y Fu(2)2746 2462 y Ft(.)50 b(So,)36 b(lo)s(cally)-8 b(,)32 b(w)m(e)k(ha)m(v)m(e)g(as)-118 2583 y(in)c(\(1\))1112 2700 y Fo(H)1193 2715 y Fu(0)1260 2700 y Ft(=)c Fo(!)1425 2715 y Fu(1)1464 2700 y Fo(\030)1507 2715 y Fu(1)1568 2700 y Ft(+)22 b Fo(!)1727 2715 y Fu(2)1766 2700 y Fo(\030)1809 2715 y Fu(2)1870 2700 y Ft(+)g Fo(\021)k Ft(+)c Fi(\001)17 b(\001)g(\001)e Fo(;)1112 2941 y(F)1175 2956 y Fu(1)1242 2941 y Ft(=)28 b Fo(\030)1389 2956 y Fu(1)1428 2941 y Fo(;)44 b(F)1562 2956 y Fu(2)1630 2941 y Ft(=)27 b Fo(\030)1776 2956 y Fu(2)1815 2941 y Fo(;)49 b Ft(and)33 b(\001)28 b(=)g(1)f Fi(6)p Ft(=)h(0)p Fo(:)-118 3096 y Ft(Th)m(us,)34 b Fo(O)h Ft(is)d(a)g(simple)f(singular)g(p)s(oin) m(t)h(for)g(the)h(in)m(tegrable)f(system.)-118 3424 y Fp(2)161 b(The)53 b(P)l(oincar)l(\023)-77 b(e)54 b(map)f(for)h(the)f (in)l(tegrable)g(system)-118 3643 y Ft(First)32 b(of)g(all,)e(w)m(e)k (will)c(distinguish)h(t)m(w)m(o)j(p)s(ossibilities)29 b(for)j(the)h(Hamiltonian)c Fo(H)2883 3658 y Fu(0)2922 3643 y Ft(:)28 3763 y(1.)43 b Fo(!)208 3778 y Fu(1)248 3763 y Fo(!)309 3778 y Fu(2)375 3763 y Fo(>)28 b Ft(0)k(\(de\014nite)h (case\);)28 3884 y(2.)43 b Fo(!)208 3899 y Fu(1)248 3884 y Fo(!)309 3899 y Fu(2)375 3884 y Fo(<)28 b Ft(0)k(\(inde\014nite)g (case\).)28 4004 y(These)43 b(t)m(w)m(o)f(cases)h(ha)m(v)m(e)f (di\013eren)m(t)f(foliation)d(in)m(to)i(lev)m(els)i Fo(H)50 b Ft(=)42 b Fo(c)f Ft(on)g(the)g(cen)m(ter)i(manifold)38 b(and)-118 4124 y(in)f(a)g(neigh)m(b)s(orho)s(o)s(d)f(of)h(homo)s (clinic)d(orbit)j(\000.)58 b(Indeed,)40 b(the)e(restriction)e(of)h Fo(X)2896 4139 y Fm(H)2954 4148 y Fl(0)3031 4124 y Ft(on)m(to)g Fo(W)3361 4088 y Fm(c)3433 4124 y Ft(giv)m(es)h(an)-118 4245 y(in)m(tegrable)32 b(system)i(with)f(t)m(w)m(o)g(degrees)i(of)d (freedom)h(ha)m(ving)g(a)f(simple)g(elliptic)e(p)s(oin)m(t)i(at)h Fo(O)s Ft(.)44 b(Its)34 b(lo)s(cal)-118 4365 y(orbit)c(structure)i(is)e (easily)g(seen)j(from)c(the)j(represen)m(tation)f(\(1\))g(\(see)h(also) e([9]\).)43 b(F)-8 b(or)30 b(the)h(de\014nite)g(case,)-118 4486 y(due)37 b(to)f(our)g(c)m(hoice)g Fo(!)724 4501 y Fu(1)798 4486 y Fo(>)d Ft(0,)k(w)m(e)h(get)e Fo(!)1395 4501 y Fu(2)1468 4486 y Fo(>)e Ft(0)i(as)g(w)m(ell.)54 b(Here)37 b(eac)m(h)g(lev)m(el)f Fo(H)2797 4501 y Fu(0)2870 4486 y Ft(=)e Fo(c;)51 b(c)33 b(>)h Ft(0,)j(on)f Fo(W)3643 4449 y Fm(c)3714 4486 y Ft(is)-118 4606 y(a)d(3-sphere)i(for)e Fo(c)h Ft(small)d(enough,)k(in)e(this)g(sphere)j(the)e(system)g(has)h (t)m(w)m(o)f(link)m(ed)g(p)s(erio)s(dic)e(orbits,)h(one)-118 4726 y(from)23 b(the)j(symplectic)e(disk)h Fo(D)1014 4741 y Fu(1)1081 4726 y Ft(=)j Fi(f)p Fo(x)1290 4741 y Fu(3)1357 4726 y Ft(=)g Fo(y)1509 4741 y Fu(3)1575 4726 y Ft(=)g Fo(x)1734 4741 y Fu(2)1801 4726 y Ft(=)g Fo(y)1953 4741 y Fu(2)2019 4726 y Ft(=)g(0)p Fi(g)c Ft(and)h(another)g (one)g(from)f(the)h(symplectic)-118 4847 y(disk)37 b Fo(D)171 4862 y Fu(2)245 4847 y Ft(=)e Fi(f)p Fo(x)461 4862 y Fu(3)536 4847 y Ft(=)g Fo(y)695 4862 y Fu(3)769 4847 y Ft(=)g Fo(x)935 4862 y Fu(1)1010 4847 y Ft(=)g Fo(y)1169 4862 y Fu(1)1243 4847 y Ft(=)g(0)p Fi(g)p Ft(.)55 b(Other)38 b(orbits)e(in)g(this)h(lev)m(el)f(b)s(elong)g(to)h(a)f (one-parameter)-118 4967 y(family)28 b(of)i(in)m(v)-5 b(arian)m(t)29 b(tori)g(foliating)e(the)k(sphere.)44 b(The)32 b(singular)c(lev)m(el)j Fo(H)2630 4982 y Fu(0)2696 4967 y Ft(=)d(0)i(consists)h(of)f(only)g(p)s(oin)m(t)-118 5088 y Fo(O)s Ft(.)43 b(Th)m(us,)34 b(for)e(the)h(de\014nite)g(case)g (all)e(tori)g(in)h Fo(W)1714 5051 y Fm(c)1781 5088 y Ft(lie)f(in)g(lev)m(els)i Fo(H)2366 5103 y Fu(0)2433 5088 y Fo(>)28 b Ft(0.)28 5208 y(F)-8 b(or)34 b(the)h(inde\014nite)f (case)h Fo(!)1074 5223 y Fu(1)1113 5208 y Fo(!)1174 5223 y Fu(2)1244 5208 y Fo(<)c Ft(0)j(the)h(singular)e(lev)m(el)h Fo(H)2280 5223 y Fu(0)2350 5208 y Ft(=)d(0)j(in)g Fo(W)2762 5172 y Fm(c)2830 5208 y Ft(is)g(top)s(ologically)d(a)j(cone)-118 5328 y(o)m(v)m(er)26 b(a)f(2-torus)481 5292 y Fu(1)520 5328 y Ft(.)41 b(Eac)m(h)27 b(lev)m(el)d Fo(H)1119 5343 y Fu(0)1186 5328 y Ft(=)k Fo(c)p Ft(,)f Fo(c)g Fi(6)p Ft(=)h(0,)e(in)f Fo(W)1874 5292 y Fm(c)1933 5328 y Ft(is)g(top)s (ologically)c(\(and)k(di\013eren)m(tially\))e(manifold)p -118 5413 1560 4 v -6 5474 a Fd(1)31 5504 y Fv(This)28 b(means)f(that)h(one)f(tak)n(es)g(the)h(direct)f(pro)r(duct)h Fe(T)1766 5474 y Fd(2)1821 5504 y Fc(\002)18 b Fb(I)7 b Fv(,)28 b Fb(I)i Fv(=)22 b([0)p Fb(;)14 b Fv(1],)27 b(and)g Fe(T)2585 5474 y Fd(2)2641 5504 y Fc(\002)18 b(f)p Fv(0)p Fc(g)26 b Fv(is)h(iden)n(ti\014ed)h(to)g(the)g(p)r(oin)n (t.)1806 5753 y Ft(5)p eop %%Page: 6 6 6 5 bop -118 407 a Fo(D)-34 371 y Fu(2)31 407 y Fi(\002)27 b Fo(S)201 371 y Fu(1)279 407 y Ft(\(solid)37 b(tori\),)h(whose)i(cen)m (ter)g(circle)e(is)g(a)g(p)s(erio)s(dic)f(orbit)g(from)h(one)g Fo(D)2975 422 y Fm(i)3004 407 y Ft(-disk,)h(other)g(orbits)-118 527 y(b)s(elong)32 b(to)g(in)m(v)-5 b(arian)m(t)31 b(tori.)28 648 y(The)43 b(lo)s(cal)d(map)h(preserv)m(es)k(the)e(in)m(tegrals)e Fo(\030)1756 663 y Fu(1)1795 648 y Ft(,)j Fo(\030)1909 663 y Fu(2)1948 648 y Ft(,)h Fo(\021)t Ft(,)f(as)e(w)m(ell)f(as)i Fo(H)2690 663 y Fu(0)2729 648 y Ft(,)h Fo(F)2863 663 y Fu(1)2903 648 y Ft(,)h Fo(F)3038 663 y Fu(2)3119 648 y Ft(b)s(eing)c(functions)-118 768 y(of)35 b Fo(\030)39 783 y Fu(1)78 768 y Ft(,)i Fo(\030)185 783 y Fu(2)224 768 y Ft(,)g Fo(\021)i Ft(with)d(nondegenerate)h(Jacobian)e Fo(D)s Ft(\()p Fo(H)1863 783 y Fu(0)1902 768 y Fo(;)17 b(F)2009 783 y Fu(1)2048 768 y Fo(;)g(F)2155 783 y Fu(2)2194 768 y Ft(\))p Fo(=D)s Ft(\()p Fo(\030)2446 783 y Fu(1)2485 768 y Fo(;)g(\030)2572 783 y Fu(2)2611 768 y Fo(;)g(\021)t Ft(\).)52 b(Let)36 b(us)h(write)e(do)m(wn)i(the)-118 888 y(Hamilton)30 b(equations)1063 1053 y(_)-44 b Fo(x)1101 1068 y Fm(i)1157 1053 y Ft(=)28 b Fi(\000)1356 985 y Fo(@)5 b(h)p 1348 1029 128 4 v 1348 1121 a(@)g(\030)1447 1136 y Fm(i)1486 1053 y Fo(y)1534 1068 y Fm(i)1562 1053 y Fo(;)131 b Ft(_)-44 b Fo(y)1751 1068 y Fm(i)1806 1053 y Ft(=)1927 985 y Fo(@)5 b(h)p 1920 1029 V 1920 1121 a(@)g(\030)2019 1136 y Fm(i)2058 1053 y Fo(x)2113 1068 y Fm(i)2141 1053 y Fo(;)114 b(i)28 b Ft(=)g(1)p Fo(;)17 b Ft(2)p Fo(;)1063 1397 y Ft(_)-44 b Fo(x)1101 1412 y Fu(3)1169 1397 y Ft(=)27 b Fi(\000)1359 1330 y Fo(@)5 b(h)p 1359 1374 114 4 v 1361 1466 a(@)g(\021)1483 1397 y(x)1538 1412 y Fu(3)1578 1397 y Fo(;)131 b Ft(_)-44 b Fo(y)1767 1412 y Fu(3)1833 1397 y Ft(=)1947 1330 y Fo(@)5 b(h)p 1947 1374 V 1949 1466 a(@)g(\021)2070 1397 y(y)2118 1412 y Fu(3)2157 1397 y Fo(:)3655 1225 y Ft(\(3\))-118 1616 y(Since)37 b(the)h(function)f Fo(h)g Ft(and)h(its)e(deriv)-5 b(ativ)m(es)38 b(dep)s(end)g(only)f(on)g Fo(\030)2365 1631 y Fu(1)2404 1616 y Ft(,)i Fo(\030)2513 1631 y Fu(2)2552 1616 y Ft(,)f Fo(\021)j Ft(and)c(are)h(constan)m(t)g(along)e(a)-118 1737 y(giv)m(en)h(orbit,)g(w)m(e)h(can)g(in)m(tegrate)e(\(3\))h(as)g (linear)f(equations)h(with)g(constan)m(t)h(co)s(e\016cien)m(ts)g(and)f (get)g(the)-118 1857 y(lo)s(cal)30 b(\015o)m(w)91 2049 y Fo(x)146 2064 y Fm(i)175 2049 y Ft(\()p Fo(t)p Ft(\))e(=)f Fo(x)472 2013 y Fu(0)472 2074 y Fm(i)529 2049 y Ft(cos\()p Fo(th)788 2013 y Fu(0)788 2074 y Fm(\030)819 2084 y Fg(i)850 2049 y Ft(\))22 b Fi(\000)h Fo(y)1062 2013 y Fu(0)1058 2074 y Fm(i)1117 2049 y Ft(sin\()p Fo(th)1366 2013 y Fu(0)1366 2074 y Fm(\030)1397 2084 y Fg(i)1428 2049 y Ft(\))p Fo(;)114 b(y)1655 2064 y Fm(i)1682 2049 y Ft(\()p Fo(t)p Ft(\))28 b(=)g Fo(x)1980 2013 y Fu(0)1980 2074 y Fm(i)2036 2049 y Ft(sin\()p Fo(th)2285 2013 y Fu(0)2285 2074 y Fm(\030)2316 2084 y Fg(i)2347 2049 y Ft(\))22 b(+)g Fo(y)2557 2013 y Fu(0)2553 2074 y Fm(i)2612 2049 y Ft(cos)q(\()p Fo(th)2872 2013 y Fu(0)2872 2074 y Fm(\030)2903 2084 y Fg(i)2934 2049 y Ft(\))p Fo(;)114 b(i)28 b Ft(=)f(1)p Fo(;)17 b Ft(2)p Fo(;)91 2292 y(x)146 2307 y Fu(3)186 2292 y Ft(\()p Fo(t)p Ft(\))28 b(=)f Fo(x)483 2255 y Fu(0)483 2316 y(3)540 2292 y Ft(exp)q([)p Fi(\000)p Fo(th)884 2255 y Fu(0)884 2316 y Fm(\021)927 2292 y Ft(])p Fo(;)114 b(y)1143 2307 y Fu(3)1182 2292 y Ft(\()p Fo(t)p Ft(\))28 b(=)f Fo(y)1476 2255 y Fu(0)1472 2316 y(3)1532 2292 y Ft(exp)q([)p Fo(th)1799 2255 y Fu(0)1799 2316 y Fm(\021)1841 2292 y Ft(])p Fo(;)3655 2172 y Ft(\(4\))-118 2506 y(where)g(lo)m(w)m (er)g(indices)f(b)m(y)h Fo(h)f Ft(mean)g(deriv)-5 b(ativ)m(es)26 b(with)g(resp)s(ect)h(to)f(related)g(v)-5 b(ariables,)26 b(and)h(upp)s(er)g(zero)s(es)-118 2627 y(mark)32 b(that)g(the)h (calculations)e(in)h Fo(h)g Ft(are)h(made)f(at)g Fo(\030)1837 2590 y Fu(0)1832 2651 y Fm(i)1904 2627 y Ft(=)27 b(\(\()p Fo(x)2138 2590 y Fu(0)2138 2651 y Fm(i)2178 2627 y Ft(\))2216 2590 y Fu(2)2278 2627 y Ft(+)22 b(\()p Fo(y)2466 2590 y Fu(0)2462 2651 y Fm(i)2504 2627 y Ft(\))2542 2590 y Fu(2)2582 2627 y Ft(\))p Fo(=)p Ft(2,)32 b Fo(i)c Ft(=)f(1)p Fo(;)17 b Ft(2,)32 b(and)h Fo(\021)3384 2590 y Fu(0)3451 2627 y Ft(=)27 b Fo(x)3609 2590 y Fu(0)3609 2651 y(3)3649 2627 y Fo(y)3701 2590 y Fu(0)3697 2651 y(3)3739 2627 y Ft(.)28 2747 y(Th)m(us,)42 b(w)m(e)e(see)g(that)f(the)g(signs)g(of)f (the)i(v)-5 b(ariables)37 b Fo(x)2019 2762 y Fu(3)2098 2747 y Ft(and)i Fo(y)2342 2762 y Fu(3)2420 2747 y Ft(are)g(preserv)m (ed)i(b)m(y)f(the)f(lo)s(cal)e(\015o)m(w.)-118 2867 y(The)d(lo)s(cal)e (map)h(on)g(b)s(oth)h Fo(N)989 2831 y Fm(s)979 2892 y Fn(\006)1071 2867 y Ft(is)g(disconnected,)h(the)f(discon)m(tin)m(uit)m (y)f(tak)m(es)i(place)f(along)e(the)i(trace)g(of)-118 2988 y Fo(W)-12 2952 y Fm(cs)91 2988 y Ft(\()p Fo(y)177 3003 y Fu(3)249 2988 y Ft(=)f(0\).)53 b(So)36 b(the)h(part)e(of)h Fo(N)1253 2952 y Fm(s)1243 3012 y Fu(+)1338 2988 y Ft(where)h Fo(y)1671 3003 y Fu(3)1743 2988 y Fo(>)c Ft(0)j(is)g(transformed)f(on)m (to)h Fo(N)2898 2952 y Fm(u)2888 3012 y Fu(+)2947 2988 y Ft(,)h(and)f(the)g(part)g(with)-118 3108 y Fo(y)-70 3123 y Fu(3)-3 3108 y Fo(<)27 b Ft(0)32 b(is)g(transformed)h(on)m(to)f Fo(N)1132 3072 y Fm(u)1122 3133 y Fn(\000)1181 3108 y Ft(.)44 b(The)33 b(same)g(is)f(true)h(for)f Fo(N)2238 3072 y Fm(s)2228 3133 y Fn(\000)2287 3108 y Ft(.)28 3229 y(First)38 b(w)m(e)h(solv)m(e)g(the)g(equation)g Fo(h)p Ft(\()p Fo(\030)1382 3244 y Fu(1)1421 3229 y Fo(;)17 b(\030)1508 3244 y Fu(2)1547 3229 y Fo(;)g(\021)t Ft(\))37 b(=)h Fo(c)g Ft(in)g Fo(U)10 b Ft(.)62 b(Since)39 b Fo(h)p Ft(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0\))36 b(=)i(0)g(and)h Fo(h)3314 3244 y Fm(\021)3356 3229 y Ft(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0\))37 b(=)-118 3349 y Fo(\025)27 b Fi(6)p 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Fn(\006)1817 3985 y Ft(b)s(elonging)26 b(to)j Fo(U)10 b Ft(,)30 b(then)f(using)f (inequalit)m(y)f Fo(@)5 b(a=@)g(c)30 b Fi(6)p Ft(=)-118 4105 y(0)f(in)f Fo(U)10 b Ft(,)30 b(w)m(e)g(can)g(in)m(tro)s(duce)f(co) s(ordinates)f(\()p Fo(x)1556 4120 y Fu(1)1596 4105 y Fo(;)17 b(y)1688 4120 y Fu(1)1727 4105 y Fo(;)g(x)1826 4120 y Fu(2)1865 4105 y Fo(;)g(y)1957 4120 y Fu(2)1996 4105 y Fo(;)g(c)p Ft(\))29 b(in)f Fo(N)2347 4069 y Fm(s)2337 4130 y Fn(\006)2425 4105 y Ft(\(and)h Fo(N)2737 4069 y Fm(u)2727 4130 y Fn(\006)2786 4105 y Ft(\))g(instead)g(of)g(co)s (ordinates)-118 4225 y(\()p Fo(x)-25 4240 y Fu(1)15 4225 y Fo(;)17 b(y)107 4240 y Fu(1)145 4225 y Fo(;)g(x)244 4240 y Fu(2)284 4225 y Fo(;)g(y)376 4240 y Fu(2)415 4225 y Fo(;)g(y)507 4240 y Fu(3)545 4225 y Ft(\))34 b(\(\()p Fo(x)748 4240 y Fu(1)788 4225 y Fo(;)17 b(y)880 4240 y Fu(1)918 4225 y Fo(;)g(x)1017 4240 y Fu(2)1057 4225 y Fo(;)g(y)1149 4240 y Fu(2)1188 4225 y Fo(;)g(x)1287 4240 y Fu(3)1326 4225 y Ft(\),)34 b(resp)s(ectiv)m(ely\))h(b)m(y)f (form)m(ulae)e Fo(y)2582 4240 y Fu(3)2651 4225 y Ft(=)d Fo(a)p Ft(\()p Fo(\030)2888 4240 y Fu(1)2927 4225 y Fo(;)17 b(\030)3014 4240 y Fu(2)3053 4225 y Fo(;)g(c)p Ft(\))p Fo(=d)32 b Ft(in)h Fo(N)3512 4189 y Fm(s)3502 4250 y Fn(\006)3561 4225 y Ft(,)h(and)-118 4346 y(similarly)24 b(in)j Fo(N)473 4310 y Fm(u)463 4370 y Fn(\006)550 4346 y Ft(:)h Fo(x)660 4361 y Fu(3)727 4346 y Ft(=)g Fo(a)p Ft(\()p Fo(\030)963 4361 y Fu(1)1002 4346 y Fo(;)17 b(\030)1089 4361 y Fu(2)1128 4346 y Fo(;)g(c)p Ft(\))p Fo(=d)p Ft(.)41 b(On)27 b(eac)m(h)i(4-disk)e Fo(H)2153 4361 y Fu(0)2220 4346 y Ft(=)h Fo(c)g Ft(in)f Fo(N)2591 4310 y Fm(s)2581 4370 y Fu(+)2640 4346 y Ft(,)h(co)s(ordinates)g(\()p Fo(x)3302 4361 y Fu(1)3342 4346 y Fo(;)17 b(y)3434 4361 y Fu(1)3472 4346 y Fo(;)g(x)3571 4361 y Fu(2)3611 4346 y Fo(;)g(y)3703 4361 y Fu(2)3742 4346 y Ft(\))-118 4466 y(are)34 b(symplectic)f(with)g(resp)s(ect)i(to)e(the)h(standard)g (symplectic)f(2-form)f Fo(dx)2682 4481 y Fu(1)2744 4466 y Fi(^)24 b Fo(dy)2933 4481 y Fu(1)2994 4466 y Ft(+)f Fo(dx)3199 4481 y Fu(2)3261 4466 y Fi(^)h Fo(dy)3450 4481 y Fu(2)3488 4466 y Ft(,)34 b(b)s(eing)-118 4587 y(the)f(restriction)f(of)g(\012)h(to)f(this)g(disk.)28 4707 y(No)m(w)25 b(w)m(e)h(describ)s(e)f(ho)m(w)g(the)f(trace)h(of)f (the)h(stable)f(manifold)e(of)h(a)i(giv)m(en)f(torus)h(in)e Fo(W)3145 4671 y Fm(c)3204 4707 y Ft(is)h(represen)m(ted)-118 4827 y(in)f Fo(N)75 4791 y Fm(s)65 4852 y Fn(\006)148 4827 y Ft(in)h(V)-8 b(ey-Eliasson)23 b(co)s(ordinates,)j(and)e(the)h (same)f(for)f(the)i(trace)f(of)g(the)g(unstable)h(manifold)c(in)i Fo(N)3703 4791 y Fm(u)3693 4852 y Fn(\006)3752 4827 y Ft(.)-118 4948 y(The)k(union)g(of)f(stable)g(manifolds)e(of)j(all)d (tori)i(in)f Fo(W)1786 4912 y Fm(c)1847 4948 y Ft(mak)m(e)i(up)g Fo(W)2337 4912 y Fm(cs)2431 4948 y Ft(itself,)f(it)g(is)g(giv)m(en)h(b) m(y)h(the)f(equation)-118 5068 y Fo(y)-70 5083 y Fu(3)-3 5068 y Ft(=)g(0,)j(the)f(same)h(is)e(true)i(for)f(the)g(union)g(of)g (unstable)g(manifolds)e(making)h(up)h Fo(W)3005 5032 y Fm(cu)3110 5068 y Ft(giv)m(en)g(as)h Fo(x)3533 5083 y Fu(3)3600 5068 y Ft(=)e(0.)-118 5189 y(The)33 b(restrictions)f(of)f Fo(H)778 5204 y Fu(0)850 5189 y Ft(on)h Fo(W)1091 5152 y Fm(cs)1190 5189 y Ft(and)g Fo(W)1485 5152 y Fm(cu)1593 5189 y Ft(in)f(these)i(co)s(ordinates)f(ha)m(v)m(e)i(the)e(same)g (represen)m(tations,)-118 5309 y(giv)m(en)38 b(b)m(y)g(putting)f Fo(\021)j Ft(=)c(0)h(in)g Fo(H)1120 5324 y Fu(0)1159 5309 y Ft(.)59 b(T)-8 b(o)37 b(\014x)h(a)g(de\014nite)f(torus)h Fo(T)2300 5273 y Fu(2)2286 5334 y(0)2377 5309 y Ft(in)f Fo(W)2602 5273 y Fm(c)2674 5309 y Ft(one)g(should)h(assign)f (de\014nite)-118 5429 y(v)-5 b(alues)38 b(to)f(the)h(functions)g Fo(\030)944 5444 y Fu(1)1019 5429 y Ft(=)f Fo(\030)1180 5393 y Fu(0)1175 5454 y(1)1218 5429 y Ft(,)j Fo(\030)1328 5444 y Fu(2)1403 5429 y Ft(=)c Fo(\030)1563 5393 y Fu(0)1558 5454 y(2)1640 5429 y Ft(and)i(put)g Fo(\021)i Ft(=)c(0.)59 b(In)38 b(co)s(ordinates)f(\()p Fo(x)3097 5444 y Fu(1)3137 5429 y Fo(;)17 b(y)3229 5444 y Fu(1)3268 5429 y Fo(;)g(x)3367 5444 y Fu(2)3406 5429 y Fo(;)g(y)3498 5444 y Fu(2)3537 5429 y Fo(;)g(c)p Ft(\))37 b(in)1806 5753 y(6)p eop %%Page: 7 7 7 6 bop -118 407 a Fo(N)-30 371 y Fm(s)-40 431 y Fn(\006)54 407 y Ft(the)36 b(trace)f(of)f Fo(W)690 371 y Fm(s)727 407 y Ft(\()p Fo(T)836 371 y Fu(2)822 431 y(0)875 407 y Ft(\))h(is)f(giv)m(en)i(as)f Fo(\030)1471 422 y Fu(1)1542 407 y Ft(=)d Fo(\030)1698 371 y Fu(0)1693 431 y(1)1736 407 y Ft(,)k Fo(\030)1842 422 y Fu(2)1913 407 y Ft(=)c Fo(\030)2069 371 y Fu(0)2064 431 y(2)2108 407 y Ft(,)j Fo(c)d Ft(=)g Fo(h)p Ft(\()p Fo(\030)2494 371 y Fu(0)2489 431 y(1)2533 407 y Fo(;)17 b(\030)2625 371 y Fu(0)2620 431 y(2)2663 407 y Fo(;)g Ft(0\).)51 b(The)36 b(same)f(set)g Fo(c;)17 b(\030)3610 371 y Fu(0)3605 431 y(1)3649 407 y Fo(;)g(\030)3741 371 y Fu(0)3736 431 y(2)-118 527 y Ft(determines)33 b(the)g(trace)f(of)h Fo(W)1005 491 y Fm(u)1049 527 y Ft(\()p Fo(T)1158 491 y Fu(2)1144 552 y(0)1197 527 y Ft(\))g(in)f Fo(N)1470 491 y Fm(u)1460 552 y Fn(\006)1519 527 y Ft(.)28 648 y(T)-8 b(o)43 b(write)f(do)m(wn)h (the)g(form)e(of)h(the)h(lo)s(cal)d(map,)k(let)e(us)h(in)m(tegrate)f (equations)h(\(3\))f(and)g(\014nd)h(the)-118 768 y(passage)d(time)f(b)m (y)h(orbits)f(from)f Fo(N)1223 732 y Fm(s)1213 793 y Fn(\006)1312 768 y Ft(till)f Fo(N)1559 732 y Fm(u)1549 793 y Fn(\006)1608 768 y Ft(.)65 b(F)-8 b(rom)38 b(form)m(ulae)g(\(4\)) h(for)g(solutions)g(w)m(e)i(immediately)-118 888 y(deriv)m(ed)33 b(that)g(the)g(passage)g(time)e(from)h Fo(N)1497 852 y Fm(s)1487 913 y Fn(\006)1578 888 y Ft(till)e Fo(N)1818 852 y Fm(u)1808 913 y Fn(\006)1900 888 y Ft(is)i(equal)g(to)1515 1128 y Fo(t)1550 1143 y Fm(p)1617 1128 y Ft(=)1756 1061 y(1)p 1731 1105 99 4 v 1731 1196 a Fo(h)1787 1168 y Fu(0)1787 1221 y Fm(\021)1856 1128 y Ft(ln)2011 1061 y Fo(d)p 1964 1105 146 4 v 1964 1196 a Fi(j)p Fo(y)2044 1162 y Fu(0)2040 1218 y(3)2082 1196 y Fi(j)2119 1128 y Fo(:)-118 1380 y Ft(Plugging)41 b(this)h(in)m(to)h(the)g(\014rst)g(four)g(equations)g (in)f(\(4\))g(w)m(e)i(get)f(the)g(lo)s(cal)e(map)h Fo(T)3080 1395 y Fu(0)3120 1380 y Ft(.)74 b(The)44 b(simplest)-118 1501 y(form)32 b(of)h(this)g(map)g(is)g(obtained)g(in)g(the)h (symplectic)f(p)s(olar)f(co)s(ordinates)h(\()p Fo(\030)2753 1516 y Fu(1)2792 1501 y Fo(;)17 b(\030)2879 1516 y Fu(2)2917 1501 y Fo(;)g(')3025 1516 y Fu(1)3065 1501 y Fo(;)g(')3173 1516 y Fu(2)3212 1501 y Ft(\))33 b(on)g Fo(N)3507 1465 y Fm(u)3497 1525 y Fn(\006)3557 1501 y Ft(\()p Fo(c)p Ft(\))c(=)-118 1621 y Fo(N)-30 1585 y Fm(u)-40 1646 y Fn(\006)41 1621 y Fi(\\)23 b(f)p Fo(H)261 1636 y Fu(0)328 1621 y Ft(=)28 b Fo(c)p Fi(g)p Ft(,)k(and)h(the)g(same)g(with)f(zeroth) i(upp)s(er)f(indices)f(on)h Fo(N)2530 1585 y Fm(s)2520 1646 y Fn(\006)2579 1621 y Ft(\()p Fo(c)p Ft(\))28 b(=)g Fo(N)2917 1585 y Fm(s)2907 1646 y Fn(\006)2988 1621 y Fi(\\)23 b(f)p Fo(H)3208 1636 y Fu(0)3275 1621 y Ft(=)28 b Fo(c)p Fi(g)p Ft(,)k(where)-118 1741 y Fo(x)-63 1756 y Fm(i)-7 1741 y Ft(=)97 1667 y Fi(p)p 180 1667 120 4 v 74 x Ft(2)p Fo(\030)272 1756 y Fm(i)316 1741 y Ft(cos)17 b Fo(')527 1756 y Fm(i)555 1741 y Fo(;)114 b(y)744 1756 y Fm(i)800 1741 y Ft(=)903 1667 y Fi(p)p 986 1667 V 74 x Ft(2)p Fo(\030)1078 1756 y Fm(i)1123 1741 y Ft(sin)16 b Fo(')1323 1756 y Fm(i)1351 1741 y Ft(:)562 1980 y Fo(\030)610 1939 y Fu(1)605 2005 y Fm(i)676 1980 y Ft(=)28 b Fo(\030)828 1939 y Fu(0)823 2005 y Fm(i)867 1980 y Fo(;)114 b(')1072 1939 y Fu(1)1072 2005 y Fm(i)1139 1980 y Ft(=)28 b Fo(')1307 1939 y Fu(0)1307 2005 y Fm(i)1368 1980 y Fi(\000)1478 1913 y Fo(@)5 b(a)p Ft(\()p Fo(\030)1671 1877 y Fu(0)1666 1938 y(1)1711 1913 y Fo(;)17 b(\030)1803 1877 y Fu(0)1798 1938 y(2)1841 1913 y Fo(;)g(c)p Ft(\))p 1478 1957 488 4 v 1649 2049 a Fo(@)5 b(\030)1753 2014 y Fu(0)1748 2071 y Fm(i)1991 1980 y Ft(ln)2269 1913 y Fo(d)2320 1877 y Fu(2)p 2099 1957 431 4 v 2099 2049 a Fo(a)p Ft(\()p Fo(\030)2236 2014 y Fu(0)2231 2070 y(1)2275 2049 y Fo(;)17 b(\030)2367 2014 y Fu(0)2362 2070 y(2)2406 2049 y Fo(;)g(c)p Ft(\))2669 1980 y(\(mo)s(d)32 b(2)p Fo(\031)t Ft(\))o Fo(:)556 b Ft(\(5\))-118 2214 y(These)31 b(p)s(olar)d(co)s(ordinates)h(degenerate) i(on)e(the)h(planes)f Fo(x)2057 2229 y Fu(1)2125 2214 y Ft(=)e Fo(y)2276 2229 y Fu(1)2343 2214 y Ft(=)h(0)h(and)g Fo(x)2766 2229 y Fu(2)2834 2214 y Ft(=)e Fo(y)2985 2229 y Fu(2)3052 2214 y Ft(=)h(0,)h(resp)s(ectiv)m(ely)-8 b(,)-118 2335 y(but)32 b(these)h(planes)f Fo(x)662 2299 y Fu(0)662 2359 y(1)730 2335 y Ft(=)27 b Fo(y)885 2299 y Fu(0)881 2359 y(1)952 2335 y Ft(=)g(0)32 b(and)g Fo(x)1380 2299 y Fu(0)1380 2359 y(2)1447 2335 y Ft(=)c Fo(y)1603 2299 y Fu(0)1599 2359 y(2)1669 2335 y Ft(=)g(0)j(are)h(transformed)g(b) m(y)h(the)f(lo)s(cal)e(\015o)m(w)j(on)m(to)e(planes)-118 2455 y Fo(x)-63 2419 y Fu(1)-63 2480 y(1)9 2455 y Ft(=)h Fo(y)169 2419 y Fu(1)165 2480 y(1)240 2455 y Ft(=)g(0)j(and)g Fo(x)679 2419 y Fu(1)679 2480 y(2)752 2455 y Ft(=)d Fo(y)912 2419 y Fu(1)908 2480 y(2)983 2455 y Ft(=)g(0,)k(resp)s(ectiv)m(ely)-8 b(,)36 b(so)g(their)f(traces)h(on)f Fo(N)2626 2419 y Fm(s)2616 2480 y Fn(\006)2710 2455 y Ft(and)h Fo(N)2991 2419 y Fm(u)2981 2480 y Fn(\006)3075 2455 y Ft(are)f(mapp)s(ed)g(in)m (to)-118 2575 y(eac)m(h)e(other.)28 2735 y(T)-8 b(o)34 b(study)i(the)e(global)e(map)h(w)m(e)j(supp)s(ose,)g(without)d(loss)h (of)g(generalit)m(y)-8 b(,)34 b(that)g(the)g(outgoing)f(lo)s(cal)-118 2856 y(piece)g(of)f(\000)g(in)g Fo(U)43 b Ft(coincides)33 b(with)f(the)h(semiaxis)f Fo(y)1793 2871 y Fu(3)1859 2856 y Fo(>)c Ft(0.)28 2976 y(Global)34 b(map)i Fo(S)628 2991 y Fu(0)702 2976 y Ft(:)f Fo(N)852 2940 y Fm(u)842 3001 y Fu(+)936 2976 y Fi(!)f Fo(N)1158 2940 y Fm(s)1148 3001 y Fn(\006)1244 2976 y Ft(is)i(de\014ned)i(b)m(y)f(the)h(\015o)m(w) f(of)f(the)h(v)m(ector)h(\014eld)e Fo(X)3085 2991 y Fm(H)3143 3000 y Fl(0)3219 2976 y Ft(near)g(a)h(global)-118 3096 y(piece)g(of)f(\000.)57 b(Along)35 b(orbits)i(of)f Fo(X)1156 3111 y Fm(H)1214 3120 y Fl(0)1289 3096 y Ft(the)i(v)-5 b(alues)36 b(of)h Fo(H)1953 3111 y Fu(0)1992 3096 y Fo(;)17 b(F)2099 3111 y Fu(1)2138 3096 y Fo(;)g(F)2245 3111 y Fu(2)2321 3096 y Ft(are)37 b(preserv)m(ed.)59 b(Inside)37 b Fo(U)48 b Ft(quadratic)-118 3217 y(functions)35 b Fo(\030)348 3232 y Fu(1)387 3217 y Fo(;)17 b(\030)474 3232 y Fu(2)512 3217 y Fo(;)g(\021)38 b Ft(are)d(also)f(in)m(tegrals)f(and)i(they)h (are)f(smo)s(oth)e(functions)i(of)f Fo(H)2942 3232 y Fu(0)2981 3217 y Fo(;)17 b(F)3088 3232 y Fu(1)3128 3217 y Fo(;)g(F)3235 3232 y Fu(2)3274 3217 y Ft(,)35 b(due)h(to)e(the)-118 3337 y(condition)22 b Fo(D)s Ft(\()p Fo(H)504 3352 y Fu(0)543 3337 y Fo(;)17 b(F)650 3352 y Fu(1)689 3337 y Fo(;)g(F)796 3352 y Fu(2)835 3337 y Ft(\))p Fo(=D)s Ft(\()p Fo(\030)1087 3352 y Fu(1)1126 3337 y Fo(;)g(\030)1213 3352 y Fu(2)1252 3337 y Fo(;)g(\021)t Ft(\))27 b Fi(6)p Ft(=)g(0)c(in)g Fo(U)10 b Ft(.)41 b(It)24 b(implies)d(that)i(if)g(at)g (a)g(p)s(oin)m(t)g(in)g Fo(N)3160 3301 y Fm(u)3150 3362 y Fu(+)3233 3337 y Ft(the)g(functions)-118 3458 y Fo(\030)-75 3473 y Fu(1)-36 3458 y Fo(;)17 b(\030)51 3473 y Fu(2)90 3458 y Fo(;)g(\021)30 b Ft(tak)m(e)e(de\014nite)g(v)-5 b(alues,)28 b(then)g(at)e(the)i(image)d(of)i(this)g(p)s(oin)m(t)f(in)h Fo(N)2579 3421 y Fm(s)2569 3482 y Fu(+)2655 3458 y Ft(\(or)g Fo(N)2895 3421 y Fm(s)2885 3482 y Fn(\000)2944 3458 y Ft(\))g(the)g(v)-5 b(alues)28 b(of)e(these)-118 3578 y(functions)32 b(are)h(the)g(same,)g(since)g(the)g(v)-5 b(alues)32 b(of)g(functions)h(the)g Fo(H)2383 3593 y Fu(0)2422 3578 y Fo(;)17 b(F)2529 3593 y Fu(1)2568 3578 y Fo(;)g(F)2675 3593 y Fu(2)2747 3578 y Ft(are)33 b(preserv)m(ed.)28 3698 y(W)-8 b(e)30 b(stress)i(that)d(the)i(c)m(hoice)f(of)f(the)h(sign) g(in)f Fo(N)1790 3662 y Fm(s)1780 3723 y Fn(\006)1868 3698 y Ft(dep)s(ends)j(on)d(whic)m(h)i(cross-section)f(of)g(t)m(w)m(o)g (ones)h(\000)-118 3819 y(returns)i(to.)43 b(In)32 b(the)g(examples)g (considered)h(in)e(Section)h(1.1,)f(it)g(is)h(not)f(hard)h(to)g(c)m (hec)m(k)i(that)e(\000)f(returns)-118 3939 y(to)h Fo(N)89 3903 y Fm(s)79 3964 y Fn(\000)171 3939 y Ft(in)g(example)g(1,)g(and)h (to)f Fo(N)1172 3903 y Fm(s)1162 3964 y Fu(+)1254 3939 y Ft(in)f(examples)i(2)f(and)h(3.)28 4060 y(F)-8 b(or)35 b(a)h(\014xed)i Fo(c)e Ft(the)g(form)f(of)h(the)g(global)e(map)h(in)g (co)s(ordinates)h(\()p Fo(\030)2531 4075 y Fu(1)2570 4060 y Fo(;)17 b(\030)2657 4075 y Fu(2)2696 4060 y Fo(;)g(')2804 4075 y Fu(1)2843 4060 y Fo(;)g(')2951 4075 y Fu(2)2990 4060 y Ft(\))36 b(is)g(de\014ned)h(b)m(y)g(the)-118 4180 y(preserv)-5 b(ation)22 b(of)f(co)s(ordinates)h Fo(\030)1073 4195 y Fu(1)1112 4180 y Fo(;)17 b(\030)1199 4195 y Fu(2)1238 4180 y Ft(;)25 b(that)d(is,)h(a)f(torus)g Fo(\030)1960 4195 y Fu(1)2027 4180 y Ft(=)28 b Fo(\030)2179 4144 y Fu(0)2174 4205 y(1)2217 4180 y Ft(,)d Fo(\030)2312 4195 y Fu(2)2378 4180 y Ft(=)j Fo(\030)2530 4144 y Fu(0)2525 4205 y(2)2591 4180 y Ft(in)21 b Fo(N)2782 4144 y Fm(u)2772 4205 y Fu(+)2831 4180 y Ft(\()p Fo(c)p Ft(\))h(is)f(transformed)h(on)m (to)-118 4300 y(the)34 b(torus)g(in)f Fo(N)505 4264 y Fm(s)495 4325 y Fn(\006)554 4300 y Ft(\()p Fo(c)p Ft(\))g(with)g(the)i (same)e(v)-5 b(alues)34 b(of)f Fo(\030)1790 4315 y Fu(1)1829 4300 y Ft(,)h Fo(\030)1933 4315 y Fu(2)1972 4300 y Ft(,)g(and)f(the)h (only)f(thing)g(w)m(e)i(need)g(to)e(determine)-118 4421 y(is)39 b(the)h(form)e(of)h(the)g(map)g(in)g Fo(')1100 4436 y Fm(i)1128 4421 y Ft(,)i Fo(i)e Ft(=)g(1)p Fo(;)17 b Ft(2.)63 b(Since)40 b(disks)g Fo(\030)2169 4436 y Fu(1)2247 4421 y Ft(=)f(0)g(and)h Fo(\030)2690 4436 y Fu(2)2768 4421 y Ft(=)f(0)g(are)h(transformed)f(to)-118 4541 y(disks)138 4515 y(\026)121 4541 y Fo(\030)164 4556 y Fu(1)231 4541 y Ft(=)27 b(0)i(and)616 4515 y(\026)599 4541 y Fo(\030)642 4556 y Fu(2)709 4541 y Ft(=)e(0,)j(resp)s(ectiv)m(ely)-8 b(,)31 b(then,)g(due)f(to)f(the)h(smo)s(othness)g(of)f(the)h(global)d (map,)i(and)h(its)-118 4661 y(symplecticit)m(y)-8 b(,)32 b(the)h(map)f(in)f Fo(')1057 4676 y Fm(i)1118 4661 y Ft(is)h(of)g(the)h(form)694 4889 y(\026)-77 b Fo(')730 4904 y Fu(1)797 4889 y Ft(=)28 b Fo(')965 4904 y Fu(1)1026 4889 y Ft(+)1138 4822 y Fo(@)5 b(R)p 1134 4866 139 4 v 1134 4957 a(@)g(\030)1233 4972 y Fu(1)1343 4889 y Ft(\(mo)s(d)32 b(2)p Fo(\031)t Ft(\))p Fo(;)141 b Ft(\026)-76 b Fo(')1952 4904 y Fu(2)2019 4889 y Ft(=)27 b Fo(')2186 4904 y Fu(2)2248 4889 y Ft(+)2359 4822 y Fo(@)5 b(R)p 2356 4866 V 2356 4957 a(@)g(\030)2455 4972 y Fu(2)2565 4889 y Ft(\(mo)s(d)31 b(2)p Fo(\031)t Ft(\))p Fo(;)-118 5123 y Ft(where)45 b Fo(R)k Ft(=)e Fo(R)q Ft(\()p Fo(\030)577 5138 y Fu(1)616 5123 y Fo(;)17 b(\030)703 5138 y Fu(2)742 5123 y Fo(;)g(c)p Ft(\).)78 b(Th)m(us,)49 b(the)44 b(global)e(map)i Fo(S)2027 5138 y Fu(0)2113 5123 y Ft(:)k Fo(N)2276 5087 y Fm(u)2266 5148 y Fu(+)2325 5123 y Ft(\()p Fo(c)p Ft(\))f Fi(\000)-16 b(!)47 b Fo(N)2786 5087 y Fm(s)2776 5148 y Fn(\006)2836 5123 y Ft(\()p Fo(c)p Ft(\))d(is)f(close)i(to)e(a)h(\014xed)-118 5243 y(rotation)31 b(of)h(angles)1545 5421 y Fo(\015)1596 5436 y Fm(i)1652 5421 y Ft(=)1775 5353 y Fo(@)5 b(R)p 1775 5397 132 4 v 1777 5489 a(@)g(\030)1876 5504 y Fm(i)1917 5271 y Fh(\014)1917 5321 y(\014)1917 5371 y(\014)1917 5421 y(\014)1917 5470 y(\014)1945 5524 y Fm(\030)s Fu(=0)2089 5421 y Fo(:)1806 5753 y Ft(7)p eop %%Page: 8 8 8 7 bop -118 407 a Ft(It)25 b(will)e(b)s(e)j(con)m(v)m(enien)m(t)h(to)e (c)m(hange)h(the)g(co)s(ordinates)f(in)f Fo(N)2051 371 y Fm(u)2041 431 y Fu(+)2100 407 y Ft(\()p Fo(c)p Ft(\))h(in)g(suc)m(h)i (a)e(w)m(a)m(y)h(that)g(this)e(map)h(b)s(ecomes)-118 527 y(close)43 b(to)f(the)h(iden)m(tit)m(y)-8 b(.)74 b(T)-8 b(o)43 b(do)g(this,)i(with)d(a)h(rotation)e(of)h(co)s(ordinates) g Fo(X)53 b Ft(=)45 b(\()p Fo(x)3089 542 y Fu(1)3129 527 y Fo(;)17 b(x)3228 542 y Fu(2)3267 527 y Fo(;)g(y)3359 542 y Fu(1)3398 527 y Fo(;)g(y)3490 542 y Fu(2)3529 527 y Ft(\))3567 491 y Fn(>)3668 527 y Ft(w)m(e)-118 648 y(in)m(tro)s(duce)32 b(new)i(co)s(ordinates)e Fo(Y)49 b Ft(=)28 b(\()p Fo(u)1338 663 y Fu(1)1376 648 y Fo(;)17 b(u)1476 663 y Fu(2)1515 648 y Fo(;)g(v)1606 663 y Fu(1)1645 648 y Fo(;)g(v)1736 663 y Fu(2)1776 648 y Ft(\))1814 611 y Fn(>)1905 648 y Ft(in)32 b Fo(N)2107 611 y Fm(u)2097 672 y Fu(+)2156 648 y Ft(\()p Fo(c)p Ft(\):)1148 859 y Fo(u)1204 874 y Fm(i)1259 859 y Ft(=)c Fo(x)1418 874 y Fm(i)1463 859 y Ft(cos)17 b Fo(\015)1661 874 y Fm(i)1711 859 y Fi(\000)23 b Fo(y)1859 874 y Fm(i)1903 859 y Ft(sin)16 b Fo(\015)2090 874 y Fm(i)2118 859 y Fo(;)1148 1004 y(v)1195 1019 y Fm(i)1251 1004 y Ft(=)27 b Fo(x)1409 1019 y Fm(i)1454 1004 y Ft(sin)17 b Fo(\015)1642 1019 y Fm(i)1692 1004 y Ft(+)22 b Fo(y)1838 1019 y Fm(i)1882 1004 y Ft(cos)17 b Fo(\015)2080 1019 y Fm(i)2108 1004 y Fo(;)212 b(i)28 b Ft(=)f(1)p Fo(;)17 b Ft(2)p Fo(:)-118 1216 y Ft(F)-8 b(or)39 b(the)h(images)f(in)g Fo(N)776 1180 y Fm(s)766 1241 y Fn(\006)825 1216 y Ft(\()p Fo(c)p Ft(\),)i(w)m(e)g(k)m(eep)h (the)e(co)s(ordinates)2120 1191 y(\026)2092 1216 y Fo(X)48 b Ft(=)40 b(\()6 b(\026)-55 b Fo(x)2430 1231 y Fu(1)2470 1216 y Fo(;)22 b Ft(\026)-54 b Fo(x)2569 1231 y Fu(2)2608 1216 y Fo(;)24 b Ft(\026)-56 b Fo(y)2700 1231 y Fu(1)2739 1216 y Fo(;)23 b Ft(\026)-55 b Fo(y)2831 1231 y Fu(2)2870 1216 y Ft(\))2908 1180 y Fn(>)2967 1216 y Ft(.)65 b(Then,)43 b(the)d(global)-118 1336 y(map)32 b(tak)m(es)i(the)f(form:)1135 1592 y(\026)-55 b Fo(x)1184 1607 y Fm(i)1240 1592 y Ft(=)28 b Fo(u)1400 1607 y Fm(i)1444 1592 y Ft(cos)1601 1525 y Fo(@)5 b(R)1731 1540 y Fu(1)p 1601 1569 171 4 v 1623 1660 a Fo(@)g(\030)1722 1675 y Fm(i)1804 1592 y Fi(\000)23 b Fo(v)1951 1607 y Fm(i)1996 1592 y Ft(sin)2142 1525 y Fo(@)5 b(R)2272 1540 y Fu(1)p 2142 1569 V 2163 1660 a Fo(@)g(\030)2262 1675 y Fm(i)2323 1592 y Fo(;)1305 b Ft(\(6\))1162 1849 y(\026)-56 b Fo(y)1203 1864 y Fm(i)1259 1849 y Ft(=)27 b Fo(u)1418 1864 y Fm(i)1462 1849 y Ft(sin)1609 1782 y Fo(@)5 b(R)1739 1797 y Fu(1)p 1609 1826 V 1630 1918 a Fo(@)g(\030)1729 1933 y Fm(i)1811 1849 y Ft(+)22 b Fo(v)1956 1864 y Fm(i)2001 1849 y Ft(cos)2159 1782 y Fo(@)5 b(R)2289 1797 y Fu(1)p 2159 1826 V 2180 1918 a Fo(@)g(\030)2279 1933 y Fm(i)2339 1849 y Fo(;)1289 b Ft(\(7\))-118 2136 y(where)36 b Fo(R)240 2151 y Fu(1)311 2136 y Ft(=)31 b Fo(R)q Ft(\()p Fo(\030)574 2151 y Fu(1)613 2136 y Fo(;)17 b(\030)700 2151 y Fu(2)739 2136 y Fo(;)g(c)p Ft(\))23 b Fi(\000)i Fo(\015)1039 2151 y Fu(1)1078 2136 y Fo(\030)1121 2151 y Fu(1)1183 2136 y Fi(\000)f Fo(\015)1335 2151 y Fu(2)1374 2136 y Fo(\030)1417 2151 y Fu(2)1457 2136 y Ft(,)1539 2097 y Fm(@)t(R)1633 2106 y Fl(1)p 1539 2113 129 4 v 1554 2171 a Fm(@)t(\030)1626 2181 y Fg(i)1678 2037 y Fh(\014)1678 2087 y(\014)1678 2136 y(\014)1705 2190 y Fm(\030)s Fu(=0)1865 2136 y Ft(=)31 b(0.)50 b(Expanding)35 b(the)g(righ)m(t)f(hand)h(side)g(of)f(\(6{7\))-118 2281 y(at)e Fo(\030)g Ft(=)c(0,)k(the)h(global)d(map)i Fo(S)1020 2296 y Fu(0)1092 2281 y Ft(casts)i(as)e(follo)m(ws)1486 2467 y(\026)1458 2492 y Fo(X)j Ft(=)28 b Fo(Y)43 b Ft(+)22 b Fo(F)14 b Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))p Fo(;)1451 b Ft(\(8\))-118 2704 y(where)34 b(the)f(function)f Fo(F)46 b Ft(b)s(egins)32 b(with)h(third)e(order)i(terms)g(in)e Fo(Y)22 b Ft(.)28 2824 y(No)m(w)36 b(w)m(e)h(are)e(able)g(to)g(study)i (prop)s(erties)e(of)g(P)m(oincar)m(\023)-46 b(e)36 b(map)f(whic)m(h)h (is)f(the)h(comp)s(osition)d(of)i(lo)s(cal)-118 2945 y(and)27 b(global)d(maps,)k Fo(S)685 2960 y Fu(0)735 2945 y Fi(\016)10 b Fo(T)852 2960 y Fu(0)891 2945 y Ft(.)42 b(W)-8 b(e)27 b(consider)g(\014rst)g(the)h(de\014nite)f(case)g Fo(!)2456 2960 y Fu(1)2495 2945 y Fo(!)2556 2960 y Fu(2)2623 2945 y Fo(>)h Ft(0,)g(that)e(is,)i(b)s(oth)e Fo(!)3442 2960 y Fm(i)3498 2945 y Fo(>)h Ft(0.)42 b(\000)-118 3065 y(returns)31 b(after)f(global)e(tra)m(v)m(el)i(either)g(on)g Fo(N)1495 3029 y Fm(s)1485 3090 y Fn(\000)1575 3065 y Ft(\(case)h Fk(A)p Ft(\))e(or)h(on)g Fo(N)2307 3029 y Fm(s)2297 3090 y Fu(+)2387 3065 y Ft(\(case)h Fk(B)p Ft(\).)e(The)j(di\013erence)e(b)s(et)m(w)m(een)-118 3185 y(these)40 b(t)m(w)m(o)g(p)s(ossibilities)c(consists)j(in)f(the)i (di\013eren)m(t)f(top)s(ology)e(of)i(the)g(lev)m(el)g Fo(H)2920 3200 y Fu(0)2997 3185 y Ft(=)g(0.)62 b(F)-8 b(or)38 b(the)i(case)-118 3306 y Fk(A)i Ft(the)h(disk)f Fo(y)448 3321 y Fu(3)531 3306 y Ft(=)j(0)d(\(the)g(trace)h(of)f Fo(W)1439 3270 y Fm(cs)1506 3306 y Ft(\))g(in)g Fo(N)1798 3270 y Fm(s)1788 3331 y Fn(\000)1889 3306 y Ft(touc)m(hes)i(the)f(disk) f Fo(H)2721 3321 y Fu(0)2805 3306 y Ft(=)i(0)e(at)g(only)g(one)g(p)s (oin)m(t)-118 3426 y Fo(\030)-75 3441 y Fu(1)0 3426 y Ft(=)36 b Fo(\030)155 3441 y Fu(2)231 3426 y Ft(=)g(0,)i(for)f(other)h (p)s(oin)m(ts)f(of)g(this)h(lev)m(el)f Fo(y)1756 3441 y Fu(3)1831 3426 y Fo(>)f Ft(0,)j(and)f(so,)h(all)c(these)k(p)s(oin)m (ts)e(are)h(transformed)-118 3547 y(b)m(y)32 b(the)g(lo)s(cal)e(map)h (to)g Fo(N)835 3510 y Fm(u)825 3571 y Fu(+)884 3547 y Ft(.)43 b(Therefore,)33 b(the)f(lo)s(cal)e(map)g(has)i(a)g(discon)m (tin)m(uit)m(y)f(only)g(at)h(the)g(trace)f(of)h(\000)-118 3667 y(on)j(this)f(lev)m(el,)i(but)f(can)g(b)s(e)g(determined)g(b)m(y)h (con)m(tin)m(uit)m(y)f(up)g(to)f(a)h(homeomorphism)e(of)h(the)h(disk.) 51 b(In)-118 3787 y(particular,)31 b(a)h(neigh)m(b)s(orho)s(o)s(d)f(of) i(\000)f(in)g(the)h(lev)m(el)f Fo(H)1842 3802 y Fu(0)1909 3787 y Ft(=)c(0)k(is)g(homeomorphic)f(to)h Fo(D)3051 3751 y Fu(4)3112 3787 y Fi(\002)23 b Fo(S)3278 3751 y Fu(1)3317 3787 y Ft(.)28 3908 y(F)-8 b(or)39 b Fo(c)h(>)f Ft(0)h(the)g(in)m(tersection)g(of)f(disks)h Fo(H)1650 3923 y Fu(0)1729 3908 y Ft(=)g Fo(c)f Ft(and)h Fo(W)2229 3872 y Fm(cs)2336 3908 y Ft(\(i.e.,)h Fo(y)2615 3923 y Fu(3)2694 3908 y Ft(=)e(0\))g(in)g Fo(N)3144 3872 y Fm(s)3134 3932 y Fn(\000)3233 3908 y Ft(is)g(a)h(3-sphere)-118 4028 y(\006)-48 4043 y Fm(c)25 4028 y Ft(giv)m(en)e(b)m(y)h(solutions)e (of)h(the)g(equation)g Fo(h)p Ft(\()p Fo(\030)1670 4043 y Fu(1)1709 4028 y Fo(;)17 b(\030)1796 4043 y Fu(2)1835 4028 y Fo(;)g Ft(0\))36 b(=)h Fo(c)p Ft(.)59 b(P)m(oin)m(ts)39 b(inside)e(\006)2901 4043 y Fm(c)2974 4028 y Ft(are)h(transformed)g(to) -118 4149 y Fo(N)-30 4112 y Fm(u)-40 4173 y Fn(\000)19 4149 y Ft(,)i(and)f(p)s(oin)m(ts)f(outside)g(\006)993 4164 y Fm(c)1067 4149 y Ft(\(but)g(inside)g(of)g(some)g(larger)g (sphere)i(de\014ned)g(b)m(y)f(the)g(size)f(of)g Fo(U)10 b Ft(\))39 b(are)-118 4269 y(transformed)f(to)g Fo(N)646 4233 y Fm(u)636 4294 y Fu(+)695 4269 y Ft(,)j(an)m(y)e(transv)m(erse)i (to)d(\006)1616 4284 y Fm(c)1689 4269 y Ft(segmen)m(t)i(through)e(the)h (p)s(oin)m(t)f(on)g(\006)3093 4284 y Fm(c)3167 4269 y Ft(breaks.)63 b(These)-118 4389 y(considerations)32 b(de\014ne)i(a)e (domain)f(of)h(the)h(P)m(oincar)m(\023)-46 b(e)33 b(map)f(on)g Fo(N)2346 4353 y Fm(s)2336 4414 y Fn(\000)2395 4389 y Ft(.)28 4510 y(F)-8 b(or)29 b Fo(c)f(<)g Ft(0)h(all)f(p)s(oin)m(ts)i (of)f(some)h(4-disk)f(cen)m(tered)j(at)e(the)g(origin)e(in)h Fo(N)2648 4474 y Fm(s)2638 4534 y Fn(\000)2727 4510 y Ft(are)h(smo)s(othly)f(mapp)s(ed)h(b)m(y)-118 4630 y(the)j(lo)s(cal)d (map)i Fo(T)554 4645 y Fu(0)626 4630 y Ft(on)m(to)h(some)f(4-disk)g(in) g Fo(N)1577 4594 y Fm(u)1567 4655 y Fu(+)1626 4630 y Ft(.)44 b(Th)m(us)34 b(w)m(e)f(get)-118 4826 y Fk(Lemma)k(1)49 b Fj(Supp)-5 b(ose)37 b Fo(!)819 4841 y Fu(1)858 4826 y Fo(!)919 4841 y Fu(2)992 4826 y Fo(>)d Ft(0)p Fj(.)54 b(Then)37 b(for)h(the)g(c)-5 b(ase)37 b(A)i(and)e Fo(c)d Ft(=)g(0)j Fj(the)i(domain)d(of)i(the)g(Poinc)-5 b(ar)n(\023)-47 b(e)-118 4947 y(map)35 b(is)h(a)g(punctur)-5 b(e)g(d)36 b(4-disk.)47 b(F)-7 b(or)35 b Fo(c)30 b(<)g Ft(0)35 b Fj(the)h(domain)f(is)h(some)f(4-disk,)g(and)h(for)g Fo(c)29 b(>)h Ft(0)36 b Fj(the)g(domain)-118 5067 y(is)e(a)h(layer)g(b)-5 b(etwe)g(en)34 b(two)h(c)-5 b(onc)g(entric)34 b(spher)-5 b(es.)28 5263 y Ft(F)d(or)33 b(the)h(case)g Fk(B)f Ft(\(\000)h(returns) g(to)f Fo(N)1371 5227 y Fm(s)1361 5288 y Fu(+)1420 5263 y Ft(\))h(the)g(situation)e(is)h(opp)s(osite.)45 b(Namely)-8 b(,)33 b(for)g Fo(c)c Ft(=)g(0)k(all)f(p)s(oin)m(ts)-118 5384 y(\(in)g Fo(N)122 5347 y Fm(s)112 5408 y Fu(+)171 5384 y Ft(\))g(excepting)i(the)f(trace)g(of)f(\000)h(in)f(4-disk)g Fo(H)1775 5399 y Fu(0)1842 5384 y Ft(=)c(0,)k(are)h(mapp)s(ed)f(b)m(y)i Fo(T)2781 5399 y Fu(0)2853 5384 y Ft(on)m(to)e Fo(N)3160 5347 y Fm(u)3150 5408 y Fn(\000)3242 5384 y Ft(and)h(lost)f(\(i.e.)-118 5504 y(go)i(out)h(from)f(the)i(neigh)m(b)s(orho)s(o)s(d)d(under)j (consideration\).)50 b(If)35 b Fo(c)d(>)g Ft(0,)j(then)h(the)f(p)s(oin) m(ts)g(inside)f(of)h(the)1806 5753 y(8)p eop %%Page: 9 9 9 8 bop -118 407 a Ft(small)34 b(sphere)k(\006)519 422 y Fm(c)591 407 y Ft(determined)f(b)m(y)g Fo(y)1292 422 y Fu(3)1366 407 y Ft(=)e(0)h(are)h(mapp)s(ed)f(on)m(to)h Fo(N)2416 371 y Fm(u)2406 431 y Fu(+)2465 407 y Ft(,)g(the)h(p)s(oin)m (ts)e(out)g(of)h(\006)3362 422 y Fm(c)3433 407 y Ft(are)g(lost.)-118 527 y(F)-8 b(or)32 b Fo(c)27 b(<)h Ft(0)k(all)f(p)s(oin)m(ts)h(of)g (the)h(4-disk)f(are)h(lost.)28 648 y(In)43 b(the)g(inde\014nite)f(case) h Fo(!)1053 663 y Fu(1)1092 648 y Fo(!)1153 663 y Fu(2)1236 648 y Fo(<)i Ft(0,)f(the)f(situaton)f(is)f(more)h(complicated.)71 b(T)-8 b(o)42 b(\014x)h(ideas,)i(sup-)-118 768 y(p)s(ose)i(that)g(\000) g(returns)h(to)e Fo(N)1022 732 y Fm(s)1012 793 y Fu(+)1071 768 y Ft(.)87 b(As)47 b(w)m(as)h(said)f(b)s(efore,)j(the)e(lo)s(cal)c (map)i(breaks)i(at)f(the)g(p)s(oin)m(ts)g(of)-118 888 y(the)41 b(trace)g(of)f Fo(W)535 852 y Fm(cs)642 888 y Ft(on)h Fo(N)874 852 y Fm(u)864 913 y Fu(+)963 888 y Ft(\(they)h(are)e(giv)m(en)h(for)f(di\013eren)m(t)h Fo(c)f Ft(b)m(y)i(the)e(equation)h Fo(h)p Ft(\()p Fo(\030)3157 903 y Fu(1)3196 888 y Fo(;)17 b(\030)3283 903 y Fu(2)3322 888 y Fo(;)g Ft(0\))40 b(=)h Fo(c)g Ft(or)-118 1009 y Fo(y)-70 1024 y Fu(3)0 1009 y Ft(=)31 b Fo(a)p Ft(\()p Fo(\030)239 1024 y Fu(1)279 1009 y Fo(;)17 b(\030)366 1024 y Fu(2)404 1009 y Fo(;)g(c)p Ft(\))p Fo(=d)31 b Ft(=)g(0\).)50 b(These)36 b(traces)g(are)f(describ)s(ed)g(for)f (di\013eren)m(t)h Fo(c)g Ft(as)g(follo)m(ws.)49 b(If)34 b Fo(c)e Ft(=)f(0,)k(then)-118 1129 y(this)d(trace)h(in)e(4-disk)h Fo(N)802 1093 y Fm(s)792 1154 y Fu(+)851 1129 y Ft(\(0\))g(is)g (di\013eomorphic)f(to)h(a)g(cone)h(o)m(v)m(er)g(torus.)44 b(This)33 b(set)g(divides)f(this)g(4-disk)-118 1249 y(in)m(to)38 b(t)m(w)m(o)i(regions,)g Fo(a)p Ft(\()p Fo(\030)775 1264 y Fu(1)815 1249 y Fo(;)17 b(\030)902 1264 y Fu(2)940 1249 y Fo(;)g Ft(0\))39 b(is)g(p)s(ositiv)m(e)f(in)g(one)i(region)e (\(these)i(p)s(oin)m(ts)f(are)g(transformed)g(b)m(y)h(the)-118 1370 y(lo)s(cal)d(map)j(to)f Fo(N)558 1334 y Fm(u)548 1394 y Fu(+)607 1370 y Ft(\))h(and)g(negativ)m(e)g(in)f(another)h(one)g (\(these)h(p)s(oin)m(ts)e(are)h(transformed)g(b)m(y)g(the)h(lo)s(cal) -118 1490 y(map)c(to)g Fo(N)316 1454 y Fm(u)306 1515 y Fn(\000)402 1490 y Ft(and)h(are)f(not)h(considered)g(further,)h (since)f(\000)f(in)m(tersects)i Fo(N)2653 1454 y Fm(u)2643 1515 y Fu(+)2702 1490 y Ft(\).)58 b(If)38 b Fo(c)e Fi(6)p Ft(=)g(0,)i(the)g(situation)-118 1611 y(is)f(the)i(same)f(as)g(for)f Fo(c)g Ft(=)g(0,)i(only)f(the)g(trace)g(of)g Fo(W)1860 1574 y Fm(cs)1965 1611 y Ft(giv)m(en)g(b)m(y)h Fo(a)e Ft(=)g(0)g(here)i(is)f(smo)s(oth)f Fo(D)3408 1574 y Fu(3)3447 1611 y Ft(,)i(it)e(also)-118 1731 y(divides)d(4-disk)g Fo(H)580 1746 y Fu(0)650 1731 y Ft(=)c Fo(c)k Ft(in)m(to)g(regions)f (with)h(opp)s(osite)g(signs)g(of)g Fo(a)p Ft(,)h(so)f(only)g(one)h(of)e (these)j(domains)d(is)-118 1851 y(mapp)s(ed)f(to)g Fo(N)460 1815 y Fm(u)450 1876 y Fu(+)510 1851 y Ft(.)28 1972 y(F)-8 b(or)36 b(the)h(problem)e(ab)s(out)h(homo)s(clinic)d(orbits)i(to)h(in)m (v)-5 b(arian)m(t)35 b(tori)g(in)h Fo(W)2767 1936 y Fm(c)2838 1972 y Ft(w)m(e)h(only)f(need)h(to)f(kno)m(w)-118 2092 y(ho)m(w)e(images)f(under)h(action)f(of)g(the)h(global)e(map)h(of)g (traces)i(of)e(unstable)h(manifolds)d(of)i(tori)g(from)f Fo(W)3745 2056 y Fm(c)-118 2213 y Ft(in)g Fo(N)84 2176 y Fm(u)74 2237 y Fu(+)165 2213 y Ft(in)m(tersect)i(traces)f(of)f(their) g(stable)h(manifolds)d(in)i Fo(N)2112 2176 y Fm(s)2102 2237 y Fn(\006)2161 2213 y Ft(.)28 2333 y(Th)m(us,)41 b(w)m(e)e(ha)m(v)m(e)g(got)e(that)h(for)f(in)m(tegrable)g(system)i(all) d(in)m(v)-5 b(arian)m(t)36 b(tori)h(in)g Fo(W)2989 2297 y Fm(c)3061 2333 y Ft(ha)m(v)m(e)i(their)f(stable)-118 2453 y(and)33 b(unstable)g(manifolds)e(coinciding,)g(that)i(is)g(all)e (orbits)h(are)h(homo)s(clinic.)42 b(As)34 b(w)m(e)g(will)d(see)j (later,)e(in)-118 2574 y(general)i(for)g(a)g(p)s(erturb)s(ed)i(system)f (only)f(a)h(\014nite)f(n)m(um)m(b)s(er)h(of)f(homo)s(clinic)e(orbits)i (to)g(in)m(v)-5 b(arian)m(t)33 b(torus)-118 2694 y(surviv)m(e.)-118 3027 y Fp(3)161 b(The)53 b(p)t(erturb)t(ed)g(lo)t(cal)i(map)-118 3246 y Ft(No)m(w)45 b(let)e(us)i(consider)g(a)f(p)s(erturb)s(ed)h (Hamiltonian)40 b(system)45 b(with)f(Hamiltonian)d Fo(H)55 b Ft(=)47 b Fo(H)3411 3261 y Fu(0)3481 3246 y Ft(+)30 b Fo("H)3714 3261 y Fu(1)3752 3246 y Ft(,)-118 3366 y(where)e Fo(H)239 3381 y Fu(0)278 3366 y Ft(,)g(as)f(ab)s(o)m(v)m(e,)i(is)d(in)m (tegrable)g(with)g(a)h(homo)s(clinic)c(orbit)j(\000)g(to)h(a)f(cen)m (ter-cen)m(ter-saddle)j(singular)-118 3487 y(p)s(oin)m(t)i Fo(O)s Ft(.)42 b(Suc)m(h)33 b(the)f(singular)e(p)s(oin)m(t)h(p)s (ersists)i(under)f(a)g(p)s(erturbation)f(and)g(smo)s(othly)g(dep)s (ends)i(on)f Fo(")p Ft(.)-118 3607 y(Without)f(loss)g(of)f(generalit)m (y)h(one)h(can)f(assume)h(that)f(this)g(singular)f(p)s(oin)m(t)g(do)s (es)i(not)f(mo)m(v)m(e)h(under)g(the)-118 3727 y(p)s(erturbation.)43 b(This)32 b(will)f(b)s(e)h(assumed)i(henceforth.)28 3848 y(Using)e(V)-8 b(ey-Eliasson)31 b(co)s(ordinates)g(in)g(some)g(neigh)m (b)s(orho)s(o)s(d)g(of)g Fo(O)j Ft(one)e(can)g(consider)g(the)g (restric-)-118 3968 y(tion)j(of)h(the)h(p)s(erturb)s(ed)g(system)h(to)e Fo(U)10 b Ft(.)56 b(Then)37 b(the)g(system)h(will)c(b)s(e)j(a)f(p)s (erturbation)f(of)h(system)i(\(3\).)-118 4089 y(W)-8 b(e)38 b(tak)m(e)g(the)g(same)g(t)m(w)m(o)g(cross-sections)h Fo(N)1580 4052 y Fm(s)1570 4113 y Fu(+)1666 4089 y Ft(\(or)e Fo(N)1916 4052 y Fm(s)1906 4113 y Fn(\000)1966 4089 y Ft(\),)h Fo(N)2157 4052 y Fm(u)2147 4113 y Fu(+)2244 4089 y Ft(as)g(for)f(the)h(in)m(tegrable)e(system.)60 b(These)-118 4209 y(cross-sections)40 b(are)g(determined)f(b)m(y)i(the) e(in)m(tegrable)g(system)h(therefore)g(w)m(e)h(omit)c(lo)m(w)m(er)j (indices)f Fi(\006)p Ft(.)-118 4329 y(The)f(lo)s(cal)d(map)h Fo(T)600 4344 y Fm(")672 4329 y Ft(:)f Fo(N)822 4293 y Fm(s)895 4329 y Fi(!)g Fo(N)1118 4293 y Fm(u)1200 4329 y Ft(is)i(a)f(p)s(erturbation)h(of)f(the)i(map)e(\(5\).)56 b(Similarly)-8 b(,)34 b(w)m(e)k(get)f(the)h(p)s(er-)-118 4450 y(turbation)32 b(of)h(the)h(global)d(map)i Fo(S)1163 4465 y Fm(")1229 4450 y Ft(:)c Fo(N)1373 4414 y Fm(u)1447 4450 y Fi(!)g Fo(N)1664 4414 y Fm(s)1701 4450 y Ft(.)46 b(W)-8 b(e)34 b(need)g(to)g(analyse)f(the)h(prop)s(erties)f(of)g(these) i(t)m(w)m(o)-118 4570 y(maps.)28 4690 y(W)-8 b(e)36 b(b)s(egin)f(with)f (the)i(study)h(of)e(the)g(lo)s(cal)e(map.)51 b(The)37 b(principal)c(di\013erence)j(with)f(the)h(in)m(tegrable)-118 4811 y(case)25 b(consists)f(in)f(that,)j(due)e(to)f(KAM)i(theory)-8 b(,)26 b(lo)s(cal)21 b(cen)m(ter)k(manifold)c(fails)h(to)i(b)s(e)g (\014lled)e(with)i(in)m(v)-5 b(arian)m(t)-118 4931 y(2-tori:)70 b(only)47 b(a)f(Can)m(tor)h(set)h(of)f(a)f(p)s(ositiv)m(e)g(measure)i (is)e(\014lled)g(with)h(tori.)85 b(Stable)46 b(and)h(unstable)-118 5052 y(manifolds)21 b(of)i(these)i(tori)d(surviv)m(ed)j(cut)e Fo(N)1458 5015 y Fm(s)1519 5052 y Ft(and)h Fo(N)1788 5015 y Fm(u)1833 5052 y Ft(,)h(resp)s(ectiv)m(ely)-8 b(,)27 b(giving)22 b(related)h(Can)m(tor)g(sets)i(of)e(tori)-118 5172 y(on)j(them.)42 b(F)-8 b(or)26 b(our)g(purp)s(oses)i(w)m(e)g(use)f (t)m(w)m(o)h(kno)m(wn)f(results:)41 b(the)27 b(theorem)g(on)f(the)i(p)s (ersistence)g(of)e(lo)s(cal)-118 5292 y(cen)m(ter-stable)33 b(and)g(cen)m(ter-unstable)h(manifolds)c([3,)i(7],)h(and)f(the)h(KAM)g (theorem)g(b)m(y)g(P\177)-49 b(osc)m(hel)34 b([13)o(].)28 5413 y(F)-8 b(or)31 b Fo(")c Ft(=)h(0)j(the)h(v)m(ector)h(\014eld)e (has)h(cen)m(ter-stable)g(manifold)d Fo(W)2381 5377 y Fm(cs)2475 5413 y Ft(:)f Fo(y)2578 5428 y Fu(3)2645 5413 y Ft(=)f(0,)32 b(cen)m(ter-unstable)h(mani-)1806 5753 y(9)p eop %%Page: 10 10 10 9 bop -118 407 a Ft(fold)30 b Fo(W)179 371 y Fm(cu)282 407 y Ft(:)e Fo(x)392 422 y Fu(3)460 407 y Ft(=)f(0,)32 b(and)g(cen)m(ter)h(manifold)28 b Fo(W)1657 371 y Fm(c)1719 407 y Ft(:)g Fo(x)1829 422 y Fu(3)1897 407 y Ft(=)f Fo(y)2048 422 y Fu(3)2115 407 y Ft(=)h(0.)42 b(They)34 b(are)d(all)f(\015at,)i (that)f(is,)h(pieces)g(of)-118 527 y(the)k(related)f(co)s(ordinate)g (subspaces.)56 b(First)35 b(what)h(w)m(e)h(do)e(is)g(the)i (\015attenning)e(of)g(the)i(corresp)s(onding)-118 648 y(p)s(erturb)s(ed)30 b(manifolds)e Fo(W)880 611 y Fm(cs)866 672 y(")947 648 y Fo(;)44 b(W)1124 611 y Fm(cu)1110 672 y(")1199 648 y Fo(;)h(W)1377 611 y Fm(c)1363 672 y(")1411 648 y Ft(.)d(Recall)28 b(that,)j(b)m(y)f(the)g(cen)m(ter)h(manifold)c (theory)j([3,)g(7],)g(these)-118 768 y(manifolds)j(exist)j(and)g(are)f Fo(C)996 732 y Fm(r)1034 768 y Ft(-smo)s(oth)f(\(with)h(arbitrary)g (high)g Fo(r)s Ft(,)h(if)e(the)i(neigh)m(b)s(orho)s(o)s(d)e(of)h Fo(O)j Ft(where)-118 888 y(these)26 b(manifolds)c(are)i(de\014ned)i(is) e(c)m(hosen)i(su\016cien)m(tly)f(small\).)38 b(In)25 b(the)f(co)s(ordinates)g(w)m(e)i(use)f(these)h(man-)-118 1009 y(ifolds)31 b(can)i(b)s(e)g(represen)m(ted)i(as)e(graphs)g(of)g (the)g(follo)m(wing)d Fo(C)2174 973 y Fm(r)2212 1009 y Ft(-functions:)43 b Fo(y)2751 1024 y Fu(3)2818 1009 y Ft(=)28 b Fo(f)11 b Ft(\()p Fo(x)3074 1024 y Fu(1)3114 1009 y Fo(;)17 b(y)3206 1024 y Fu(1)3244 1009 y Fo(;)g(x)3343 1024 y Fu(2)3383 1009 y Fo(;)g(y)3475 1024 y Fu(2)3514 1009 y Fo(;)g(x)3613 1024 y Fu(3)3652 1009 y Fo(;)g(")p Ft(\))-118 1129 y(\(for)34 b Fo(W)177 1093 y Fm(cs)244 1129 y Ft(\),)h Fo(x)399 1144 y Fu(3)470 1129 y Ft(=)c Fo(g)t Ft(\()p Fo(x)721 1144 y Fu(1)760 1129 y Fo(;)17 b(y)852 1144 y Fu(1)891 1129 y Fo(;)g(x)990 1144 y Fu(2)1029 1129 y Fo(;)g(y)1121 1144 y Fu(2)1160 1129 y Fo(;)g(y)1252 1144 y Fu(3)1291 1129 y Fo(;)g(")p Ft(\))34 b(\(for)g Fo(W)1748 1093 y Fm(cu)1823 1129 y Ft(\),)h(\()p Fo(x)2016 1144 y Fu(3)2056 1129 y Fo(;)17 b(y)2148 1144 y Fu(3)2187 1129 y Ft(\))31 b(=)g Fo(w)s Ft(\()p Fo(x)2529 1144 y Fu(1)2568 1129 y Fo(;)17 b(y)2660 1144 y Fu(1)2699 1129 y Fo(;)g(x)2798 1144 y Fu(2)2837 1129 y Fo(;)g(y)2929 1144 y Fu(2)2968 1129 y Fo(;)g(")p Ft(\))34 b(\(for)g Fo(W)3425 1093 y Fm(c)3459 1129 y Ft(\).)50 b(One-)-118 1249 y(dimensional)33 b(stable)j(and)f(unstable)h(manifolds)d(also)i(p) s(ersist,)i(they)g(sta)m(y)f(as)g(smo)s(oth)f(as)h(the)g(p)s(ertur-) -118 1370 y(bation.)28 1490 y(As)h(w)m(as)g(said)f(b)s(efore,)h(w)m(e)h (assume)e(that)g Fo(\025)e(>)g Ft(0)i(and)g Fo(!)2139 1505 y Fu(1)2213 1490 y Fo(>)d Ft(0.)55 b(The)37 b(sp)s(ectrum)g(of)e (the)i(linearized)-118 1611 y(v)m(ector)25 b(\014eld)e(at)h Fo(O)h Ft(is)f(\()p Fi(\000)p Fo(\025)p Ft(\()p Fo(")p Ft(\))p Fo(;)17 b Fi(\006)p Fo(i!)1178 1626 y Fu(1)1217 1611 y Ft(\()p Fo(")p Ft(\))p Fo(;)g Fi(\006)p Fo(i!)1554 1626 y Fu(2)1593 1611 y Ft(\()p Fo(")p Ft(\))p Fo(;)g(\025)p Ft(\()p Fo(")p Ft(\)\).)39 b(Th)m(us,)28 b(there)c(are)g(in)m(v)-5 b(arian)m(t)22 b(linear)g(subspaces)-118 1731 y(of)33 b(the)i(tangen)m(t)f(space)h(at)f Fo(O)i Ft(for)e(eigen)m(v)-5 b(alues)34 b(\()p Fi(\000)p Fo(\025)p Ft(\()p Fo(")p Ft(\))p Fo(;)17 b Fi(\006)p Fo(i!)2180 1746 y Fu(1)2219 1731 y Ft(\()p Fo(")p Ft(\))p Fo(;)g Fi(\006)p Fo(i!)2556 1746 y Fu(2)2596 1731 y Ft(\()p Fo(")p Ft(\)\))33 b(\(it)g(is)g(the)i (tangen)m(t)f(space)-118 1851 y(to)k Fo(W)113 1815 y Fm(cs)99 1876 y(")218 1851 y Ft(at)g Fo(O)s Ft(\),)h(the)g(same)f(for)g (\()p Fi(\006)p Fo(i!)1313 1866 y Fu(1)1353 1851 y Ft(\()p Fo(")p Ft(\))p Fo(;)17 b Fi(\006)p Fo(i!)1690 1866 y Fu(2)1729 1851 y Ft(\()p Fo(")p Ft(\))p Fo(;)g(\025)p Ft(\()p Fo(")p Ft(\)\))37 b(\(it)g(is)h(the)h(tangen)m(t)f(space)i(to)e Fo(W)3426 1815 y Fm(cu)3412 1876 y(")3539 1851 y Ft(at)g Fo(O)s Ft(\))-118 1972 y(and)h(\()p Fi(\006)p Fo(i!)287 1987 y Fu(1)327 1972 y Ft(\()p Fo(")p Ft(\))p Fo(;)17 b Fi(\006)p Fo(i!)664 1987 y Fu(2)703 1972 y Ft(\()p Fo(")p Ft(\)\))39 b(\(tangen)m(t)g(space)i(to)d Fo(W)1800 1936 y Fm(c)1786 1996 y(")1874 1972 y Ft(at)h Fo(O)s Ft(\).)62 b(Making)39 b(previously)g(the)h(linear)e(c)m(hange)-118 2092 y(of)h(v)-5 b(ariables)39 b(one)h(can)g(supp)s(ose)i(that)d(the)i (related)e(linear)f(in)m(v)-5 b(arian)m(t)39 b(subspaces)j(coincide)e (with)f(the)-118 2213 y(co)s(ordinate)c(planes.)55 b(Then,)38 b(the)f(functions)f Fo(f)5 b(;)17 b(g)t(;)g(w)38 b Ft(b)s(egin)e(with)g (the)g(terms)h(of)e(the)i(second)h(order)e(in)-118 2333 y(their)c(v)-5 b(ariables.)42 b(W)-8 b(e)33 b(assume)g(this)f(b)s(elo)m (w.)28 2453 y(T)-8 b(o)33 b(\015atten)g(related)f(manifolds)e(w)m(e)k (use)f(the)g(follo)m(wing)d(assertion.)-118 2651 y Fk(Prop)s(osition)35 b(1)49 b Fj(Ther)-5 b(e)34 b(is)h(a)g(c)-5 b(anonic)g(al)33 b(change)h(of)g(variables)861 2865 y Ft(\()p Fo(x)954 2880 y Fu(1)994 2865 y Fo(;)17 b(y)1086 2880 y Fu(1)1125 2865 y Fo(;)g(x)1224 2880 y Fu(2)1263 2865 y Fo(;)g(y)1355 2880 y Fu(2)1394 2865 y Fo(;)g(x)1493 2880 y Fu(3)1533 2865 y Fo(;)g(y)1625 2880 y Fu(3)1663 2865 y Ft(\))28 b Fi(!)f Ft(\()p Fo(X)1975 2880 y Fu(1)2015 2865 y Fo(;)17 b(Y)2116 2880 y Fu(1)2154 2865 y Fo(;)g(X)2279 2880 y Fu(2)2319 2865 y Fo(;)g(Y)2420 2880 y Fu(2)2458 2865 y Fo(;)g(X)2583 2880 y Fu(3)2622 2865 y Fo(;)g(Y)2723 2880 y Fu(3)2762 2865 y Ft(\))-118 3079 y Fj(such)35 b(that)g(the)g(Hamiltonian)f(takes)g(the)h(form)549 3284 y Fo(H)8 b Ft(\()p Fo(X)757 3299 y Fu(1)796 3284 y Fo(;)17 b(Y)897 3299 y Fu(1)935 3284 y Fo(;)g(X)1060 3299 y Fu(2)1100 3284 y Fo(;)g(Y)1201 3299 y Fu(2)1239 3284 y Fo(;)g(X)1364 3299 y Fu(3)1403 3284 y Fo(;)g(Y)1504 3299 y Fu(3)1543 3284 y Fo(;)g(")p Ft(\))27 b(=)h Fi(H)1886 3299 y Fu(1)1925 3284 y Ft(\()p Fo(X)2044 3299 y Fu(1)2084 3284 y Fo(;)17 b(Y)2185 3299 y Fu(1)2223 3284 y Fo(;)g(X)2348 3299 y Fu(2)2387 3284 y Fo(;)g(Y)2488 3299 y Fu(2)2527 3284 y Fo(;)g(")p Ft(\)+)549 3404 y Fo(X)630 3419 y Fu(3)669 3404 y Fo(Y)726 3419 y Fu(3)765 3404 y Fi(H)849 3419 y Fu(2)889 3404 y Ft(\()p Fo(X)1008 3419 y Fu(1)1047 3404 y Fo(;)g(Y)1148 3419 y Fu(1)1187 3404 y Fo(;)g(X)1312 3419 y Fu(2)1351 3404 y Fo(;)g(Y)1452 3419 y Fu(2)1491 3404 y Fo(;)g(X)1616 3419 y Fu(3)1655 3404 y Fo(;)g(Y)1756 3419 y Fu(3)1795 3404 y Fo(;)g(")p Ft(\))p Fo(;)115 b Fi(H)2149 3419 y Fu(2)2189 3404 y Ft(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0\))24 b(=)k Fo(\025)p Ft(\()p Fo(")p Ft(\))p Fo(;)-118 3621 y Fj(wher)-5 b(e)53 b(the)g(exp)-5 b(ansion)52 b(of)h Fi(H)1040 3636 y Fu(1)1079 3621 y Ft(\()p Fo(X)1198 3636 y Fu(1)1237 3621 y Fo(;)17 b(Y)1338 3636 y Fu(1)1377 3621 y Fo(;)g(X)1502 3636 y Fu(2)1541 3621 y Fo(;)g(Y)1642 3636 y Fu(2)1681 3621 y Fo(;)g(")p Ft(\))53 b Fj(in)g(sp)-5 b(atial)52 b(variables)h(b)-5 b(e)g(gins)52 b(with)h Fo(!)3341 3636 y Fu(1)3380 3621 y Ft(\()p Fo(")p Ft(\)\()p Fo(X)3629 3585 y Fu(2)3621 3645 y(1)3704 3621 y Ft(+)-118 3741 y Fo(Y)-40 3705 y Fu(2)-61 3766 y(1)0 3741 y Ft(\))p Fo(=)p Ft(2)21 b(+)h Fo(!)316 3756 y Fu(2)355 3741 y Ft(\()p Fo(")p Ft(\)\()p Fo(X)604 3705 y Fu(2)596 3766 y(2)665 3741 y Ft(+)g Fo(Y)842 3705 y Fu(2)820 3766 y(2)881 3741 y Ft(\))p Fo(=)p Ft(2)p Fj(.)-118 3939 y Fk(Pro)s(of)p Ft(.)86 b(W)-8 b(e)48 b(only)e(mak)m(e)i(one)f(step,)52 b(namely)46 b(w)m(e)i(will)c(\015atten)k Fo(W)2532 3903 y Fm(cs)2599 3939 y Ft(.)87 b(One)47 b(can)g(use)h(a)f(standard)-118 4059 y(result)c(from)f(symplectic)h(geometry)g(\(see,)k(for)42 b(instance,)47 b([10)o(]\))c(but)h(here)g(w)m(e)g(are)f(able)g(to)f (giv)m(e)i(an)-118 4180 y(easy)f(direct)f(pro)s(of.)72 b(The)43 b(idea)f(is)g(the)g(same)h(as)f(for)g(pro)m(ving)g(the)g(Darb) s(oux)g(theorem)g(in)g([2].)72 b(Let)-118 4300 y Fo(y)-70 4315 y Fu(3)31 4300 y Ft(=)61 b Fo(f)11 b Ft(\()p Fo(x)320 4315 y Fu(1)360 4300 y Fo(;)17 b(y)452 4315 y Fu(1)490 4300 y Fo(;)g(x)589 4315 y Fu(2)629 4300 y Fo(;)g(y)721 4315 y Fu(2)760 4300 y Fo(;)g(x)859 4315 y Fu(3)898 4300 y Fo(;)g(")p Ft(\))52 b(b)s(e)h(the)g(represen)m(tation)g(of)f Fo(W)2313 4264 y Fm(cs)2380 4300 y Ft(.)104 b(In)m(tro)s(duce)53 b(the)g(v)-5 b(ariable)51 b Fo(Y)3603 4315 y Fu(3)3704 4300 y Ft(=)-118 4421 y Fo(y)-70 4436 y Fu(3)1 4421 y Fi(\000)33 b Fo(f)11 b Ft(\()p Fo(x)263 4436 y Fu(1)303 4421 y Fo(;)17 b(y)395 4436 y Fu(1)433 4421 y Fo(;)g(x)532 4436 y Fu(2)572 4421 y Fo(;)g(y)664 4436 y Fu(2)703 4421 y Fo(;)g(x)802 4436 y Fu(3)841 4421 y Fo(;)g(")p Ft(\))47 b(\(recall)f(that)h Fo(f)58 b Ft(is)47 b(of)g(the)h(order)g(2)f(in)g (spatial)f(v)-5 b(ariables\).)87 b(T)-8 b(ak)m(e)48 b Fo(Y)3740 4436 y Fu(3)-118 4541 y Ft(as)31 b(a)g(Hamilton)d(function)i (and)h(consider)g(the)g(related)g(Hamiltonian)c(v)m(ector)32 b(\014eld.)43 b(The)31 b(h)m(yp)s(ersurface)-118 4661 y Fo(x)-63 4676 y Fu(3)20 4661 y Ft(=)44 b Fo(g)t Ft(\()p Fo(x)284 4676 y Fu(1)323 4661 y Fo(;)17 b(y)415 4676 y Fu(1)454 4661 y Fo(;)g(x)553 4676 y Fu(2)592 4661 y Fo(;)g(y)684 4676 y Fu(2)723 4661 y Fo(;)g(y)815 4676 y Fu(3)854 4661 y Fo(;)g(")p Ft(\))41 b(is)g(transv)m(ersal)i(to)f(the) g(\015o)m(w)h(near)f Fo(O)s Ft(,)h(since)g(at)e(this)h(p)s(oin)m(t)58 b(_)-44 b Fo(x)3423 4676 y Fu(3)3507 4661 y Ft(=)43 b Fi(\000)p Ft(1.)-118 4782 y(T)-8 b(ak)m(e)42 b(as)g(the)f(conjugated)h (co)s(ordinate)e(to)h Fo(Y)1609 4797 y Fu(3)1689 4782 y Ft(the)h(function)f Fo(X)2338 4797 y Fu(3)2377 4782 y Ft(\()p Fo(x)2470 4797 y Fu(1)2510 4782 y Fo(;)17 b(y)2602 4797 y Fu(1)2641 4782 y Fo(;)g(x)2740 4797 y Fu(2)2779 4782 y Fo(;)g(y)2871 4797 y Fu(2)2910 4782 y Fo(;)g(x)3009 4797 y Fu(3)3048 4782 y Fo(;)g(y)3140 4797 y Fu(3)3179 4782 y Fo(;)g(")p Ft(\))41 b(de\014ned)h(as)-118 4902 y(the)32 b(time)e(of)h(reac)m(hing)h(the)f(p)s(oin)m(t)g(\()p Fo(x)1280 4917 y Fu(1)1320 4902 y Fo(;)17 b(y)1412 4917 y Fu(1)1451 4902 y Fo(;)g(x)1550 4917 y Fu(2)1589 4902 y Fo(;)g(y)1681 4917 y Fu(2)1720 4902 y Fo(;)g(x)1819 4917 y Fu(3)1858 4902 y Fo(;)g(y)1950 4917 y Fu(3)1989 4902 y Ft(\))32 b(for)f(an)g(orbit)f(starting)h(on)h(the)f(h)m(yp)s (ersurface)-118 5023 y Fo(x)-63 5038 y Fu(3)8 5023 y Ft(=)h Fo(g)t Ft(.)49 b(It)35 b(is)g(eviden)m(t)g(that)g Fo(X)1090 5038 y Fu(3)1161 5023 y Ft(=)c(0)k(on)g(this)f(h)m(yp)s (ersurface.)53 b(The)35 b(P)m(oisson)h(brac)m(k)m(et)g(of)e(these)i (func-)-118 5143 y(tions)c(satis\014es)h Fi(f)p Fo(X)611 5158 y Fu(3)651 5143 y Fo(;)17 b(Y)752 5158 y Fu(3)790 5143 y Fi(g)28 b(\021)g Ft(1.)44 b(Therefore,)34 b(the)f(functions)g Fo(X)2232 5158 y Fu(3)2271 5143 y Ft(,)g Fo(Y)2388 5158 y Fu(3)2460 5143 y Ft(are)f(indep)s(enden)m(t)i(and)f(comm)m(ute.)-118 5263 y(The)d(common)e(lev)m(el)h Fo(X)766 5278 y Fu(3)833 5263 y Ft(=)f Fo(Y)994 5278 y Fu(3)1060 5263 y Ft(=)g(0)h(is)g(a)g(smo) s(oth)f(symplectic)h(4-disk)g Fo(D)s Ft(.)42 b(One)29 b(can)h(tak)m(e)g(a)f(symplectic)-118 5384 y(c)m(hart)35 b(\()p Fo(X)255 5399 y Fu(1)294 5384 y Fo(;)17 b(Y)395 5399 y Fu(1)434 5384 y Fo(;)g(X)559 5399 y Fu(2)598 5384 y Fo(;)g(Y)699 5399 y Fu(2)738 5384 y Ft(\))34 b(at)g(the)g(p)s(oin)m (t)g Fo(O)i Ft(in)e Fo(D)s Ft(.)48 b(Observ)m(e)37 b(that)d(the)g(comm) m(uting)f(indep)s(enden)m(t)i(func-)-118 5504 y(tions)j Fo(X)208 5519 y Fu(3)248 5504 y Fo(;)17 b(Y)349 5519 y Fu(3)426 5504 y Ft(generate)40 b(a)f(P)m(oisson)g(action)f(of)g(the)i (group)f Fe(R)2197 5468 y Fu(2)2242 5504 y Ft(,)i(with)d(t)m(w)m (o-dimensional)f(orbits)h(b)s(eing)1782 5753 y(10)p eop %%Page: 11 11 11 10 bop -118 407 a Ft(transv)m(ersal)31 b(to)e Fo(D)s Ft(.)42 b(This)31 b(means)f(that)f(one)h(can)h(extend)g(Darb)s(oux)f (co)s(ordinate)f(functions)h(giv)m(en)g(in)f Fo(D)-118 527 y Ft(to)36 b(some)g(neigh)m(b)s(orho)s(o)s(d)f(of)h Fo(D)i Ft(in)e(transv)m(erse)i(direction)d(obtaining)g(the)h(complete)g (set)h(of)e(functions)-118 648 y(giving)41 b(Darb)s(oux)i(co)s (ordinates)g(near)g Fo(O)i Ft(\(see)g(details)d(in)g([2]\).)75 b(In)44 b(these)g(co)s(ordinates,)h(due)f(to)f(the)-118 768 y(construction,)33 b Fo(W)574 732 y Fm(cs)673 768 y Ft(is)f(giv)m(en)h(b)m(y)h(the)f(equation)f Fo(Y)1785 783 y Fu(3)1852 768 y Ft(=)27 b(0,)33 b(as)f(desired.)p 2552 768 42 42 v 28 888 a(Th)m(us,)41 b(after)d(these)i (transformations)c(w)m(e)j(get)g(the)f(Hamiltonian)d(in)i(the)i(form)d (w)m(e)k(sough)m(t.)61 b(The)-118 1009 y(corresp)s(onding)32 b(di\013eren)m(tial)f(system)j(casts)f(as)g(follo)m(ws)738 1235 y(_)699 1260 y Fo(X)780 1275 y Fu(1)847 1260 y Ft(=)27 b Fi(\000)1037 1192 y Fo(@)5 b Fi(H)1177 1207 y Fu(1)p 1037 1236 181 4 v 1052 1328 a Fo(@)g(Y)1165 1343 y Fu(1)1250 1260 y Fi(\000)23 b Fo(X)1431 1275 y Fu(3)1470 1260 y Fo(Y)1527 1275 y Fu(3)1576 1192 y Fo(@)5 b Fi(H)1716 1207 y Fu(2)p 1576 1236 V 1590 1328 a Fo(@)g(Y)1703 1343 y Fu(1)1767 1260 y Fo(;)1934 1235 y Ft(_)1908 1260 y Fo(Y)1965 1275 y Fu(1)2032 1260 y Ft(=)2145 1192 y Fo(@)g Fi(H)2285 1207 y Fu(1)p 2145 1236 V 2147 1328 a Fo(@)g(X)2284 1343 y Fu(1)2358 1260 y Ft(+)22 b Fo(X)2537 1275 y Fu(3)2576 1260 y Fo(Y)2633 1275 y Fu(3)2682 1192 y Fo(@)5 b Fi(H)2822 1207 y Fu(2)p 2682 1236 V 2684 1328 a Fo(@)g(X)2821 1343 y Fu(1)2873 1260 y Fo(;)738 1575 y Ft(_)699 1600 y Fo(X)780 1615 y Fu(2)847 1600 y Ft(=)27 b Fi(\000)1037 1533 y Fo(@)5 b Fi(H)1177 1548 y Fu(1)p 1037 1577 V 1052 1668 a Fo(@)g(Y)1165 1683 y Fu(2)1250 1600 y Fi(\000)23 b Fo(X)1431 1615 y Fu(3)1470 1600 y Fo(Y)1527 1615 y Fu(3)1576 1533 y Fo(@)5 b Fi(H)1716 1548 y Fu(2)p 1576 1577 V 1590 1668 a Fo(@)g(Y)1703 1683 y Fu(2)1767 1600 y Fo(;)1934 1575 y Ft(_)1908 1600 y Fo(Y)1965 1615 y Fu(2)2032 1600 y Ft(=)2145 1533 y Fo(@)g Fi(H)2285 1548 y Fu(1)p 2145 1577 V 2147 1668 a Fo(@)g(X)2284 1683 y Fu(2)2358 1600 y Ft(+)22 b Fo(X)2537 1615 y Fu(3)2576 1600 y Fo(Y)2633 1615 y Fu(3)2682 1533 y Fo(@)5 b Fi(H)2822 1548 y Fu(2)p 2682 1577 V 2684 1668 a Fo(@)g(X)2821 1683 y Fu(2)2873 1600 y Fo(;)738 1915 y Ft(_)699 1940 y Fo(X)780 1955 y Fu(3)847 1940 y Ft(=)27 b Fi(\000)p Fo(X)1108 1955 y Fu(3)1148 1940 y Fi(H)1232 1955 y Fu(2)1294 1940 y Fi(\000)c Fo(X)1475 1955 y Fu(3)1514 1940 y Fo(Y)1571 1955 y Fu(3)1620 1873 y Fo(@)5 b Fi(H)1760 1888 y Fu(2)p 1620 1917 V 1634 2009 a Fo(@)g(Y)1747 2024 y Fu(3)1810 1940 y Fo(;)1977 1915 y Ft(_)1952 1940 y Fo(Y)2009 1955 y Fu(3)2075 1940 y Ft(=)28 b Fo(Y)2236 1955 y Fu(3)2275 1940 y Fi(H)2359 1955 y Fu(2)2421 1940 y Ft(+)22 b Fo(X)2600 1955 y Fu(3)2639 1940 y Fo(Y)2696 1955 y Fu(3)2745 1873 y Fo(@)5 b Fi(H)2885 1888 y Fu(2)p 2745 1917 V 2747 2009 a Fo(@)g(X)2884 2024 y Fu(3)2936 1940 y Fo(:)28 2239 y Ft(Next)51 b(w)m(e)g(apply)f(the)g(KAM)h(results)f(of)g(P\177)-49 b(osc)m(hel)51 b([13)o(])f(to)g(the)h(restriction)e(of)g(the)i(system)g (to)-118 2360 y(the)i(cen)m(ter)h(manifold)c Fo(X)885 2375 y Fu(3)986 2360 y Ft(=)62 b Fo(Y)1181 2375 y Fu(3)1282 2360 y Ft(=)g(0:)84 b(there)53 b(is)f(a)h(smo)s(oth)f(symplectic)g(c)m (hange)i(of)e(v)-5 b(ariables)-118 2480 y(\()p Fo(X)1 2495 y Fu(1)40 2480 y Fo(;)17 b(Y)141 2495 y Fu(1)180 2480 y Fo(;)g(X)305 2495 y Fu(2)344 2480 y Fo(;)g(Y)445 2495 y Fu(2)484 2480 y Ft(\))54 b Fi(!)f Ft(\()p Fo(I)810 2495 y Fu(1)850 2480 y Fo(;)17 b(\022)939 2495 y Fu(1)978 2480 y Fo(;)g(I)1065 2495 y Fu(2)1104 2480 y Fo(;)g(\022)1193 2495 y Fu(2)1233 2480 y Ft(\))48 b(smo)s(othly)f(dep)s(ending)h(on)g Fo(")f Ft(suc)m(h)j(that)e(in)f(new)i(co)s(ordinates)-118 2601 y(Hamiltonian)29 b Fi(H)527 2616 y Fu(1)599 2601 y Ft(turns)k(in)m(to)1081 2575 y(^)1052 2601 y Fi(H)1136 2616 y Fu(1)1203 2601 y Ft(=)28 b Fo(h)1363 2616 y Fu(1)1402 2601 y Ft(\()p Fo(I)1483 2616 y Fu(1)1523 2601 y Fo(;)17 b(\022)1612 2616 y Fu(1)1651 2601 y Fo(;)g(I)1738 2616 y Fu(2)1777 2601 y Fo(;)g(\022)1866 2616 y Fu(2)1906 2601 y Fo(;)g(")p Ft(\),)32 b(where)i(the)f(deriv)-5 b(ativ)m(es)1646 2790 y Fo(@)5 b(h)1758 2805 y Fu(1)p 1646 2834 153 4 v 1651 2925 a Fo(@)g(\022)1752 2940 y Fu(1)1809 2857 y Fo(;)1863 2790 y(@)g(h)1975 2805 y Fu(1)p 1863 2834 V 1868 2925 a Fo(@)g(\022)1969 2940 y Fu(2)-118 3110 y Ft(v)-5 b(anish)29 b(on)g(a)g(Can)m(torian)f(set)i(of)f(\()p Fo(I)1182 3125 y Fu(1)1221 3110 y Fo(;)17 b(I)1308 3125 y Fu(2)1348 3110 y Ft(\))29 b(of)f(p)s(ositiv)m(e)h(measure)g(near)h (the)f(origin,)f(that)h(is,)h(at)e(these)j(tori)-118 3231 y(w)m(e)39 b(get)228 3205 y(^)199 3231 y Fi(H)283 3246 y Fu(1)359 3231 y Ft(=)e Fo(h)528 3246 y Fu(1)568 3231 y Ft(\()p Fo(I)649 3246 y Fu(1)688 3231 y Fo(;)17 b(I)775 3246 y Fu(2)814 3231 y Ft(\).)60 b(In)38 b([13])g(it)f(is)h (required)g(the)h(necessary)h(smo)s(othness)f(to)e(b)s(e)h(greater)g (than)-118 3351 y(3)p Fo(n)22 b Fi(\000)g Ft(1,)33 b(where)g Fo(n)g Ft(is)f(the)h(n)m(um)m(b)s(er)g(of)f(degrees)i(of)e(freedom.)43 b(F)-8 b(or)31 b(our)i(case,)g(w)m(e)h(ha)m(v)m(e)g Fo(n)27 b Ft(=)h(2)k(since)h(the)-118 3471 y(KAM)d(result)f(is)g(applied)f(on)i (the)g(cen)m(ter)h(manifold.)39 b(This)30 b(necessary)i(smo)s(othness)e (can)g(b)s(e)f(reac)m(hed)i(in)-118 3592 y(a)h(su\016cien)m(tly)h (small)e(neigh)m(b)s(orho)s(o)s(d)g(of)h Fo(O)s Ft(.)28 3712 y(W)-8 b(e)40 b(stress)h(that)f(the)g(new)g(co)s(ordinates)f(\()p Fo(I)1684 3727 y Fu(1)1723 3712 y Fo(;)17 b(\022)1812 3727 y Fu(1)1852 3712 y Fo(;)g(I)1939 3727 y Fu(2)1978 3712 y Fo(;)g(\022)2067 3727 y Fu(2)2107 3712 y Ft(\))39 b(can)h(b)s(e)g(tak)m(en)g Fi(O)s Ft(\()p Fo(")p Ft(\)-close)f(to)g (the)h(p)s(olar)-118 3832 y(co)s(ordinates)32 b(\(5\).)28 3953 y(In)h(the)g(KAM)g(co)s(ordinates)f(\()p Fo(I)1188 3968 y Fu(1)1227 3953 y Fo(;)17 b(\022)1316 3968 y Fu(1)1356 3953 y Fo(;)g(I)1443 3968 y Fu(2)1482 3953 y Fo(;)g(\022)1571 3968 y Fu(2)1611 3953 y Fo(;)g(X)1736 3968 y Fu(3)1775 3953 y Fo(;)g(Y)1876 3968 y Fu(3)1915 3953 y Ft(\),)32 b(the)h(di\013eren)m(tial)e(system)j(turns)f(in)m(to)737 4184 y(_)714 4209 y Fo(I)757 4224 y Fu(1)824 4209 y Ft(=)28 b Fi(\000)1015 4142 y Fo(@)5 b(h)1127 4157 y Fu(1)p 1015 4186 V 1020 4278 a Fo(@)g(\022)1121 4293 y Fu(1)1200 4209 y Fi(\000)23 b Fo(X)1381 4224 y Fu(3)1420 4209 y Fo(Y)1477 4224 y Fu(3)1526 4142 y Fo(@)5 b Fi(H)1666 4157 y Fu(2)p 1526 4186 181 4 v 1545 4278 a Fo(@)g(\022)1646 4293 y Fu(1)1717 4209 y Fo(;)1984 4183 y Ft(_)1965 4209 y Fo(\022)2010 4224 y Fu(1)2078 4209 y Ft(=)2191 4142 y Fo(@)g(h)2303 4157 y Fu(1)p 2191 4186 153 4 v 2198 4278 a Fo(@)g(I)2297 4293 y Fu(1)2376 4209 y Ft(+)22 b Fo(X)2555 4224 y Fu(3)2594 4209 y Fo(Y)2651 4224 y Fu(3)2700 4142 y Fo(@)5 b Fi(H)2840 4157 y Fu(2)p 2700 4186 181 4 v 2721 4278 a Fo(@)g(I)2820 4293 y Fu(1)2891 4209 y Fo(;)737 4524 y Ft(_)714 4550 y Fo(I)757 4565 y Fu(2)824 4550 y Ft(=)28 b Fi(\000)1015 4482 y Fo(@)5 b(h)1127 4497 y Fu(1)p 1015 4526 153 4 v 1020 4618 a Fo(@)g(\022)1121 4633 y Fu(2)1200 4550 y Fi(\000)23 b Fo(X)1381 4565 y Fu(3)1420 4550 y Fo(Y)1477 4565 y Fu(3)1526 4482 y Fo(@)5 b Fi(H)1666 4497 y Fu(2)p 1526 4526 181 4 v 1545 4618 a Fo(@)g(\022)1646 4633 y Fu(2)1717 4550 y Fo(;)1984 4523 y Ft(_)1965 4550 y Fo(\022)2010 4565 y Fu(2)2078 4550 y Ft(=)2191 4482 y Fo(@)g(h)2303 4497 y Fu(1)p 2191 4526 153 4 v 2198 4618 a Fo(@)g(I)2297 4633 y Fu(2)2376 4550 y Ft(+)22 b Fo(X)2555 4565 y Fu(3)2594 4550 y Fo(Y)2651 4565 y Fu(3)2700 4482 y Fo(@)5 b Fi(H)2840 4497 y Fu(2)p 2700 4526 181 4 v 2721 4618 a Fo(@)g(I)2820 4633 y Fu(2)2891 4550 y Fo(;)753 4865 y Ft(_)714 4890 y Fo(X)795 4905 y Fu(3)862 4890 y Ft(=)28 b Fi(\000)p Fo(X)1124 4905 y Fu(3)1164 4890 y Fi(H)1248 4905 y Fu(2)1310 4890 y Fi(\000)22 b Fo(X)1490 4905 y Fu(3)1530 4890 y Fo(Y)1587 4905 y Fu(3)1636 4822 y Fo(@)5 b Fi(H)1776 4837 y Fu(2)p 1636 4867 V 1649 4958 a Fo(@)g(Y)1762 4973 y Fu(3)1826 4890 y Fo(;)1962 4865 y Ft(_)1936 4890 y Fo(Y)1993 4905 y Fu(3)2060 4890 y Ft(=)27 b Fo(Y)2220 4905 y Fu(3)2259 4890 y Fi(H)2343 4905 y Fu(2)2405 4890 y Ft(+)22 b Fo(X)2584 4905 y Fu(3)2624 4890 y Fo(Y)2681 4905 y Fu(3)2730 4822 y Fo(@)5 b Fi(H)2870 4837 y Fu(2)p 2730 4867 V 2732 4958 a Fo(@)g(X)2869 4973 y Fu(3)2920 4890 y Fo(:)3655 4548 y Ft(\(9\))-118 5143 y(Let)38 b(us)g(consider)g (a)f(torus)h(from)f(the)h(Can)m(torian)f(set)h(in)f Fo(W)2172 5107 y Fm(c)2206 5143 y Ft(.)59 b(F)-8 b(rom)36 b(\(9\),)i(setting)g Fo(X)3150 5158 y Fu(3)3225 5143 y Ft(=)f Fo(Y)3395 5158 y Fu(3)3470 5143 y Ft(=)f(0)h(w)m(e)-118 5263 y(get)69 5238 y(_)46 5263 y Fo(I)89 5278 y Fu(1)160 5263 y Ft(=)31 b(0,)401 5238 y(_)378 5263 y Fo(I)421 5278 y Fu(2)491 5263 y Ft(=)g(0.)50 b(Th)m(us,)36 b(w)m(e)g(see)g(that)e Fo(Y)1576 5278 y Fu(3)1646 5263 y Ft(=)d(0)p Fo(;)79 b(I)1951 5278 y Fu(1)2022 5263 y Ft(=)31 b Fo(I)2180 5227 y Fu(0)2172 5288 y(1)2219 5263 y Fo(;)79 b(I)2368 5278 y Fu(2)2438 5263 y Ft(=)31 b Fo(I)2596 5227 y Fu(0)2588 5288 y(2)2670 5263 y Ft(de\014ne)36 b(the)f(stable)f(manifold)-118 5384 y(of)j(a)g(preserv)m(ed)j(torus)e Fo(X)859 5399 y Fu(3)934 5384 y Ft(=)e Fo(Y)1103 5399 y Fu(3)1178 5384 y Ft(=)g(0)p Fo(;)88 b(I)1497 5399 y Fu(1)1573 5384 y Ft(=)36 b Fo(I)1736 5347 y Fu(0)1728 5408 y(1)1775 5384 y Fo(;)89 b(I)1934 5399 y Fu(2)2009 5384 y Ft(=)36 b Fo(I)2172 5347 y Fu(0)2164 5408 y(2)2211 5384 y Ft(.)58 b(The)39 b(same)e(is)g(v)-5 b(alid)36 b(for)g(the)i(unstable)-118 5504 y(manifold)30 b(of)i(suc)m(h)i(a)e(torus,)h(one)g(only)f(needs)i (to)e(replace)h Fo(Y)2141 5519 y Fu(3)2212 5504 y Ft(b)m(y)h Fo(X)2429 5519 y Fu(3)2468 5504 y Ft(.)1782 5753 y(11)p eop %%Page: 12 12 12 11 bop 28 407 a Ft(So,)31 b(in)f(order)g(to)g(\014nd)h(homo)s (clinic)d(orbits)h(to)h(the)h(torus)g(preserv)m(ed,)i(one)e(needs)h(to) e(pro)m(v)m(e)i(that)e(the)-118 527 y(image)38 b(under)k(the)e(global)e (mapping)h Fo(S)1395 542 y Fm(")1472 527 y Ft(of)h(the)h(trace)g(on)f Fo(N)2250 491 y Fm(u)2336 527 y Ft(of)g(lo)s(cal)e(unstable)i(manifold) e(of)i(this)-118 648 y(torus)33 b(in)m(tersects)h(the)f(trace)g(of)f (lo)s(cal)e(stable)i(manifold)e(on)j Fo(N)2227 611 y Fm(s)2296 648 y Ft(of)f(the)h(same)g(torus.)-118 980 y Fp(4)161 b(The)53 b(p)t(erturb)t(ed)g(global)i(map)-118 1199 y Ft(Under)26 b(a)f(p)s(erturbation,)h(in)e(general,)i(the)g (Hamiltonian)21 b(system)26 b(loses)g(in)m(tegrabilit)m(y)-8 b(.)38 b(In)26 b(particular,)e(it)-118 1320 y(can)h(lead)f(to)h(the)g (splitting)d(of)j(homo)s(clinic)c(lo)s(op)i(to)i(the)g(cen)m(ter-cen)m (ter-saddle)i Fo(O)s Ft(.)40 b(Nev)m(ertheless,)29 b(there)-118 1440 y(are)f(p)s(erturbations)f(whic)m(h)h(preserv)m(e)i(the)f(homo)s (clinic)24 b(lo)s(op,)k(that)f(is,)h(p)s(erturb)s(ed)h(stable)e(and)h (unstable)-118 1561 y(manifolds)40 b(of)i(the)g(singular)f(p)s(oin)m(t) h(coalesce.)73 b(These)44 b(p)s(erturbations)e(b)s(elong)f(to)h(a)g(co) s(dimension)f(4)-118 1681 y(submanifold)d(near)j(the)g(in)m(tegrable)f (Hamiltonian)c Fo(H)1951 1696 y Fu(0)2031 1681 y Ft(in)k(the)h(space)h (of)e(all)e(smo)s(oth)i(Hamiltonians)-118 1801 y(with)j Fo(C)192 1765 y Fm(r)230 1801 y Fi(\000)p Ft(top)s(ology)-8 b(,)46 b Fo(r)g Ft(is)e(greater)g(than)g(3)p Fo(n)29 b Fi(\000)i Ft(1)43 b(\(with)h Fo(n)j Ft(=)g(2\).)77 b(Indeed,)48 b(the)c(lo)s(cal)e(stable)h(and)-118 1922 y(unstable)34 b(manifolds)e(are)j(one-dimensional,)d(their)i(traces)h Fo(m)2232 1937 y Fm(s)2303 1922 y Ft(on)f Fo(N)2528 1886 y Fm(s)2600 1922 y Ft(and)g Fo(n)2849 1937 y Fm(u)2929 1922 y Ft(on)g Fo(N)3154 1886 y Fm(u)3200 1922 y Ft(,)g(resp)s(ectiv)m (ely)-8 b(,)-118 2042 y(are)26 b(p)s(oin)m(ts)g(on)g(the)h(4-disks)f Fo(N)1021 2006 y Fm(s)1068 2042 y Fi(\\)9 b(f)p Fo(H)35 b Ft(=)28 b Fo(H)8 b Ft(\()p Fo(O)s Ft(\))p Fi(g)25 b Ft(and)h Fo(N)2002 2006 y Fm(u)2057 2042 y Fi(\\)9 b(f)p Fo(H)36 b Ft(=)27 b Fo(H)8 b Ft(\()p Fo(O)s Ft(\))p Fi(g)p Ft(.)40 b(The)27 b(p)s(erturbation)f(mo)m(v)m(es)-118 2162 y(these)40 b(traces)g(on)e(disks.)64 b(The)39 b(global)e(map)h (transforms)g Fo(n)2141 2177 y Fm(u)2226 2162 y Ft(to)g(a)h(p)s(oin)m (t)f Fo(m)2785 2177 y Fm(u)2869 2162 y Ft(on)h Fo(N)3099 2126 y Fm(s)3136 2162 y Ft(.)62 b(The)40 b(distance)-118 2283 y(b)s(et)m(w)m(een)34 b Fo(m)343 2298 y Fm(s)412 2283 y Ft(and)f Fo(m)687 2298 y Fm(u)764 2283 y Ft(c)m(haracterizes)g (the)g(splitting)d(of)i(the)g(lo)s(op.)42 b(It)32 b(is)g(clear)g(that)g (in)f(order)h(to)g(mak)m(e)-118 2403 y(related)c(p)s(oin)m(ts)g Fo(m)577 2418 y Fm(s)642 2403 y Ft(and)g Fo(m)912 2418 y Fm(u)986 2403 y Ft(coinciden)m(t)g(one)h(has)f(to)g(ha)m(v)m(e)i(at)e (least)g(a)g(4-parametric)e(unfolding)g(of)i(the)-118 2524 y(initial)37 b(Hamiltonian.)65 b(Th)m(us,)44 b(co)s(dimension)c (of)g(Hamiltonians)e(with)i(homo)s(clinic)e(lo)s(ops)i(to)h(cen)m(ter-) -118 2644 y(cen)m(ter-saddle)c(is)f(equal)h(to)f(four.)55 b(This)37 b(can)f(b)s(e)h(pro)m(v)m(ed)h(rigorously)d(but)i(w)m(e)g (omit)e(it)h(here)h(\(see)g(also)-118 2764 y(Section)32 b(6)h(for)f(a)g(related)g(result\).)44 b(Let)33 b(us)g(denote)h(the)f (set)g(of)f(Hamiltonians)e(with)i(lo)s(ops)g(near)g Fo(H)3626 2779 y Fu(0)3698 2764 y Ft(in)-118 2885 y(the)h(space)h(of)e(all)e Fo(C)634 2849 y Fm(r)672 2885 y Fi(\000)p Ft(Hamiltonians)g(as)j Fi(L)p Ft(.)28 3005 y(The)h(p)s(erturb)s(ed)f(global)d(map)i(has)h(the) g(form)e(of)h(a)h(p)s(erturbation)e(of)i(\(8\).)43 b(So)32 b(w)m(e)i(get)1208 3200 y(\026)1180 3225 y Fo(X)h Ft(=)28 b Fo(Y)43 b Ft(+)22 b Fo(F)14 b Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))22 b(+)g Fo("G)p Ft(\()p Fo(Y)5 b(;)17 b(c;)g(")p Ft(\))p Fo(:)-118 3445 y Ft(Recall)33 b(that)i Fo(Y)52 b Ft(=)31 b(0)k(corresp)s(onds)h(to)e(the)h(trace)g(of)f(\000)h(on)g Fo(N)2201 3409 y Fm(u)2246 3445 y Ft(.)50 b(Let)35 b(us)g(expand)h (function)e Fo(G)h Ft(in)f Fo(Y)56 b Ft(at)-118 3566 y Fo(Y)d Ft(=)32 b(0)j(up)h(to)f(third)f(order)i(terms.)51 b(The)36 b(linear)e(term)g(in)h Fo(Y)56 b Ft(tak)m(es)37 b(the)e(form)f(\()p Fo(E)c Ft(+)24 b Fo("U)10 b Ft(\()p Fo(";)17 b(c)p Ft(\)\))p Fo(Y)56 b Ft(with)-118 3686 y(symplectic)41 b(matrix)e Fo(E)34 b Ft(+)28 b Fo("U)10 b Ft(.)69 b(Therefore)42 b(this)f(close)g(to)g(iden)m(tit)m(y)g(matrix) e(can)j(b)s(e)f(written)g(as)g(the)-118 3806 y(exp)s(onen)m(t)f(of)e (some)g(Hamiltonian)d(matrix)i Fo(I)8 b(A)38 b Ft(with)g(a)g(symmetric) g Fo(A)g Ft(\(w)m(e)i(denote)f Fo(I)46 b Ft(the)39 b(standard)-118 3927 y(symplectic)32 b(matrix\).)42 b(Th)m(us)681 4121 y(\026)652 4147 y Fo(X)36 b Ft(=)27 b Fo("\013)q Ft(\()p Fo(";)17 b(c)p Ft(\))k(+)h(exp)q([)p Fo("I)8 b(A)p Ft(\()p Fo(";)17 b(c)p Ft(\)])p Fo(Y)43 b Ft(+)22 b Fo(F)14 b Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))k(+)h Fo("G)2630 4162 y Fu(1)2669 4147 y Ft(\()p Fo(Y)5 b(;)17 b(c;)g(")p Ft(\))p Fo(;)596 b Ft(\(10\))-118 4367 y(where)34 b(the)f(expansion)g(of)f (function)g Fo(G)1355 4382 y Fu(1)1427 4367 y Ft(b)s(egins)g(with)g (the)h(second)h(order)f(terms)f(in)g Fo(Y)22 b Ft(.)28 4487 y(The)34 b(case)f(of)f(unfolding)f(lying)g(in)h Fi(L)g Ft(is)g(c)m(haracterized)i(b)m(y)f(the)g(condition)-118 4657 y(\()p Fk(L)p Ft(\))228 b Fo(\013)q Ft(\()p Fo(";)17 b Ft(0\))26 b Fi(\021)i Ft(0.)-118 4935 y Fk(Theorem)37 b(1)49 b Fj(F)-7 b(or)42 b(the)i(c)-5 b(ase)43 b Ft(\()p Fk(L)p Ft(\))h Fj(and)f Fo(")g Fj(smal)5 b(l)43 b(enough)g(ther)-5 b(e)44 b(exists)f(a)g(r)-5 b(e)g(gion)43 b Fo(V)65 b Fj(in)44 b(the)f(p)-5 b(ositive)-118 5056 y(quadr)g(ant)36 b(of)g(the)g(plane)f Ft(\()p Fo(I)908 5071 y Fu(1)947 5056 y Fo(;)17 b(I)1034 5071 y Fu(2)1073 5056 y Ft(\))36 b Fj(ne)-5 b(ar)36 b(the)g(p)-5 b(oint)36 b Ft(\(0)p Fo(;)17 b Ft(0\))35 b Fj(such)h(that)g(the)g(p)-5 b(erturb)g(e)g(d)36 b(system)g(has)g(either)-118 5176 y(8,)k(12)g(or)f(16)g(tr)-5 b(ansverse)39 b(\(on)g(the)h(r)-5 b(elate)g(d)39 b(level)g(of)g(the)h (Hamiltonian\))f(homo)-5 b(clinic)37 b(orbits)j(to)g(every)-118 5297 y(p)-5 b(erturb)g(e)g(d)36 b(p)-5 b(ersistent)36 b(Diophantine)f(tori)h Fo(I)1533 5312 y Fu(1)1603 5297 y Ft(=)30 b Fo(I)1760 5260 y Fu(0)1752 5321 y(1)1799 5297 y Fj(,)37 b Fo(I)1909 5312 y Fu(2)1978 5297 y Ft(=)30 b Fo(I)2135 5260 y Fu(0)2127 5321 y(2)2174 5297 y Fj(,)37 b Ft(\()p Fo(I)2330 5260 y Fu(0)2322 5321 y(1)2369 5297 y Fo(;)17 b(I)2464 5260 y Fu(0)2456 5321 y(2)2503 5297 y Ft(\))30 b Fi(2)h Fo(V)21 b Fj(,)37 b(on)f(the)g(c)-5 b(enter)36 b(manifold)-118 5417 y Fo(W)-12 5381 y Fm(c)-26 5442 y(")22 5417 y Fj(.)1782 5753 y Ft(12)p eop %%Page: 13 13 13 12 bop -118 407 a Fk(Remark)37 b(2)49 b Fj(The)44 b(r)-5 b(e)g(gion)45 b Fo(V)67 b Fj(c)-5 b(onsists)45 b(of)g(the)g(p)-5 b(art)45 b(of)g(the)h(quadr)-5 b(ant)45 b Fo(I)2684 422 y Fu(1)2771 407 y Fo(>)i Ft(0)p Fj(,)g Fo(I)3063 422 y Fu(2)3150 407 y Fo(>)g Ft(0)e Fj(wher)-5 b(e)45 b(the)-118 527 y(function)33 b Fo(R)i Fj(de\014ne)-5 b(d)33 b(in)g(\(14\))h(is)f(a)h(Morse)g(function,)f(i.e.)g(al)5 b(l)34 b(its)g(critic)-5 b(al)33 b(p)-5 b(oints)34 b(ar)-5 b(e)33 b(nonde)-5 b(gener)g(ate.)-118 648 y(This)28 b(me)-5 b(ans)28 b(that)h Fo(V)50 b Fj(is)28 b(the)h(whole)f(quadr)-5 b(ant)29 b(exc)-5 b(ept)28 b(for)g(smal)5 b(l)28 b(neighb)-5 b(orho)g(o)g(ds)27 b(of)i(some)e(r)-5 b(ays)29 b Fo(I)3505 663 y Fu(1)3545 648 y Fo(=I)3637 663 y Fu(2)3704 648 y Ft(=)-118 768 y(const)q Fj(.)50 b(In)36 b(fact,)h(this)f(nonde)-5 b(gener)g(acy)35 b(c)-5 b(ondition)36 b(for)g Fo(R)i Fj(c)-5 b(an)36 b(b)-5 b(e)36 b(interpr)-5 b(ete)g(d)37 b(as)f(a)h(c)-5 b(ondition)35 b(on)i(the)-118 888 y(\014rst)i(or)-5 b(der)38 b(deviation)g(in)g Fo(")h Fj(for)f(the)h(glob)-5 b(al)38 b(map)g(at)h(the)f(tr)-5 b(ac)g(e)39 b(of)g Ft(\000)f Fj(on)h Fo(N)2776 852 y Fm(u)2821 888 y Ft(\(0\))p Fj(.)56 b(In)38 b(Se)-5 b(ction)38 b(5,)i(this)-118 1009 y(deviation)34 b(is)g(written)h(in)g(terms)g(of)f(the)h(Melnikov)f(p)-5 b(otential.)-118 1233 y Fk(Pro)s(of)p Ft(.)85 b(Let)46 b(us)i(\014x)f(a)f(torus)h(in)e Fo(W)1338 1196 y Fm(c)1324 1257 y(")1373 1233 y Ft(:)71 b Fo(I)1514 1248 y Fu(1)1605 1233 y Ft(=)51 b Fo(I)1783 1196 y Fu(0)1775 1257 y(1)1822 1233 y Fo(;)17 b(I)1909 1248 y Fu(2)2000 1233 y Ft(=)51 b Fo(I)2178 1196 y Fu(0)2170 1257 y(2)2217 1233 y Ft(,)f(in)c(the)h (Can)m(torian)e(set)j(of)e(p)s(ersisten)m(t)-118 1353 y(tori.)h(T)-8 b(races)36 b(of)e(its)g(stable)g(and)g(unstable)h (manifolds)d(on)i Fo(N)2206 1317 y Fm(s)2243 1353 y Ft(\()p Fo(c)p Ft(\))g(and)h Fo(N)2675 1317 y Fm(u)2720 1353 y Ft(\()p Fo(c)p Ft(\),)g(as)g(w)m(as)g(sho)m(wn)h(ab)s(o)m(v)m(e,)-118 1473 y(ha)m(v)m(e)42 b(the)f(same)f(represen)m(tation)i Fo(T)1260 1437 y Fu(2)1246 1498 y Fm(s)1341 1473 y Ft(=)f Fi(f)1520 1448 y Ft(\026)1508 1473 y Fo(I)1551 1488 y Fu(1)1632 1473 y Ft(=)g Fo(I)1800 1437 y Fu(0)1792 1498 y(1)1839 1473 y Fo(;)1895 1448 y Ft(\026)1883 1473 y Fo(I)1926 1488 y Fu(2)2007 1473 y Ft(=)g Fo(I)2175 1437 y Fu(0)2167 1498 y(2)2214 1473 y Fi(g)p Ft(,)i Fo(T)2405 1437 y Fu(2)2391 1498 y Fm(u)2486 1473 y Ft(=)e Fi(f)p Fo(I)2696 1488 y Fu(1)2777 1473 y Ft(=)g Fo(I)2945 1437 y Fu(0)2937 1498 y(1)2984 1473 y Fo(;)17 b(I)3071 1488 y Fu(2)3152 1473 y Ft(=)41 b Fo(I)3320 1437 y Fu(0)3312 1498 y(2)3359 1473 y Fi(g)p Ft(.)68 b(Let)41 b(us)-118 1594 y(\014nd)c Fo(T)152 1558 y Fu(2)138 1618 y Fm(s)216 1594 y Fi(\\)25 b Fo(S)367 1609 y Fm(")404 1594 y Ft(\()p Fo(T)513 1558 y Fu(2)499 1618 y Fm(u)552 1594 y Ft(\).)56 b(T)-8 b(o)37 b(this)f(end,)i(square)g(b)s(oth)f(sides)g(of)f(the)h(ev) m(ery)i(equalit)m(y)d(in)g(\(10\))g(and)h(sum)f(up)-118 1714 y(separately)d(left)f(hand)g(sides)h(and)g(righ)m(t)f(hand)h (sides)g(for)f(symplectically)f(conjugated)h(co)s(ordinates)635 1902 y(1)p 635 1946 49 4 v 635 2037 a(2)694 1969 y(\()6 b(\026)-55 b Fo(x)787 1928 y Fu(2)787 1994 y(1)849 1969 y Ft(+)28 b(\026)-55 b Fo(y)999 1928 y Fu(2)995 1994 y(1)1037 1969 y Ft(\))28 b(=)1217 1902 y(1)p 1217 1946 V 1217 2037 a(2)1275 1969 y(\()p Fo(u)1369 1928 y Fu(2)1369 1994 y(1)1430 1969 y Ft(+)22 b Fo(v)1579 1928 y Fu(2)1575 1994 y(1)1618 1969 y Ft(\))h(+)f Fo(")p Ft(\()p Fo(A)1934 1984 y Fu(1)1973 1969 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))k(+)h Fo("B)2436 1984 y Fu(1)2476 1969 y Ft(\()p Fo(Y)5 b(;)17 b(";)g(c)p Ft(\)\))p Fo(;)652 2135 y Ft(1)p 652 2179 V 652 2271 a(2)710 2202 y(\()6 b(\026)-55 b Fo(x)803 2161 y Fu(2)803 2227 y(2)865 2202 y Ft(+)29 b(\026)-56 b Fo(y)1015 2161 y Fu(2)1011 2227 y(2)1054 2202 y Ft(\))28 b(=)1233 2135 y(1)p 1233 2179 V 1233 2271 a(2)1292 2202 y(\()p Fo(u)1386 2161 y Fu(2)1386 2227 y(2)1447 2202 y Ft(+)22 b Fo(v)1596 2161 y Fu(2)1592 2227 y(2)1635 2202 y Ft(\))g(+)g Fo(")p Ft(\()p Fo(A)1950 2217 y Fu(2)1989 2202 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))22 b(+)g Fo("B)2453 2217 y Fu(2)2492 2202 y Ft(\()p Fo(Y)5 b(;)17 b(";)g(c)p Ft(\)\))p Fo(;)-118 2444 y Ft(where)400 2536 y Fo(A)473 2551 y Fu(1)512 2536 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))27 b(=)83 b Fo(u)978 2551 y Fu(1)1017 2536 y Ft(\()p Fo(\013)1117 2551 y Fu(1)1157 2536 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))k Fi(\000)i Fo(a)1540 2551 y Fu(13)1615 2536 y Fo(u)1671 2551 y Fu(1)1732 2536 y Fi(\000)f Fo(a)1882 2551 y Fu(23)1957 2536 y Fo(u)2013 2551 y Fu(2)2074 2536 y Fi(\000)h Fo(a)2225 2551 y Fu(33)2300 2536 y Fo(v)2347 2551 y Fu(1)2409 2536 y Fi(\000)f Fo(a)2559 2551 y Fu(34)2634 2536 y Fo(v)2681 2551 y Fu(2)2721 2536 y Ft(\)+)922 2657 y Fo(v)969 2672 y Fu(1)1009 2657 y Ft(\()p Fo(\013)1109 2672 y Fu(3)1148 2657 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))22 b(+)g Fo(a)1530 2672 y Fu(11)1605 2657 y Fo(u)1661 2672 y Fu(1)1722 2657 y Ft(+)g Fo(a)1871 2672 y Fu(12)1946 2657 y Fo(u)2002 2672 y Fu(2)2063 2657 y Ft(+)g Fo(a)2212 2672 y Fu(13)2287 2657 y Fo(v)2334 2672 y Fu(1)2396 2657 y Ft(+)g Fo(a)2545 2672 y Fu(14)2620 2657 y Fo(v)2667 2672 y Fu(2)2706 2657 y Ft(\))g(+)g Fo(O)s Ft(\()p Fi(k)p Fo(Y)e Fi(k)3157 2620 y Fu(3)3197 2657 y Ft(\))p Fo(;)400 2777 y(A)473 2792 y Fu(2)512 2777 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))27 b(=)83 b Fo(u)978 2792 y Fu(2)1017 2777 y Ft(\()p Fo(\013)1117 2792 y Fu(2)1157 2777 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))k Fi(\000)i Fo(a)1540 2792 y Fu(14)1615 2777 y Fo(u)1671 2792 y Fu(1)1732 2777 y Fi(\000)f Fo(a)1882 2792 y Fu(24)1957 2777 y Fo(u)2013 2792 y Fu(2)2074 2777 y Fi(\000)h Fo(a)2225 2792 y Fu(34)2300 2777 y Fo(v)2347 2792 y Fu(1)2409 2777 y Fi(\000)f Fo(a)2559 2792 y Fu(44)2634 2777 y Fo(v)2681 2792 y Fu(2)2721 2777 y Ft(\)+)922 2897 y Fo(v)969 2912 y Fu(2)1009 2897 y Ft(\()p Fo(\013)1109 2912 y Fu(4)1148 2897 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))22 b(+)g Fo(a)1530 2912 y Fu(12)1605 2897 y Fo(u)1661 2912 y Fu(1)1722 2897 y Ft(+)g Fo(a)1871 2912 y Fu(22)1946 2897 y Fo(u)2002 2912 y Fu(2)2063 2897 y Ft(+)g Fo(a)2212 2912 y Fu(23)2287 2897 y Fo(v)2334 2912 y Fu(1)2396 2897 y Ft(+)g Fo(a)2545 2912 y Fu(24)2620 2897 y Fo(v)2667 2912 y Fu(2)2706 2897 y Ft(\))g(+)g Fo(O)s Ft(\()p Fi(k)p Fo(Y)e Fi(k)3157 2861 y Fu(3)3197 2897 y Ft(\))p Fo(:)-118 3076 y Ft(T)-8 b(o)47 b(\014nd)h(the)f(in)m(tersection)g Fo(T)1041 3039 y Fu(2)1027 3100 y Fm(s)1127 3076 y Ft(and)h Fo(S)1392 3091 y Fm(")1428 3076 y Ft(\()p Fo(T)1537 3039 y Fu(2)1523 3100 y Fm(u)1576 3076 y Ft(\))f(w)m(e)h(set)g(in)f(the)g(equalities)f (obtained)3170 3050 y(\026)3159 3076 y Fo(I)3202 3091 y Fm(i)3282 3076 y Ft(=)52 b Fo(I)3453 3091 y Fm(i)3534 3076 y Ft(=)g Fo(I)3713 3039 y Fu(0)3705 3100 y Fm(i)3752 3076 y Ft(,)-118 3196 y Fo(I)-75 3211 y Fm(i)-19 3196 y Ft(=)27 b(\()p Fo(u)178 3160 y Fu(2)178 3221 y Fm(i)239 3196 y Ft(+)22 b Fo(v)388 3160 y Fu(2)384 3221 y Fm(i)428 3196 y Ft(\))p Fo(=)p Ft(2,)32 b Fo(i)c Ft(=)f(1)p Fo(;)17 b Ft(2.)43 b(Th)m(us)34 b(w)m(e)g(get)1111 3412 y Fo(")p Ft(\()p Fo(A)1268 3427 y Fu(1)1307 3412 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))22 b(+)g Fo("B)1771 3427 y Fu(1)1810 3412 y Ft(\()p Fo(Y)5 b(;)17 b(";)g(c)p Ft(\)\))26 b(=)i(0)p Fo(;)1127 3557 y(")p Ft(\()p Fo(A)1284 3572 y Fu(2)1324 3557 y Ft(\()p Fo(Y)5 b(;)17 b(c)p Ft(\))k(+)h Fo("B)1787 3572 y Fu(2)1827 3557 y Ft(\()p Fo(Y)5 b(;)17 b(";)g(c)p Ft(\)\))26 b(=)i(0)p Fo(;)1221 b Ft(\(11\))-118 3773 y(Divide)37 b(b)s(oth)i(sides)g(of)f(the)h(equations)g(\(11\))f(b)m(y)h Fo(")f Ft(and)h(tak)m(e)g(the)h(limit)35 b Fo(")i Fi(!)h Ft(0.)61 b(After)39 b(in)m(tro)s(ducing)-118 3915 y(symplectic)32 b(p)s(olar)f(co)s(ordinates)h Fo(u)1188 3930 y Fm(i)1244 3915 y Ft(=)1347 3813 y Fh(q)p 1430 3813 140 4 v 102 x Ft(2)p Fo(I)1530 3881 y Fu(0)1522 3938 y Fm(i)1586 3915 y Ft(cos)17 b Fo(\022)1778 3930 y Fm(i)1807 3915 y Ft(,)32 b Fo(v)1913 3930 y Fm(i)1969 3915 y Ft(=)2073 3813 y Fh(q)p 2156 3813 V 102 x Ft(2)p Fo(I)2256 3881 y Fu(0)2248 3938 y Fm(i)2311 3915 y Ft(sin)17 b Fo(\022)2493 3930 y Fm(i)2521 3915 y Ft(,)33 b(the)g(system)h(tak)m(es)f(the)g(form) 165 4173 y(2)214 4070 y Fh(q)p 297 4070 181 4 v 103 x Fo(I)348 4138 y Fu(0)340 4195 y(1)387 4173 y Fo(I)438 4138 y Fu(0)430 4195 y(2)477 4173 y Ft(\()p Fo(a)566 4188 y Fu(12)658 4173 y Ft(sin)16 b Fo(\022)839 4188 y Fu(1)895 4173 y Ft(cos)i Fo(\022)1088 4188 y Fu(2)1128 4173 y Ft(+)82 b Fo(a)1337 4188 y Fu(14)1429 4173 y Ft(sin)16 b Fo(\022)1610 4188 y Fu(1)1667 4173 y Ft(sin)g Fo(\022)1848 4188 y Fu(2)1910 4173 y Fi(\000)23 b Fo(a)2061 4188 y Fu(23)2152 4173 y Ft(cos)18 b Fo(\022)2345 4188 y Fu(1)2401 4173 y Ft(cos)f Fo(\022)2593 4188 y Fu(2)2655 4173 y Fi(\000)23 b Fo(a)2806 4188 y Fu(34)2898 4173 y Ft(cos)17 b Fo(\022)3090 4188 y Fu(1)3146 4173 y Ft(sin)g Fo(\022)3328 4188 y Fu(2)3368 4173 y Ft(\))22 b(+)1286 4256 y Fh(q)p 1370 4256 140 4 v 1370 4364 a Ft(2)p Fo(I)1470 4329 y Fu(0)1462 4385 y(1)1509 4364 y Ft(\()p Fo(\013)1609 4379 y Fu(1)1648 4364 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))g(cos)f Fo(\022)2067 4379 y Fu(1)2129 4364 y Ft(+)22 b Fo(\013)2289 4379 y Fu(3)2329 4364 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))g(sin)e Fo(\022)2737 4379 y Fu(1)2777 4364 y Ft(\))22 b(+)693 b(\(12\))1286 4526 y Fo(I)1337 4485 y Fu(0)1329 4551 y(1)1377 4526 y Ft(\(\()p Fo(a)1504 4541 y Fu(11)1601 4526 y Fi(\000)22 b Fo(a)1751 4541 y Fu(33)1826 4526 y Ft(\))17 b(sin)f(2)p Fo(\022)2111 4541 y Fu(1)2173 4526 y Fi(\000)23 b Ft(2)p Fo(a)2373 4541 y Fu(13)2464 4526 y Ft(cos)17 b(2)p Fo(\022)2705 4541 y Fu(1)2745 4526 y Ft(\))22 b(+)g Fo(O)s Ft(\()p Fi(k)p Fo(I)3120 4485 y Fu(0)3158 4526 y Fi(k)3208 4485 y Fu(3)p Fm(=)p Fu(2)3318 4526 y Ft(\))28 b(=)f(0)p Fo(;)165 4694 y Ft(2)214 4591 y Fh(q)p 297 4591 181 4 v 103 x Fo(I)348 4659 y Fu(0)340 4716 y(1)387 4694 y Fo(I)438 4659 y Fu(0)430 4716 y(2)477 4694 y Ft(\()p Fo(a)566 4709 y Fu(12)658 4694 y Ft(cos)17 b Fo(\022)850 4709 y Fu(1)906 4694 y Ft(sin)g Fo(\022)1088 4709 y Fu(2)1128 4694 y Ft(+)82 b Fo(a)1337 4709 y Fu(23)1429 4694 y Ft(sin)16 b Fo(\022)1610 4709 y Fu(1)1667 4694 y Ft(sin)g Fo(\022)1848 4709 y Fu(2)1910 4694 y Fi(\000)23 b Fo(a)2061 4709 y Fu(14)2152 4694 y Ft(cos)18 b Fo(\022)2345 4709 y Fu(1)2401 4694 y Ft(cos)f Fo(\022)2593 4709 y Fu(2)2655 4694 y Fi(\000)23 b Fo(a)2806 4709 y Fu(34)2898 4694 y Ft(sin)16 b Fo(\022)3079 4709 y Fu(1)3135 4694 y Ft(cos)i Fo(\022)3328 4709 y Fu(2)3368 4694 y Ft(\))k(+)1286 4777 y Fh(q)p 1370 4777 140 4 v 1370 4885 a Ft(2)p Fo(I)1470 4850 y Fu(0)1462 4906 y(2)1509 4885 y Ft(\()p Fo(\013)1609 4900 y Fu(2)1648 4885 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))g(cos)f Fo(\022)2067 4900 y Fu(2)2129 4885 y Ft(+)22 b Fo(\013)2289 4900 y Fu(4)2329 4885 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))g(sin)e Fo(\022)2737 4900 y Fu(2)2777 4885 y Ft(\))22 b(+)1286 5047 y Fo(I)1337 5006 y Fu(0)1329 5072 y(2)1377 5047 y Ft(\(\()p Fo(a)1504 5062 y Fu(22)1601 5047 y Fi(\000)g Fo(a)1751 5062 y Fu(44)1826 5047 y Ft(\))17 b(sin)f(2)p Fo(\022)2111 5062 y Fu(2)2173 5047 y Fi(\000)23 b Ft(2)p Fo(a)2373 5062 y Fu(24)2464 5047 y Ft(cos)17 b(2)p Fo(\022)2705 5062 y Fu(2)2745 5047 y Ft(\))22 b(+)g Fo(O)s Ft(\()p Fi(k)p Fo(I)3120 5006 y Fu(0)3158 5047 y Fi(k)3208 5006 y Fu(3)p Fm(=)p Fu(2)3318 5047 y Ft(\))28 b(=)f(0)p Fo(:)-118 5263 y Ft(F)-8 b(or)39 b(the)h(case)h(\()p Fk(L)p Ft(\),)f(the)g(homo)s (clinic)d(orbit)h(to)i Fo(O)s Ft(,)h(p)s(ersists)f(so)g Fo(\013)2401 5278 y Fm(i)2429 5263 y Ft(\()p Fo(";)17 b Ft(0\))39 b Fi(\021)i Ft(0,)g(therefore,)h Fo(\013)3425 5278 y Fm(i)3453 5263 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))40 b(=)-118 5384 y Fo(c)10 b Ft(\026)-59 b Fo(\013)-14 5399 y Fm(i)14 5384 y Ft(\(0)p Fo(;)17 b(c)p Ft(\).)52 b(Since)36 b(w)m(e)h(ha)m(v)m(e)g(\014xed)g(the)f(p)s(ersisten)m(t)h(torus,)g (then)f(in)f(lo)s(cal)f(co)s(ordinates)h(the)h(restriction)-118 5504 y(of)h(the)i(Hamiltonian)34 b(on)k Fo(W)985 5468 y Fm(c)971 5529 y(")1057 5504 y Ft(dep)s(ends)i(on)e Fo(I)1632 5468 y Fu(0)1624 5529 y(1)1671 5504 y Fo(;)17 b(I)1766 5468 y Fu(0)1758 5529 y(2)1843 5504 y Ft(only)-8 b(,)39 b(that)f(is,)h Fo(c)e Ft(=)g Fo(h)2680 5519 y Fu(1)2719 5504 y Ft(\()p Fo(I)2808 5468 y Fu(0)2800 5529 y(1)2848 5504 y Fo(;)17 b(I)2943 5468 y Fu(0)2935 5529 y(2)2982 5504 y Ft(\))36 b(=)h Fo(!)3230 5519 y Fu(1)3269 5504 y Fo(I)3320 5468 y Fu(0)3312 5529 y(1)3386 5504 y Ft(+)25 b Fo(!)3548 5519 y Fu(2)3588 5504 y Fo(I)3639 5468 y Fu(0)3631 5529 y(2)3704 5504 y Ft(+)1782 5753 y(13)p eop %%Page: 14 14 14 13 bop -118 407 a Fo(O)s Ft(\()p Fi(k)p Fo(I)99 371 y Fu(0)137 407 y Fi(k)187 371 y Fu(2)226 407 y Ft(\).)52 b(Therefore,)38 b(the)e(terms)f(con)m(taining)f Fo(\013)1800 422 y Fm(i)1828 407 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))35 b(can)h(b)s(e)f(included)g(in)m(to)g Fo(O)3079 311 y Fh(\020)3129 407 y Fi(k)p Fo(I)3230 371 y Fu(0)3269 407 y Fi(k)3319 371 y Fu(3)p Fm(=)p Fu(2)3428 311 y Fh(\021)3513 407 y Ft(terms.)-118 574 y(Divide)i(b)s(oth)i(sides)h(of)e(the)h (equations)g(b)m(y)1552 472 y Fh(q)p 1635 472 181 4 v 102 x Fo(I)1686 540 y Fu(0)1678 596 y(1)1726 574 y Fo(I)1777 540 y Fu(0)1769 596 y(2)1855 574 y Ft(and)g(denote)g Fo(r)i Ft(=)2571 473 y Fh(q)p 2654 473 230 4 v 101 x Fo(I)2705 540 y Fu(0)2697 596 y(1)2744 574 y Fo(=I)2844 540 y Fu(0)2836 596 y(2)2883 574 y Ft(.)63 b(As)39 b(a)g(result)g (system)-118 695 y(\(12\))32 b(tak)m(es)i(the)f(form)345 905 y Fo(r)s Ft(\(\()p Fo(a)519 920 y Fu(11)616 905 y Fi(\000)22 b Fo(a)766 920 y Fu(33)841 905 y Ft(\))17 b(sin)f(2)p Fo(\022)1126 920 y Fu(1)1188 905 y Fi(\000)22 b Ft(2)p Fo(a)1387 920 y Fu(13)1479 905 y Ft(cos)17 b(2)p Fo(\022)1720 920 y Fu(1)1760 905 y Ft(\))22 b(+)g(2\()p Fo(a)2056 920 y Fu(12)2147 905 y Ft(sin)16 b Fo(\022)2328 920 y Fu(1)2385 905 y Ft(cos)h Fo(\022)2577 920 y Fu(2)2639 905 y Ft(+)22 b Fo(a)2788 920 y Fu(14)2880 905 y Ft(sin)16 b Fo(\022)3061 920 y Fu(1)3117 905 y Ft(sin)h Fo(\022)3299 920 y Fu(2)3361 905 y Fi(\000)345 1063 y Fo(a)396 1078 y Fu(23)487 1063 y Ft(cos)h Fo(\022)680 1078 y Fu(1)736 1063 y Ft(cos)f Fo(\022)928 1078 y Fu(2)990 1063 y Fi(\000)23 b Fo(a)1141 1078 y Fu(34)1233 1063 y Ft(cos)17 b Fo(\022)1425 1078 y Fu(1)1481 1063 y Ft(sin)g Fo(\022)1663 1078 y Fu(2)1703 1063 y Ft(\))22 b(+)g Fo(O)1954 967 y Fh(\020)2004 1063 y Fi(k)p Fo(I)2105 1022 y Fu(0)2144 1063 y Fi(k)2194 1022 y Fu(1)p Fm(=)p Fu(2)2304 967 y Fh(\021)2381 1063 y Ft(=)28 b(0)p Fo(;)1045 b Ft(\(13\))355 1210 y(1)p 355 1254 49 4 v 356 1346 a Fo(r)414 1278 y Ft(\(\()p Fo(a)541 1293 y Fu(22)638 1278 y Fi(\000)22 b Fo(a)788 1293 y Fu(44)863 1278 y Ft(\))17 b(sin)f(2)p Fo(\022)1148 1293 y Fu(2)1210 1278 y Fi(\000)22 b Ft(2)p Fo(a)1409 1293 y Fu(24)1501 1278 y Ft(cos)17 b(2)p Fo(\022)1742 1293 y Fu(2)1782 1278 y Ft(\))22 b(+)g(2\()p Fo(a)2078 1293 y Fu(12)2169 1278 y Ft(cos)17 b Fo(\022)2361 1293 y Fu(1)2418 1278 y Ft(sin)f Fo(\022)2599 1293 y Fu(2)2661 1278 y Ft(+)22 b Fo(a)2810 1293 y Fu(23)2902 1278 y Ft(sin)16 b Fo(\022)3083 1293 y Fu(1)3140 1278 y Ft(sin)g Fo(\022)3321 1293 y Fu(2)3383 1278 y Fi(\000)345 1479 y Fo(a)396 1494 y Fu(14)487 1479 y Ft(cos)i Fo(\022)680 1494 y Fu(1)736 1479 y Ft(cos)f Fo(\022)928 1494 y Fu(2)990 1479 y Fi(\000)23 b Fo(a)1141 1494 y Fu(34)1233 1479 y Ft(sin)16 b Fo(\022)1414 1494 y Fu(1)1470 1479 y Ft(cos)i Fo(\022)1663 1494 y Fu(2)1703 1479 y Ft(\))k(+)g Fo(O)1954 1382 y Fh(\020)2004 1479 y Fi(k)p Fo(I)2105 1438 y Fu(0)2144 1479 y Fi(k)2194 1438 y Fu(1)p Fm(=)p Fu(2)2304 1382 y Fh(\021)2381 1479 y Ft(=)28 b(0)p Fo(:)-118 1706 y Ft(Let)37 b(us)g(\014x)g Fo(r)j Ft(and)d(tend)g Fo(I)887 1670 y Fu(0)879 1731 y(1)961 1706 y Fi(!)d Ft(0)j(and)g Fo(I)1426 1670 y Fu(0)1418 1731 y(2)1500 1706 y Fi(!)d Ft(0)i(along)g(a)g(ra)m(y)h Fo(I)2292 1670 y Fu(0)2284 1731 y(1)2366 1706 y Ft(=)e Fo(r)2524 1670 y Fu(2)2563 1706 y Fo(I)2614 1670 y Fu(0)2606 1731 y(2)2690 1706 y Ft(in)h(the)h(plane)f(\()p Fo(I)3325 1721 y Fu(1)3365 1706 y Fo(;)17 b(I)3452 1721 y Fu(2)3491 1706 y Ft(\).)56 b(The)-118 1826 y(equations)33 b(obtained)f(are)g (written)h(as)1648 1947 y Fi(r)p Fo(R)c Ft(=)e(0)p Fo(;)-118 2117 y Ft(with)144 2347 y Fo(R)84 b Ft(=)f Fi(\000)p Fo(r)602 2226 y Fh(\022)673 2280 y Fo(a)724 2295 y Fu(11)821 2280 y Fi(\000)23 b Fo(a)972 2295 y Fu(33)p 673 2324 374 4 v 835 2415 a Ft(2)1073 2347 y(cos)17 b(2)p Fo(\022)1314 2362 y Fu(1)1376 2347 y Ft(+)22 b Fo(a)1525 2362 y Fu(13)1617 2347 y Ft(sin)16 b(2)p Fo(\022)1847 2362 y Fu(1)1887 2226 y Fh(\023)1970 2347 y Fi(\000)2079 2280 y Ft(1)p 2079 2324 49 4 v 2080 2415 a Fo(r)2155 2226 y Fh(\022)2226 2280 y Fo(a)2277 2295 y Fu(22)2374 2280 y Fi(\000)23 b Fo(a)2525 2295 y Fu(44)p 2226 2324 374 4 v 2388 2415 a Ft(2)2626 2347 y(cos)17 b(2)p Fo(\022)2867 2362 y Fu(2)2929 2347 y Ft(+)22 b Fo(a)3078 2362 y Fu(24)3170 2347 y Ft(sin)16 b(2)p Fo(\022)3400 2362 y Fu(2)3440 2226 y Fh(\023)461 2530 y Fi(\000)p Ft(2\()p Fo(a)676 2545 y Fu(12)768 2530 y Ft(cos)h Fo(\022)960 2545 y Fu(1)1016 2530 y Ft(cos)h Fo(\022)1209 2545 y Fu(2)1271 2530 y Ft(+)k Fo(a)1420 2545 y Fu(23)1511 2530 y Ft(sin)17 b Fo(\022)1693 2545 y Fu(1)1749 2530 y Ft(cos)g Fo(\022)1941 2545 y Fu(2)2003 2530 y Ft(+)22 b Fo(a)2152 2545 y Fu(14)2244 2530 y Ft(cos)17 b Fo(\022)2436 2545 y Fu(1)2493 2530 y Ft(sin)f Fo(\022)2674 2545 y Fu(2)2736 2530 y Ft(+)22 b Fo(a)2885 2545 y Fu(34)2977 2530 y Ft(sin)16 b Fo(\022)3158 2545 y Fu(1)3214 2530 y Ft(sin)h Fo(\022)3396 2545 y Fu(2)3436 2530 y Ft(\))302 2719 y(=)83 b Fi(\000)p Fo(r)602 2598 y Fh(\022)673 2652 y Fo(a)724 2667 y Fu(11)821 2652 y Fi(\000)23 b Fo(a)972 2667 y Fu(33)p 673 2696 V 835 2788 a Ft(2)1073 2719 y(cos)17 b(2)p Fo(\022)1314 2734 y Fu(1)1376 2719 y Ft(+)22 b Fo(a)1525 2734 y Fu(13)1617 2719 y Ft(sin)16 b(2)p Fo(\022)1847 2734 y Fu(1)1887 2598 y Fh(\023)1970 2719 y Fi(\000)2079 2652 y Ft(1)p 2079 2696 49 4 v 2080 2788 a Fo(r)2155 2598 y Fh(\022)2226 2652 y Fo(a)2277 2667 y Fu(22)2374 2652 y Fi(\000)23 b Fo(a)2525 2667 y Fu(44)p 2226 2696 374 4 v 2388 2788 a Ft(2)2626 2719 y(cos)17 b(2)p Fo(\022)2867 2734 y Fu(2)2929 2719 y Ft(+)22 b Fo(a)3078 2734 y Fu(24)3170 2719 y Ft(sin)16 b(2)p Fo(\022)3400 2734 y Fu(2)3440 2598 y Fh(\023)461 2902 y Fi(\000)p Ft(\(\()p Fo(a)665 2917 y Fu(12)762 2902 y Fi(\000)23 b Fo(a)913 2917 y Fu(34)988 2902 y Ft(\))17 b(cos\()p Fo(\022)1256 2917 y Fu(1)1318 2902 y Ft(+)22 b Fo(\022)1461 2917 y Fu(2)1501 2902 y Ft(\))g(+)g(\()p Fo(a)1748 2917 y Fu(23)1845 2902 y Ft(+)g Fo(a)1994 2917 y Fu(14)2069 2902 y Ft(\))17 b(sin)o(\()p Fo(\022)2326 2917 y Fu(1)2388 2902 y Ft(+)22 b Fo(\022)2531 2917 y Fu(2)2571 2902 y Ft(\))599 3047 y(+)g(\()p Fo(a)786 3062 y Fu(12)883 3047 y Ft(+)g Fo(a)1032 3062 y Fu(34)1107 3047 y Ft(\))17 b(cos\()p Fo(\022)1375 3062 y Fu(1)1437 3047 y Fi(\000)22 b Fo(\022)1581 3062 y Fu(2)1621 3047 y Ft(\))g(+)g(\()p Fo(a)1868 3062 y Fu(23)1965 3047 y Fi(\000)h Fo(a)2116 3062 y Fu(14)2191 3047 y Ft(\))17 b(sin)o(\()p Fo(\022)2448 3062 y Fu(1)2510 3047 y Fi(\000)23 b Fo(\022)2655 3062 y Fu(2)2695 3047 y Ft(\)\))302 3193 y(=)83 b Fo(s)507 3208 y Fu(1)563 3193 y Ft(cos)q(\(2)p Fo(\022)826 3208 y Fu(1)887 3193 y Fi(\000)23 b Fo(\033)1042 3208 y Fu(1)1082 3193 y Ft(\))f(+)g Fo(s)1286 3208 y Fu(2)1342 3193 y Ft(cos)q(\()p Fo(\022)1556 3208 y Fu(1)1618 3193 y Ft(+)g Fo(\022)1761 3208 y Fu(2)1823 3193 y Fi(\000)g Fo(\033)1977 3208 y Fu(2)2017 3193 y Ft(\))461 3338 y(+)p Fo(s)583 3353 y Fu(3)639 3338 y Ft(cos\()p Fo(\022)852 3353 y Fu(1)914 3338 y Fi(\000)h Fo(\022)1059 3353 y Fu(2)1121 3338 y Fi(\000)g Fo(\033)1276 3353 y Fu(3)1316 3338 y Ft(\))f(+)g Fo(s)1520 3353 y Fu(4)1576 3338 y Ft(cos\(2)p Fo(\022)1838 3353 y Fu(2)1900 3338 y Fi(\000)h Fo(\033)2055 3353 y Fu(4)2095 3338 y Ft(\))p Fo(;)1446 b Ft(\(14\))-118 3548 y(where)32 b(the)g (\(non-negativ)m(e\))e(co)s(e\016cien)m(ts)j Fo(s)1516 3563 y Fm(i)1575 3548 y Ft(dep)s(end)f(on)f Fo(r)s Ft(,)g(and)g(the)h (phases)g Fo(\033)2869 3563 y Fm(i)2929 3548 y Ft(do)f(not)g(dep)s(end) h(on)f Fo(r)s Ft(.)-118 3669 y(It)k(is)g(clear)f(that)h(this)g(gradien) m(t)g(form)e(of)i(the)h(system)g(of)e(equations)i(is)e(a)h(direct)g (consequence)j(of)d(the)-118 3789 y(prop)s(ert)m(y)e(that)g(tori)e Fo(T)747 3753 y Fu(2)733 3814 y Fm(s)819 3789 y Ft(and)h Fo(S)1068 3804 y Fm(")1105 3789 y Ft(\()p Fo(T)1214 3753 y Fu(2)1200 3814 y Fm(u)1253 3789 y Ft(\))g(are)h(Lagrangian.)28 3909 y(W)-8 b(e)37 b(are)f(seeking)h(nondegenerate)h(critical)33 b(p)s(oin)m(ts)j(of)g(function)g Fo(R)q Ft(\()p Fo(\022)2642 3924 y Fu(1)2682 3909 y Fo(;)17 b(\022)2771 3924 y Fu(2)2810 3909 y Fo(;)g(r)s Ft(\))36 b(at)g(a)g(\014xed)h Fo(r)i Ft(on)d(the)-118 4030 y(torus)22 b(\()p Fo(\022)204 4045 y Fu(1)244 4030 y Fo(;)17 b(\022)333 4045 y Fu(2)373 4030 y Ft(\))27 b(\(mo)s(d)g(2)p Fo(\031)t Ft(\).)40 b(The)23 b(nondegeneracy)h(at)e(a)g(critical)e(p)s(oin)m(t)h Fo(p)h Ft(is)g(equiv)-5 b(alen)m(t)22 b(to)g(the)h(inequalit)m(y)1507 4300 y(det)1669 4232 y Fo(@)1725 4196 y Fu(2)1765 4232 y Fo(R)p 1669 4276 172 4 v 1682 4368 a(@)5 b(\022)1786 4339 y Fu(2)1850 4300 y Ft(\()p Fo(p)p Ft(\))28 b Fi(6)p Ft(=)f(0)-118 4541 y(where)34 b Fo(\022)d Ft(=)c(\()p Fo(\022)426 4556 y Fu(1)466 4541 y Fo(;)17 b(\022)555 4556 y Fu(2)595 4541 y Ft(\).)28 4661 y(Let)30 b(us)h(observ)m(e)h (that)d(the)i(function)e Fo(R)i Ft(dep)s(ends)h(only)d(on)h Fo(\022)2290 4676 y Fu(1)2330 4661 y Ft(,)g Fo(\022)2432 4676 y Fu(2)2502 4661 y Ft(and)g Fo(r)s Ft(.)42 b(This)30 b(means)g(that)g(for)g(all)-118 4782 y(tori)k(on)h(the)h(ra)m(y)g Fo(I)600 4746 y Fu(0)592 4806 y(1)671 4782 y Ft(=)c Fo(r)826 4746 y Fu(2)865 4782 y Fo(I)916 4746 y Fu(0)908 4806 y(2)991 4782 y Ft(this)j(function)g(is)f(the)i(same.)52 b(In)35 b(particular,)g(if)f(w)m(e)i(ha)m(v)m(e)h(found)e(a)g(torus) -118 4902 y Fo(I)-75 4917 y Fu(1)-6 4902 y Ft(=)30 b Fo(I)151 4866 y Fu(0)143 4927 y(1)190 4902 y Ft(,)k Fo(I)294 4917 y Fu(2)363 4902 y Ft(=)29 b Fo(I)519 4866 y Fu(0)511 4927 y(2)592 4902 y Ft(suc)m(h)36 b(that)d(all)f(critical)f(p)s(oin)m (ts)i(of)h Fo(R)g Ft(on)g(this)f(torus)h(are)g(nondegenerate,)i(then)e (the)-118 5023 y(same)e(is)g(true)h(for)f(all)f(tori)g(on)i(this)f(ra)m (y)h(and)g(for)f(ra)m(ys)h(\(i.e.)43 b Fo(r)s Ft(\))32 b(su\016cien)m(tly)h(close)g(to)f(this)g(one.)28 5143 y(Applying)42 b(Lemma)f(2)i(\(see)g(b)s(elo)m(w\))g(to)f(the)h (function)f Fo(R)q Ft(\()p Fi(\001)p Fo(;)17 b Fi(\001)p Fo(;)g(r)s Ft(\))41 b(w)m(e)j(see)g(that,)h(if)c(all)g(its)h(critical) -118 5263 y(p)s(oin)m(ts)36 b(are)g(nondegenerate,)i(the)f(n)m(um)m(b)s (er)g(of)e(suc)m(h)j(critical)c(p)s(oin)m(ts)h(m)m(ust)i(b)s(e)f(8,)h (12)f(or)g(16.)53 b(No)m(w)37 b(let)-118 5384 y(us)j(restrict)f Fo(r)s Ft(,)i(0)f Fo(<)f(r)720 5399 y Fu(0)799 5384 y Fo(<)g(r)j(<)d(r)1159 5399 y Fu(1)1199 5384 y Ft(.)64 b(If,)41 b(for)e(a)g(\014xed)i Fo(r)1954 5399 y Fn(\003)2033 5384 y Ft(in)e(this)g(in)m(terv)-5 b(al,)40 b(the)g(function)f Fo(R)h Ft(has)g(only)-118 5504 y(nondegenerate)e(critical)c(p)s(oin)m (ts,)j(then)g(the)g(same)f(is)g(true)h(for)f(some)g(op)s(en)h(in)m (terv)-5 b(al)35 b(of)h Fo(r)j Ft(con)m(taining)1782 5753 y(14)p eop %%Page: 15 15 15 14 bop -118 407 a Fo(r)-74 422 y Fn(\003)-35 407 y Ft(.)64 b(Therefore,)42 b(the)e(set)g Fi(C)46 b Ft(of)39 b(all)e Fo(r)42 b Ft(suc)m(h)f(that)e(the)g(function)g Fo(R)i Ft(has)e(only)g(nondegenerate)i(critical)-118 527 y(p)s(oin)m(ts)29 b(is)g(op)s(en)g(and)h(b)s(elongs)f(to)g(the)g (in)m(terv)-5 b(al)29 b(\()p Fo(r)1745 542 y Fu(0)1784 527 y Fo(;)17 b(r)1872 542 y Fu(1)1911 527 y Ft(\).)42 b(F)-8 b(or)29 b Fo(r)j Ft(b)s(elonging)27 b(to)i(the)h(b)s(oundary)g (of)f Fi(C)6 b Ft(,)30 b(the)-118 648 y(function)i Fo(R)i Ft(has)f(at)f(least)g(one)h(degenerate)g(critical)e(p)s(oin)m(t.)28 768 y(Let)24 b(us)g(c)m(ho)s(ose)h(a)e(compact)g(set)h(in)f Fi(C)30 b Ft(\(for)23 b(instance,)j(a)d(collection)f(of)h(a)g(\014nite) g(n)m(um)m(b)s(er)h(of)f(segmen)m(ts\).)-118 888 y(Then)31 b(for)e(these)j Fo(r)e Fi(2)e(C)36 b Ft(the)31 b(implicit)26 b(function)j(theorem)h(can)g(b)s(e)g(applied)f(to)h(system)h(\(11\),)f (expressed)-118 1009 y(in)35 b(p)s(olar)e(co)s(ordinates)i(and)h (divided)f(b)m(y)h Fo(")p Ft(,)g(uniformly)e(in)g Fo(r)k Ft(giving)c(solutions)h Fo(\022)2956 1024 y Fm(i)2984 1009 y Ft(\()p Fo(I)3073 973 y Fu(0)3065 1033 y(1)3112 1009 y Fo(;)17 b(I)3207 973 y Fu(0)3199 1033 y(2)3246 1009 y Fo(;)g(")p Ft(\),)36 b Fo(i)d Ft(=)f(1)p Fo(;)17 b Ft(2,)-118 1129 y(for)29 b(all)f(\()p Fo(I)250 1093 y Fu(0)242 1154 y(1)289 1129 y Fo(;)17 b(I)384 1093 y Fu(0)376 1154 y(2)423 1129 y Fo(;)g(")p Ft(\))29 b(close)h(enough)h(to) e(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0\).)41 b(The)31 b(union)e(of)h(all) d(ra)m(ys)k(with)f Fo(r)g Fi(2)e(C)36 b Ft(giv)m(es)31 b(us)f(the)g(set)h Fo(V)-118 1249 y Ft(of)h(the)h(theorem.)28 1370 y(Finally)-8 b(,)48 b(w)m(e)g(select)g(an)m(y)f(torus)h Fo(I)60 b Ft(=)52 b Fo(I)1581 1334 y Fu(0)1573 1394 y(1)1620 1370 y Ft(,)f Fo(I)60 b Ft(=)52 b Fo(I)1980 1334 y Fu(0)1972 1394 y(2)2066 1370 y Ft(in)46 b Fo(W)2300 1334 y Fm(c)2381 1370 y Ft(of)h(the)g(in)m(tegrable)f(system)i(with)f(a)-118 1490 y(diophan)m(tine)e(set)h(of)f(frequencies)j(whic)m(h)e(p)s (ersists)g(for)f(0)50 b Fi(\024)g Fo(")g(<)g(")2543 1505 y Fu(0)2582 1490 y Ft(.)83 b(If)45 b(these)i Fo(I)3116 1454 y Fu(0)3108 1515 y(1)3156 1490 y Ft(,)i Fo(I)3283 1454 y Fu(0)3275 1515 y(2)3368 1490 y Ft(suc)m(h)e(that)-118 1631 y Fo(r)58 b Ft(=)115 1530 y Fh(q)p 198 1530 230 4 v 101 x Fo(I)249 1597 y Fu(0)241 1653 y(1)288 1631 y Fo(=I)388 1597 y Fu(0)380 1653 y(2)482 1631 y Fi(2)d(C)g Ft(then)50 b(w)m(e)f(can)g(obtain)f(in)m(tersection)g(on)h Fo(N)2402 1595 y Fm(s)2439 1631 y Ft(\()p Fo(c)p Ft(\))g(of)f(the)h (stable)g(and)f(unstable)-118 1763 y(manifolds)30 b(of)i(this)g(torus)h (\()p Fo(\022)958 1778 y Fu(1)998 1763 y Ft(\()p Fo(I)1087 1727 y Fu(0)1079 1788 y(1)1126 1763 y Fo(;)17 b(I)1221 1727 y Fu(0)1213 1788 y(2)1260 1763 y Fo(;)g(")p Ft(\),)32 b Fo(\022)1492 1778 y Fu(2)1532 1763 y Ft(\()p Fo(I)1621 1727 y Fu(0)1613 1788 y(1)1660 1763 y Fo(;)17 b(I)1755 1727 y Fu(0)1747 1788 y(2)1794 1763 y Fo(;)g(")p Ft(\),)32 b Fo(c)c Ft(=)f Fo(h)p Ft(\()p Fo(I)2299 1727 y Fu(0)2291 1788 y(1)2339 1763 y Fo(;)17 b(I)2434 1727 y Fu(0)2426 1788 y(2)2473 1763 y Ft(\)\).)p 2619 1763 42 42 v -118 1992 a Fk(Lemma)37 b(2)49 b Fj(L)-5 b(et)35 b Fo(R)q Ft(\()p Fo(\022)710 2007 y Fu(1)750 1992 y Fo(;)17 b(\022)839 2007 y Fu(2)879 1992 y Ft(\))34 b Fj(b)-5 b(e)35 b(a)g(sc)-5 b(alar)34 b(function)g(on)h Fe(T)2015 1955 y Fu(2)2088 1992 y Fj(of)g(the)g(form)59 2212 y Fo(R)29 b Ft(=)e Fo(s)311 2227 y Fu(1)367 2212 y Ft(cos)q(\(2)p Fo(\022)630 2227 y Fu(1)692 2212 y Fi(\000)22 b Fo(\033)846 2227 y Fu(1)886 2212 y Ft(\))g(+)g Fo(s)1090 2227 y Fu(2)1146 2212 y Ft(cos)q(\()p Fo(\022)1360 2227 y Fu(1)1422 2212 y Ft(+)g Fo(\022)1565 2227 y Fu(2)1627 2212 y Fi(\000)h Fo(\033)1782 2227 y Fu(2)1821 2212 y Ft(\))g(+)f Fo(s)2026 2227 y Fu(3)2082 2212 y Ft(cos\()p Fo(\022)2295 2227 y Fu(1)2357 2212 y Fi(\000)h Fo(\022)2502 2227 y Fu(2)2564 2212 y Fi(\000)f Fo(\033)2718 2227 y Fu(3)2758 2212 y Ft(\))g(+)g Fo(s)2962 2227 y Fu(4)3018 2212 y Ft(cos)q(\(2)p Fo(\022)3281 2227 y Fu(2)3343 2212 y Fi(\000)g Fo(\033)3497 2227 y Fu(4)3537 2212 y Ft(\))p Fo(;)-118 2432 y Fj(with)45 b(given)g(non-ne)-5 b(gative)44 b(c)-5 b(o)g(e\016cients)44 b Fo(s)1503 2447 y Fm(i)1531 2432 y Fj(,)k(and)d(phases)g Fo(\033)2184 2447 y Fm(i)2212 2432 y Fj(.)77 b(If)44 b Fo(R)j Fj(is)e(a)g(Morse)h(function,)h(i.e.)e(its)-118 2552 y(critic)-5 b(al)34 b(p)-5 b(oints)35 b(ar)-5 b(e)34 b(al)5 b(l)35 b(nonde)-5 b(gener)g(ate,)33 b(then)i(the)g(numb)-5 b(er)34 b(of)h(such)f(critic)-5 b(al)35 b(p)-5 b(oints)34 b(is)h(8,)g(12)f(or)h(16.)-118 2780 y Fk(Pro)s(of.)43 b Ft(In)33 b(view)g(of)f(the)h(in)m(v)-5 b(ariance)32 b(of)g Fo(R)h Ft(under)h(the)f(shift)1271 3000 y(\()p Fo(\022)1354 3015 y Fu(1)1394 3000 y Fo(;)17 b(\022)1483 3015 y Fu(2)1523 3000 y Ft(\))27 b Fi(7!)h Ft(\()p Fo(\022)1799 3015 y Fu(1)1861 3000 y Ft(+)22 b Fo(\031)t(;)17 b(\022)2107 3015 y Fu(2)2168 3000 y Ft(+)22 b Fo(\031)t Ft(\))p Fo(;)1216 b Ft(\(15\))-118 3220 y(w)m(e)34 b(carry)f(out)f(the)h(follo)m(wing)d (2-to-1)h(linear)g(c)m(hange)i(on)g Fe(T)2098 3184 y Fu(2)2136 3220 y Ft(:)1084 3440 y Fo( )1147 3455 y Fu(1)1215 3440 y Ft(=)27 b(2)p Fo(\022)1412 3455 y Fu(1)1474 3440 y Fi(\000)c Fo(\033)1629 3455 y Fu(1)1668 3440 y Fo(;)115 b( )1873 3455 y Fu(2)1940 3440 y Ft(=)28 b Fo(\022)2089 3455 y Fu(1)2151 3440 y Ft(+)22 b Fo(\022)2294 3455 y Fu(2)2356 3440 y Fi(\000)g Fo(\033)2510 3455 y Fu(2)2550 3440 y Fo(:)-118 3660 y Ft(Its)33 b(in)m(v)m(erse)h(is)e(a)g(\\1-to-2)f (map":)546 3924 y Fo(\022)591 3939 y Fu(1)658 3924 y Ft(=)772 3857 y Fo( )835 3872 y Fu(1)897 3857 y Ft(+)22 b Fo(\033)1050 3872 y Fu(1)p 772 3901 318 4 v 906 3992 a Ft(2)1122 3924 y(+)g Fo(\027)q(;)114 b(\022)1455 3939 y Fu(2)1523 3924 y Ft(=)27 b Fo( )1689 3939 y Fu(2)1751 3924 y Fi(\000)1861 3857 y Fo( )1924 3872 y Fu(1)1986 3857 y Ft(+)22 b Fo(\033)2139 3872 y Fu(1)p 1861 3901 V 1995 3992 a Ft(2)2211 3924 y(+)g Fo(\033)2364 3939 y Fu(2)2426 3924 y Ft(+)g Fo(\027)104 b Ft(\()p Fo(\027)34 b Ft(=)27 b(0)p Fo(;)17 b(\031)t Ft(\))p Fo(:)490 b Ft(\(16\))-118 4170 y(With)32 b(this)g(c)m(hange)i(w)m(e)f(obtain)165 4357 y Fh(e)142 4390 y Fo(R)q Ft(\()p Fo( )318 4405 y Fu(1)358 4390 y Fo(;)17 b( )465 4405 y Fu(2)505 4390 y Ft(\))27 b(=)h Fo(s)720 4405 y Fu(1)776 4390 y Ft(cos)17 b Fo( )986 4405 y Fu(1)1048 4390 y Ft(+)22 b Fo(s)1192 4405 y Fu(2)1248 4390 y Ft(cos)17 b Fo( )1458 4405 y Fu(2)1520 4390 y Ft(+)22 b Fo(s)1664 4405 y Fu(3)1720 4390 y Ft(cos)q(\()p Fo( )1952 4405 y Fu(1)2014 4390 y Fi(\000)g Fo( )2176 4405 y Fu(2)2238 4390 y Fi(\000)28 b Ft(^)-54 b Fo(\033)2393 4405 y Fu(1)2433 4390 y Ft(\))22 b(+)g Fo(s)2637 4405 y Fu(4)2693 4390 y Ft(cos\()p Fo( )2924 4405 y Fu(1)2986 4390 y Fi(\000)h Ft(2)p Fo( )3198 4405 y Fu(2)3260 4390 y Fi(\000)28 b Ft(^)-55 b Fo(\033)3414 4405 y Fu(2)3454 4390 y Ft(\))p Fo(;)-118 4610 y Ft(with)38 b(^)-54 b Fo(\033)160 4625 y Fu(1)228 4610 y Ft(=)29 b Fi(\000)p Fo(\033)465 4625 y Fu(1)528 4610 y Ft(+)23 b Fo(\033)682 4625 y Fu(2)744 4610 y Ft(+)f Fo(\033)897 4625 y Fu(3)971 4610 y Ft(and)38 b(^)-54 b Fo(\033)1216 4625 y Fu(2)1285 4610 y Ft(=)28 b Fi(\000)p Fo(\033)1521 4625 y Fu(1)1584 4610 y Ft(+)23 b(2)p Fo(\033)1787 4625 y Fu(2)1849 4610 y Fi(\000)g Fo(\033)2004 4625 y Fu(4)2044 4610 y Ft(.)45 b(Since)34 b(eac)m(h)g(critical)d(p)s(oin)m(t)h(of)3307 4577 y Fh(e)3285 4610 y Fo(R)i Ft(giv)m(es)g(rise)-118 4730 y(to)e(2)g(critical)f(p)s(oin)m(ts)h(of)g Fo(R)q Ft(,)h(w)m(e)g(need)h(to)e(pro)m(v)m(e)i(that)1934 4698 y Fh(e)1911 4730 y Fo(R)g Ft(has)f(4,)f(6)g(or)h(8)f(critical)e(p)s (oin)m(ts.)28 4850 y(One)43 b(can)g(giv)m(e)f(a)g(lo)m(w)m(er)h(b)s (ound)f(for)g(the)h(n)m(um)m(b)s(er)g(of)e(critical)f(p)s(oin)m(ts)i (of)2913 4818 y Fh(e)2890 4850 y Fo(R)i Ft(from)d(some)h(results)-118 4971 y(of)37 b(Morse)h(theory)g([11].)58 b(Recall)36 b(that)h(the)h(index)g(of)f(a)g(critical)e(p)s(oin)m(t)i Fo(p)g Ft(is)g(the)h(n)m(um)m(b)s(er)g(of)f(negativ)m(e)-118 5091 y(eigen)m(v)-5 b(alues)24 b(for)g(the)h(Hessian)g(matrix)e(for) 1499 5059 y Fh(e)1477 5091 y Fo(R)i Ft(at)f Fo(p)p Ft(,)i(and)f(denote) g(as)g Fo(\027)2437 5106 y Fm(k)2507 5091 y Ft(=)j Fo(\027)2659 5106 y Fm(k)2702 5091 y Ft(\()2762 5059 y Fh(e)2740 5091 y Fo(R)q Ft(\))c(the)h(n)m(um)m(b)s(er)g(of)f(critical)-118 5212 y(p)s(oin)m(ts)36 b(of)g(the)h(index)g Fo(k)s Ft(,)g(0)d Fi(\024)h Fo(k)i Fi(\024)e Ft(2)h(\(then,)i(the)f(amoun)m(ts)f Fo(\027)2240 5227 y Fu(0)2280 5212 y Ft(,)h Fo(\027)2392 5227 y Fu(1)2432 5212 y Ft(,)h Fo(\027)2545 5227 y Fu(2)2621 5212 y Ft(are)e(the)h(n)m(um)m(b)s(er)g(of)f(minima,)-118 5332 y(saddle)44 b(p)s(oin)m(ts)f(and)h(maxima,)g(resp)s(ectiv)m (ely\).)78 b(The)45 b(Morse)f(inequalities)e(connect)j(the)f(n)m(um)m (b)s(er)g(of)-118 5452 y(critical)29 b(p)s(oin)m(ts)j(of)f(a)g(giv)m (en)h(index)g(and)g(the)g(Betti)f(n)m(um)m(b)s(ers)i(\(ranks)f(of)g (the)g(homology)e(groups)i(of)f(the)1782 5753 y(15)p eop %%Page: 16 16 16 15 bop -118 407 a Ft(manifold)28 b(under)k(consideration\).)42 b(F)-8 b(or)30 b(the)h(case)h(of)e(the)i(2-torus,)f(the)g(Morse)h (inequalities)d(are)i(of)f(the)-118 527 y(form)778 648 y Fo(\027)826 663 y Fu(0)893 648 y Fi(\025)f Ft(1)p Fo(;)114 b(\027)1237 663 y Fu(1)1298 648 y Fi(\000)23 b Fo(\027)1446 663 y Fu(0)1513 648 y Fi(\025)28 b Ft(1)p Fo(;)114 b(\027)1856 663 y Fu(2)1918 648 y Fi(\000)23 b Fo(\027)2066 663 y Fu(1)2128 648 y Ft(+)f Fo(\027)2274 663 y Fu(0)2341 648 y Ft(=)27 b Fo(\037)p Ft(\()p Fe(T)2599 606 y Fu(2)2638 648 y Ft(\))h(=)f(0)p Fo(;)-118 821 y Ft(where)36 b Fo(\037)e Ft(is)g(the)h(Euler)g(c)m(haracteristics.)49 b(Therefore,)37 b(the)e(minimal)30 b(n)m(um)m(b)s(er)35 b(of)f(critical)e(p)s(oin)m(ts) i(of)3727 789 y Fh(e)3704 821 y Fo(R)-118 941 y Ft(\(assuming)25 b(that)i(they)g(are)f(all)e(nondegenerate\))k(is)e(equal)g(to)g(4,)i (and)e(for)g(a)g(function)g(with)g(the)h(minimal)-118 1062 y(n)m(um)m(b)s(er)32 b(of)f(critical)e(p)s(oin)m(ts)i(one)h(gets)g (the)g(follo)m(wing)c(distribution)i(of)h(their)g(t)m(yp)s(es:)44 b(a)31 b(minim)m(um,)e(t)m(w)m(o)-118 1182 y(saddles)f(and)g(a)f(maxim) m(um.)39 b(W)-8 b(e)28 b(also)f(deduce)i(that)e(the)h(total)e(n)m(um)m (b)s(er)i(of)f(critical)e(p)s(oin)m(ts)i(for)g(a)h(Morse)-118 1302 y(function)k(on)g Fe(T)455 1266 y Fu(2)527 1302 y Ft(is)g(ev)m(en,)i(since)f Fo(\027)1161 1317 y Fu(0)1223 1302 y Ft(+)22 b Fo(\027)1369 1317 y Fu(1)1431 1302 y Ft(+)g Fo(\027)1577 1317 y Fu(2)1644 1302 y Ft(=)28 b(2)p Fo(\027)1845 1317 y Fu(1)1884 1302 y Ft(.)28 1423 y(T)-8 b(o)38 b(\014nd)g(an)g(upp)s(er)g(b)s(ound)g(for)g(the)g(n)m(um)m(b)s (er)g(of)f(critical)f(p)s(oin)m(ts)h(of)2680 1391 y Fh(e)2657 1423 y Fo(R)q Ft(,)i(w)m(e)g(shall)e(sho)m(w)h(that)g(the)-118 1543 y(critical)25 b(p)s(oin)m(ts)i(can)g(b)s(e)h(reduced)h(to)d(zero)s (es)j(of)d(a)h(trigonometric)e(p)s(olynomial)f(of)j(one)g(angle.)41 b(W)-8 b(e)28 b(write)943 1729 y Fh(e)920 1761 y Fo(R)h Ft(=)e Fo(\025)1183 1776 y Fu(0)1223 1761 y Ft(\()p Fo( )1324 1776 y Fu(2)1363 1761 y Ft(\))c(+)f Fo(\025)1579 1776 y Fu(1)1618 1761 y Ft(\()p Fo( )1719 1776 y Fu(2)1759 1761 y Ft(\))17 b(cos)f Fo( )2023 1776 y Fu(1)2085 1761 y Ft(+)22 b Fo(\025)2240 1776 y Fu(2)2280 1761 y Ft(\()p Fo( )2381 1776 y Fu(2)2421 1761 y Ft(\))17 b(sin)e Fo( )2674 1776 y Fu(1)2714 1761 y Fo(;)865 b Ft(\(17\))-118 1979 y(where)34 b(w)m(e)f(de\014ne)875 2196 y Fo(\025)932 2211 y Fu(0)971 2196 y Ft(\()p Fo( )1072 2211 y Fu(2)1112 2196 y Ft(\))28 b(=)f Fo(s)1327 2211 y Fu(2)1383 2196 y Ft(cos)18 b Fo( )1594 2211 y Fu(2)1633 2196 y Fo(;)875 2342 y(\025)932 2357 y Fu(1)971 2342 y Ft(\()p Fo( )1072 2357 y Fu(2)1112 2342 y Ft(\))28 b(=)f Fo(s)1327 2357 y Fu(1)1389 2342 y Ft(+)22 b Fo(s)1533 2357 y Fu(3)1589 2342 y Ft(cos)q(\()p Fo( )1821 2357 y Fu(2)1883 2342 y Ft(+)27 b(^)-54 b Fo(\033)2036 2357 y Fu(1)2075 2342 y Ft(\))22 b(+)g Fo(s)2279 2357 y Fu(4)2335 2342 y Ft(cos)q(\(2)p Fo( )2616 2357 y Fu(2)2678 2342 y Ft(+)27 b(^)-54 b Fo(\033)2831 2357 y Fu(2)2871 2342 y Ft(\))p Fo(;)670 b Ft(\(18\))875 2487 y Fo(\025)932 2502 y Fu(2)971 2487 y Ft(\()p Fo( )1072 2502 y Fu(2)1112 2487 y Ft(\))28 b(=)f Fo(s)1327 2502 y Fu(3)1383 2487 y Ft(sin\()p Fo( )1604 2502 y Fu(2)1666 2487 y Ft(+)g(^)-54 b Fo(\033)1819 2502 y Fu(1)1859 2487 y Ft(\))22 b(+)g Fo(s)2063 2502 y Fu(4)2119 2487 y Ft(sin\(2)p Fo( )2389 2502 y Fu(2)2451 2487 y Ft(+)27 b(^)-54 b Fo(\033)2604 2502 y Fu(2)2643 2487 y Ft(\))p Fo(:)898 b Ft(\(19\))-118 2705 y(The)33 b(critical)e(p)s(oin)m(ts)h(of)834 2672 y Fh(e)812 2705 y Fo(R)h Ft(are)g(the)g(solutions)f(of)g(the)h(follo)m (wing)d(system)j(of)f(equations:)885 2922 y Fo(@)964 2890 y Fh(e)941 2922 y Fo(R)p 871 2966 160 4 v 871 3058 a(@)5 b( )990 3073 y Fu(1)1068 2989 y Ft(=)28 b Fo(\025)1229 3004 y Fu(2)1268 2989 y Ft(\()p Fo( )1369 3004 y Fu(2)1409 2989 y Ft(\))17 b(cos)g Fo( )1674 3004 y Fu(1)1736 2989 y Fi(\000)22 b Fo(\025)1892 3004 y Fu(1)1931 2989 y Ft(\()p Fo( )2032 3004 y Fu(2)2072 2989 y Ft(\))17 b(sin)f Fo( )2326 3004 y Fu(1)2394 2989 y Ft(=)27 b(0)p Fo(;)1033 b Ft(\(20\))885 3203 y Fo(@)964 3170 y Fh(e)941 3203 y Fo(R)p 871 3247 V 871 3338 a(@)5 b( )990 3353 y Fu(2)1068 3270 y Ft(=)28 b Fo(\025)1229 3229 y Fn(0)1229 3295 y Fu(1)1268 3270 y Ft(\()p Fo( )1369 3285 y Fu(2)1409 3270 y Ft(\))17 b(cos)g Fo( )1674 3285 y Fu(1)1736 3270 y Ft(+)22 b Fo(\025)1891 3229 y Fn(0)1891 3295 y Fu(2)1930 3270 y Ft(\()p Fo( )2031 3285 y Fu(2)2071 3270 y Ft(\))17 b(sin)f Fo( )2325 3285 y Fu(1)2387 3270 y Ft(+)22 b Fo(\025)2542 3229 y Fn(0)2542 3295 y Fu(0)2581 3270 y Ft(\()p Fo( )2682 3285 y Fu(2)2722 3270 y Ft(\))27 b(=)h(0)p Fo(:)639 b Ft(\(21\))-118 3532 y(This)33 b(can)f(b)s(e)h(in)m(terpreted)h(as)e(a)h(linear)e(system)i (for)f(cos)17 b Fo( )2072 3547 y Fu(1)2112 3532 y Ft(,)33 b(sin)16 b Fo( )2371 3547 y Fu(1)2411 3532 y Ft(,)33 b(whose)g(determinan)m(t)g(is)1063 3789 y(\001\()p Fo( )1245 3804 y Fu(2)1285 3789 y Ft(\))28 b(=)f Fo(\025)1511 3804 y Fu(1)1550 3789 y Fo(\025)1607 3748 y Fn(0)1607 3814 y Fu(1)1669 3789 y Ft(+)22 b Fo(\025)1824 3804 y Fu(2)1863 3789 y Fo(\025)1920 3748 y Fn(0)1920 3814 y Fu(2)1987 3789 y Ft(=)2101 3722 y(1)p 2101 3766 49 4 v 2101 3857 a(2)2159 3789 y(\()p Fo(\025)2254 3748 y Fu(2)2254 3814 y(1)2316 3789 y Ft(+)g Fo(\025)2471 3748 y Fu(2)2471 3814 y(2)2510 3789 y Ft(\))2548 3748 y Fn(0)2571 3789 y Fo(:)-118 4038 y Ft(If)32 b(a)h(critical)d(p)s(oin)m(t)i(\()p Fo( )742 4053 y Fu(1)781 4038 y Fo(;)17 b( )888 4053 y Fu(2)928 4038 y Ft(\))32 b(satis\014es)i(\001\()p Fo( )1540 4053 y Fu(2)1580 4038 y Ft(\))27 b Fi(6)p Ft(=)h(0,)k(solving)g(the)h (linear)e(system)i(\(20{21\))f(w)m(e)h(get)768 4309 y(cos)17 b Fo( )978 4324 y Fu(1)1045 4309 y Ft(=)28 b Fi(\000)1393 4241 y Fo(\025)1450 4205 y Fn(0)1450 4266 y Fu(0)1489 4241 y Fo(\025)1546 4256 y Fu(1)p 1236 4285 506 4 v 1236 4377 a Fo(\025)1293 4392 y Fu(1)1333 4377 y Fo(\025)1390 4343 y Fn(0)1390 4399 y Fu(1)1451 4377 y Ft(+)22 b Fo(\025)1606 4392 y Fu(2)1646 4377 y Fo(\025)1703 4343 y Fn(0)1703 4399 y Fu(2)1752 4309 y Fo(;)114 b Ft(sin)17 b Fo( )2093 4324 y Fu(1)2160 4309 y Ft(=)28 b Fi(\000)2508 4241 y Fo(\025)2565 4205 y Fn(0)2565 4266 y Fu(0)2604 4241 y Fo(\025)2661 4256 y Fu(2)p 2351 4285 V 2351 4377 a Fo(\025)2408 4392 y Fu(1)2447 4377 y Fo(\025)2504 4343 y Fn(0)2504 4399 y Fu(1)2566 4377 y Ft(+)22 b Fo(\025)2721 4392 y Fu(2)2760 4377 y Fo(\025)2817 4343 y Fn(0)2817 4399 y Fu(2)2867 4309 y Fo(;)712 b Ft(\(22\))-118 4587 y(and)48 b(the)g(equalit)m(y)f(cos)787 4551 y Fu(2)843 4587 y Fo( )906 4602 y Fu(1)978 4587 y Ft(+)33 b(sin)1206 4547 y Fu(2)1263 4587 y Fo( )1326 4602 y Fu(1)1419 4587 y Ft(=)53 b(1)48 b(implies)d(that)j Fo( )2281 4602 y Fu(2)2368 4587 y Ft(m)m(ust)g(b)s(e)g(a)g(zero)g(of)f(the)h(follo)m(wing)-118 4707 y(trigonometric)30 b(p)s(olynomial:)876 4925 y Fo(g)t Ft(\()p Fo( )1028 4940 y Fu(2)1067 4925 y Ft(\))d(=)h(\()p Fo(\025)1331 4940 y Fu(1)1370 4925 y Fo(\025)1427 4884 y Fn(0)1427 4950 y Fu(1)1489 4925 y Ft(+)22 b Fo(\025)1644 4940 y Fu(2)1683 4925 y Fo(\025)1740 4884 y Fn(0)1740 4950 y Fu(2)1779 4925 y Ft(\))1817 4884 y Fu(2)1879 4925 y Fi(\000)g Ft(\()p Fo(\025)2073 4884 y Fn(0)2073 4950 y Fu(0)2113 4925 y Ft(\))2151 4884 y Fu(2)2190 4925 y Ft(\()p Fo(\025)2285 4884 y Fu(2)2285 4950 y(1)2346 4925 y Ft(+)g Fo(\025)2501 4884 y Fu(2)2501 4950 y(2)2541 4925 y Ft(\))27 b(=)h(0)p Fo(:)820 b Ft(\(23\))-118 5143 y(One)30 b(can)g(easily)e(c)m(hec)m(k)k(that)e Fo(\025)1052 5107 y Fu(2)1052 5168 y(1)1107 5143 y Ft(+)16 b Fo(\025)1256 5107 y Fu(2)1256 5168 y(2)1325 5143 y Ft(is)29 b(a)g(trigonometric)e(p) s(olynomial)f(in)j Fo( )2777 5158 y Fu(2)2846 5143 y Ft(of)g(degree)i Fi(\024)d Ft(2)h(\(indeed,)-118 5263 y(one)45 b(can)g(use)h(\(18{19\))d(and)i(c)m(hec)m(k)i(that)e(all)e (the)i(terms)g(of)f(degrees)i(3)f(and)g(4)f(cancel\).)80 b(T)-8 b(aking)45 b(a)-118 5384 y(deriv)-5 b(ativ)m(e)28 b(it)f(is)g(clear)h(that)g(the)g(same)g(is)g(true)g(for)g Fo(\025)1847 5399 y Fu(1)1886 5384 y Fo(\025)1943 5347 y Fn(0)1943 5408 y Fu(1)1995 5384 y Ft(+)13 b Fo(\025)2141 5399 y Fu(2)2180 5384 y Fo(\025)2237 5347 y Fn(0)2237 5408 y Fu(2)2277 5384 y Ft(.)42 b(Then,)30 b(w)m(e)f(see)g(from)e (\(23\))h(that)g Fo(g)t Ft(\()p Fo( )3703 5399 y Fu(2)3742 5384 y Ft(\))-118 5504 y(is)39 b(a)g(trigonometric)d(p)s(olynomial)g (of)i(degree)j Fi(\024)e Ft(4,)i(whic)m(h)e(giv)m(es)h(at)f(most)f(8)h (p)s(ossible)g(v)-5 b(alues)39 b(for)g Fo( )3740 5519 y Fu(2)1782 5753 y Ft(16)p eop %%Page: 17 17 17 16 bop -118 407 a Ft(\(unless)32 b Fo(g)t Ft(\()p Fo( )359 422 y Fu(2)398 407 y Ft(\))f(is)f(iden)m(tically)f(zero;)j (see)h(b)s(elo)m(w\).)42 b(Replacing)30 b(eac)m(h)i(v)-5 b(alue)31 b(of)f Fo( )2864 422 y Fu(2)2935 407 y Ft(in)m(to)g(\(22\),)h (one)h(single)-118 527 y(v)-5 b(alue)39 b(of)h Fo( )323 542 y Fu(1)402 527 y Ft(is)g(determined.)65 b(This)40 b(implies)d(that)j(the)g(function)2451 495 y Fh(e)2429 527 y Fo(R)h Ft(has)f(at)f(most)h(8)f(critical)f(p)s(oin)m(ts)-118 648 y(with)32 b(\001\()p Fo( )286 663 y Fu(2)326 648 y Ft(\))c Fi(6)p Ft(=)f(0.)28 768 y(Of)j(course,)i(additional)27 b(critical)h(p)s(oin)m(ts)i(\()p Fo( )1661 783 y Fu(1)1701 768 y Fo(;)17 b( )1808 783 y Fu(2)1847 768 y Ft(\))31 b(with)e(\001\()p Fo( )2317 783 y Fu(2)2358 768 y Ft(\))e(=)h(0)i(ma)m (y)g(o)s(ccur.)43 b(But)30 b(in)g(this)g(case)-118 888 y(w)m(e)41 b(m)m(ust)f(ha)m(v)m(e)i(in)d(\(20{21\))g(that)h Fo(\025)1272 903 y Fu(1)1312 888 y Ft(\()p Fo( )1413 903 y Fu(2)1452 888 y Ft(\))h(=)f Fo(\025)1704 903 y Fu(2)1744 888 y Ft(\()p Fo( )1845 903 y Fu(2)1885 888 y Ft(\))g(=)h(0)f(or)f Fo(\025)2352 852 y Fn(0)2352 913 y Fu(0)2392 888 y Ft(\()p Fo( )2493 903 y Fu(2)2533 888 y Ft(\))h(=)h(0.)66 b(In)40 b(b)s(oth)g(sub)s(cases,)k(w)m(e)-118 1009 y(see)35 b(that)f(\(23\))g(is)g(also)f(satis\014ed,)i(no)m(w)g (with)f Fo( )1655 1024 y Fu(2)1729 1009 y Ft(as)g(a)g(double)g(zero.)49 b(Replacing)33 b Fo( )3014 1024 y Fu(2)3088 1009 y Ft(in)m(to)g (\(20{21\),)g(w)m(e)-118 1129 y(obtain)h(at)h(most)g(t)m(w)m(o)h(v)-5 b(alues)35 b(for)g Fo( )1247 1144 y Fu(1)1322 1129 y Ft(\(unless)h Fo(\025)1708 1144 y Fu(1)1747 1129 y Ft(,)g Fo(\025)1867 1093 y Fn(0)1867 1154 y Fu(1)1906 1129 y Ft(,)g Fo(\025)2026 1144 y Fu(2)2065 1129 y Ft(,)g Fo(\025)2185 1093 y Fn(0)2185 1154 y Fu(2)2260 1129 y Ft(sim)m(ultaneously)e(v)-5 b(anish)35 b(at)g Fo( )3414 1144 y Fu(2)3454 1129 y Ft(,)h(but)f(in) -118 1249 y(this)f(case)i(one)f(has)g(a)f(line)f(of)h(critical)e(p)s (oin)m(ts\).)50 b(Then,)36 b(ev)m(en)g(if)e(\001\()p Fo( )2507 1264 y Fu(2)2547 1249 y Ft(\))d(=)g(0)j(for)g(some)h(of)f (the)h(critical)-118 1370 y(p)s(oin)m(ts,)d(w)m(e)i(deduce)g(that)e (the)h(total)e(n)m(um)m(b)s(er)i(of)f(critical)f(p)s(oin)m(ts)h(for) 2539 1338 y Fh(e)2517 1370 y Fo(R)h Ft(cannot)g(b)s(e)g(larger)e(than)i (8.)28 1490 y(In)g(the)f(preceding)g(discussion,)h(one)f(should)g (consider)g(the)h(p)s(ossibilit)m(y)d(that)h Fo(g)t Ft(\()p Fo( )3099 1505 y Fu(2)3138 1490 y Ft(\))h(b)s(e)h(iden)m(tically)-118 1611 y(zero.)57 b(W)-8 b(e)38 b(are)f(going)e(to)i(sho)m(w)h(that,)g (if)e(this)h(happ)s(ens,)i(then)f(there)g(app)s(ears)f(a)g(curv)m(e)h (formed)f(with)-118 1731 y(critical)g(p)s(oin)m(ts.)65 b(Assuming)39 b Fo(g)t Ft(\()p Fo( )1174 1746 y Fu(2)1213 1731 y Ft(\))g Fi(\021)i Ft(0,)g(if)d(\001\()p Fo( )1803 1746 y Fu(2)1844 1731 y Ft(\))h Fi(6)p Ft(=)h(0)f(in)g(some)h(in)m (terv)-5 b(al,)40 b(w)m(e)g(deduce)i(from)c(\(22\))-118 1851 y(that)c(for)g(eac)m(h)i Fo( )531 1866 y Fu(2)605 1851 y Ft(in)e(this)g(in)m(terv)-5 b(al)34 b(one)h(v)-5 b(alue)34 b(of)g Fo( )1879 1866 y Fu(1)1953 1851 y Ft(is)g(determined,) i(obtaining)c(a)j(curv)m(e)h(of)e(critical)-118 1972 y(p)s(oin)m(ts.)66 b(Otherwise,)43 b(if)38 b Fo(g)t Ft(\()p Fo( )982 1987 y Fu(2)1022 1972 y Ft(\))i Fi(\021)h Ft(0)f(and)g(\001\() p Fo( )1686 1987 y Fu(2)1726 1972 y Ft(\))h Fi(\021)g Ft(0,)h(w)m(e)f(deduce)h(that)e Fo(\025)2801 1987 y Fu(1)2840 1972 y Ft(\()p Fo( )2941 1987 y Fu(2)2981 1972 y Ft(\))g(=)h Fo(\025)3233 1987 y Fu(2)3272 1972 y Ft(\()p Fo( )3373 1987 y Fu(2)3413 1972 y Ft(\))f Fi(\021)h Ft(0)f(in)-118 2092 y(some)g(in)m(terv)-5 b(al,)41 b(or)f Fo(\025)707 2056 y Fn(0)707 2117 y Fu(0)746 2092 y Ft(\()p Fo( )847 2107 y Fu(2)887 2092 y Ft(\))g Fi(\021)h Ft(0)f(in)g(some)g(in)m(terv) -5 b(al.)65 b(In)40 b(the)h(\014rst)f(case)h(w)m(e)h(see)f(from)e (\(18{19\))g(that)-118 2213 y Fo(s)-72 2228 y Fu(1)-5 2213 y Ft(=)27 b Fo(s)144 2228 y Fu(3)212 2213 y Ft(=)g Fo(s)361 2228 y Fu(4)428 2213 y Ft(=)h(0,)j(and)g(clearly)g(there)h (are)f(lines)f(of)h(critical)e(p)s(oin)m(ts.)43 b(In)31 b(the)h(second)h(case,)f(w)m(e)h(obtain)-118 2333 y(from)40 b(\(20{21\))g(that)h(for)f(eac)m(h)j Fo( )1151 2348 y Fu(2)1232 2333 y Ft(t)m(w)m(o)f(v)-5 b(alues)41 b(of)f Fo( )1906 2348 y Fu(1)1987 2333 y Ft(are)i(determined,)h(giving)d(rise) h(to)g(t)m(w)m(o)g(curv)m(es)-118 2453 y(of)h(critical)e(p)s(oin)m(ts.) 73 b(Th)m(us,)46 b(if)c Fo(g)t Ft(\()p Fo( )1237 2468 y Fu(2)1276 2453 y Ft(\))i Fi(\021)h Ft(0)d(there)i(are)e(alw)m(a)m(ys) h(curv)m(es)i(of)d(critical)e(p)s(oin)m(ts,)k(and)f(this)-118 2574 y(con)m(tradicts)27 b(the)h(assumption)f(that)1281 2541 y Fh(e)1258 2574 y Fo(R)h Ft(is)f(a)g(Morse)h(function)f(\(for)g (suc)m(h)i(a)e(function,)h(all)d(critical)g(p)s(oin)m(ts)-118 2694 y(m)m(ust)33 b(b)s(e)f(isolated\).)28 2814 y(W)-8 b(e)39 b(deduce)g(from)e(these)i(argumen)m(ts)g(that)f(the)g(n)m(um)m (b)s(er)g(of)g(critical)e(p)s(oin)m(ts)h(of)3140 2782 y Fh(e)3117 2814 y Fo(R)i Ft(m)m(ust)f(b)s(e)h(4,)g(6)-118 2935 y(or)29 b(8.)43 b(After)30 b(the)g(linear)e(c)m(hange)j(\(16\),)f (the)g(n)m(um)m(b)s(er)g(of)f(critical)f(p)s(oin)m(ts)h(of)h Fo(R)g Ft(is)g(doubled:)42 b(8,)30 b(12)f(or)h(16.)p -118 3055 42 42 v 28 3176 a(In)j(an)g(analogous)e(w)m(a)m(y)j(to)e (Theorem)h(1,)f(one)h(can)g(study)h(the)f(case)g(when,)h(for)e(an)m(y)h Fo(c)p Ft(,)-118 3346 y(\()p Fk(LL)p Ft(\))228 b Fo(\013)q Ft(\(0)p Fo(;)17 b(c)p Ft(\))27 b(=)761 3307 y Fm(d)p 744 3323 69 4 v 744 3380 a(d")823 3246 y Fh(\014)823 3296 y(\014)823 3346 y(\014)851 3400 y Fm(")p Fu(=0)994 3346 y Ft(\()p Fo("\013)q Ft(\()p Fo(";)17 b(c)p Ft(\)\))26 b Fi(\021)j Ft(0)1600 3310 y Fu(2)-118 3533 y Ft(\(see)42 b(\(10\)\).)66 b(This)41 b(case)h(is)e(reduced)i(to)e(the)h(preceding)g (one,)i(as)e(terms)f(con)m(taining)g Fo(\013)3232 3548 y Fm(i)3260 3533 y Ft(\(0)p Fo(;)17 b(c)p Ft(\))40 b(v)-5 b(anish)-118 3675 y(and,)33 b(after)f(dividing)f(b)m(y)841 3572 y Fh(q)p 924 3572 181 4 v 103 x Fo(I)975 3641 y Fu(0)967 3697 y(1)1014 3675 y Fo(I)1065 3641 y Fu(0)1057 3697 y(2)1105 3675 y Ft(,)h(w)m(e)i(get)f(the)g(same)f(system)i (\(13\).)28 3845 y(No)m(w)h(consider)f(a)f(generic)h(one-parameter)f (unfolding.)45 b(W)-8 b(e)34 b(will)e(sho)m(w)j(here)f(that)g(the)g(p)s (erturb)s(ed)-118 3966 y(system)42 b(has)g(four)f(transv)m(erse)j(\(on) d(the)h(related)f(lev)m(el)g(of)f(the)i(Hamiltonian\))c(homo)s(clinic)g (orbits)j(to)-118 4086 y(ev)m(ery)c(p)s(erturb)s(ed)e(p)s(ersisten)m(t) h(Diophan)m(tine)e(tori)g Fo(I)1834 4101 y Fu(1)1905 4086 y Ft(=)e Fo(I)2064 4050 y Fu(0)2056 4111 y(1)2103 4086 y Ft(,)j Fo(I)2208 4101 y Fu(2)2280 4086 y Ft(=)c Fo(I)2438 4050 y Fu(0)2430 4111 y(2)2513 4086 y Ft(on)k(the)g(cen)m (ter)h(manifold)d Fo(W)3626 4050 y Fm(c)3660 4086 y Ft(,)j(if)-118 4207 y(\()p Fo(I)-29 4170 y Fu(0)-37 4231 y(1)10 4207 y Fo(;)17 b(I)105 4170 y Fu(0)97 4231 y(2)144 4207 y Ft(\))33 b(b)s(elong)e(to)h(some)h(set)g(in)f(the)h(plane)f(\()p Fo(I)1665 4222 y Fu(1)1704 4207 y Fo(;)17 b(I)1791 4222 y Fu(2)1831 4207 y Ft(\).)-118 4435 y Fk(Theorem)37 b(2)49 b Fj(Ther)-5 b(e)31 b(ar)-5 b(e)31 b Fo(")947 4450 y Fu(0)1014 4435 y Fo(>)c Ft(0)32 b Fj(and)f(some)g(op)-5 b(en)31 b(set)h Fo(D)i Fj(in)d(the)h(quadr)-5 b(ant)31 b Fo(I)2839 4450 y Fu(1)2907 4435 y Fo(>)c Ft(0)p Fo(;)17 b(I)3146 4450 y Fu(2)3213 4435 y Fo(>)27 b Ft(0)32 b Fj(such)f(that)-118 4555 y(for)k Ft(\()p Fo(I)127 4519 y Fu(0)119 4580 y(1)166 4555 y Fo(;)17 b(I)261 4519 y Fu(0)253 4580 y(2)300 4555 y Ft(\))29 b Fi(2)h Fo(D)38 b Fj(a)d(p)-5 b(ersistent)36 b(Diophantine)e(torus)i Fo(I)1957 4570 y Fu(1)2025 4555 y Ft(=)29 b Fo(I)2181 4519 y Fu(0)2173 4580 y(1)2220 4555 y Fo(;)17 b(I)2307 4570 y Fu(2)2375 4555 y Ft(=)29 b Fo(I)2531 4519 y Fu(0)2523 4580 y(2)2606 4555 y Fj(has)35 b(exactly)h(4)f(tr)-5 b(ansverse)35 b(\(in)-118 4676 y(the)g(r)-5 b(elate)g(d)34 b(level)h(of)f(Hamiltonian\))g(homo)-5 b(clinic)33 b(orbits)i(to)g (this)g(torus.)-118 4904 y Fk(Pro)s(of.)47 b Ft(T)-8 b(o)34 b(\014nd)g(homo)s(clinic)d(orbits)i(to)h(a)f(p)s(ersisten)m(t)i (in)m(v)-5 b(arian)m(t)32 b(torus,)j(giv)m(en)f(b)m(y)h Fo(I)3118 4919 y Fu(1)3187 4904 y Ft(=)30 b Fo(I)3344 4868 y Fu(0)3336 4929 y(1)3383 4904 y Ft(,)k Fo(I)3487 4919 y Fu(2)3556 4904 y Ft(=)c Fo(I)3713 4868 y Fu(0)3705 4929 y(2)3752 4904 y Ft(,)-118 5024 y(w)m(e)g(need)g(to)f(\014nd)g(in)m (tersections)h(b)s(et)m(w)m(een)h(its)d(stable)h(and)g(unstable)h (manifolds)c(or)j(equiv)-5 b(alen)m(tly)d(,)29 b(as)g(in)-118 5145 y(Theorem)k(1,)f(solutions)g(of)g(system)h(\(12\).)p -118 5238 1560 4 v -6 5299 a Fd(2)31 5329 y Fv(The)c(sense)e(of)h(this) h(iden)n(tit)n(y)f(is)g(that)h(one)f(considers)f(an)g(one-parameter)f (unfolding)j(of)f Fb(H)2953 5341 y Fd(0)3018 5329 y Fv(suc)n(h)g(that)h (the)f(related)-118 5429 y(path)g(in)f(the)h(space)f(of)h(all)f (Hamiltonians)g(is)h(tangen)n(t)f(at)g(the)h(p)r(oin)n(t)g Fb(H)2173 5441 y Fd(0)2238 5429 y Fv(to)g Fc(L)p Fv(.)1782 5753 y Ft(17)p eop %%Page: 18 18 18 17 bop 28 417 a Ft(Dividing)38 b(the)j(\014rst)g(equation)g(of)f (system)i(\(12\))e(b)m(y)2033 314 y Fh(q)p 2116 314 140 4 v 103 x Ft(2)p Fo(I)2216 382 y Fu(0)2208 438 y(1)2255 417 y Ft(,)j(and)e(the)g(second)h(one)f(b)m(y)3352 314 y Fh(q)p 3435 314 V 103 x Ft(2)p Fo(I)3535 382 y Fu(0)3527 438 y(2)3574 417 y Ft(,)i(the)-118 537 y(system)33 b(casts)h(in)m(to)e (the)h(form)707 757 y Fo(\013)769 772 y Fu(1)809 757 y Ft(\(0)p Fo(;)17 b Ft(0\))g(cos)f Fo(\022)1235 772 y Fu(1)1297 757 y Ft(+)22 b Fo(\013)1457 772 y Fu(3)1496 757 y Ft(\(0)p Fo(;)17 b Ft(0\))g(sin)e Fo(\022)1911 772 y Fu(1)1973 757 y Ft(+)22 b Fo(O)2165 661 y Fh(\020)2215 757 y Fi(k)p Fo(I)2316 716 y Fu(0)2355 757 y Fi(k)2405 716 y Fu(1)p Fm(=)p Fu(2)2515 661 y Fh(\021)2592 757 y Ft(=)28 b(0)p Fo(;)724 940 y(\013)786 955 y Fu(2)825 940 y Ft(\(0)p Fo(;)17 b Ft(0\))g(cos)f Fo(\022)1251 955 y Fu(2)1313 940 y Ft(+)22 b Fo(\013)1473 955 y Fu(4)1513 940 y Ft(\(0)p Fo(;)17 b Ft(0\))g(sin)e Fo(\022)1928 955 y Fu(2)1990 940 y Ft(+)22 b Fo(O)2182 843 y Fh(\020)2231 940 y Fi(k)p Fo(I)2332 899 y Fu(0)2372 940 y Fi(k)2422 899 y Fu(1)p Fm(=)p Fu(2)2531 843 y Fh(\021)2609 940 y Ft(=)27 b(0)p Fo(:)818 b Ft(\(24\))-118 1176 y(T)-8 b(ending)33 b(in)e(\(24\))h(to)h(the)g(limit)c Fo(I)1150 1140 y Fu(0)1142 1201 y(1)1217 1176 y Fi(!)e Ft(0,)32 b Fo(I)1503 1140 y Fu(0)1495 1201 y(2)1570 1176 y Fi(!)27 b Ft(0,)33 b(w)m(e)g(obtain)f(the)h(system)1014 1396 y Fo(\013)1076 1411 y Fu(1)1115 1396 y Ft(\(0)p Fo(;)17 b Ft(0\))g(cos)f Fo(\022)1541 1411 y Fu(1)1603 1396 y Ft(+)22 b Fo(\013)1763 1411 y Fu(3)1803 1396 y Ft(\(0)p Fo(;)17 b Ft(0\))g(sin)e Fo(\022)2218 1411 y Fu(1)2285 1396 y Ft(=)28 b(0)p Fo(;)1058 1542 y(\013)1120 1557 y Fu(2)1159 1542 y Ft(\(0)p Fo(;)17 b Ft(0\))g(cos)f Fo(\022)1585 1557 y Fu(2)1647 1542 y Ft(+)22 b Fo(\013)1807 1557 y Fu(4)1847 1542 y Ft(\(0)p Fo(;)17 b Ft(0\))g(sin)e Fo(\022)2262 1557 y Fu(2)2329 1542 y Ft(=)28 b(0)-118 1762 y(\(notice)37 b(that,)j(as)e(in)f(the)h(pro)s(of)f(of)h(Theorem)g (1,)h(this)f(system)h(can)f(b)s(e)g(put)g(in)f(gradien)m(t)h(form\).)58 b(W)-8 b(e)-118 1882 y(assume)33 b(\(it)e(is)i(the)g(condition)e(of)h (the)h(general)f(p)s(osition\))f(that)-118 2052 y(\()p Fk(LLL)p Ft(\))228 b Fo(\013)449 2067 y Fu(1)489 2052 y Ft(\(0)p Fo(;)17 b Ft(0\))707 2016 y Fu(2)768 2052 y Ft(+)22 b Fo(\013)928 2067 y Fu(3)967 2052 y Ft(\(0)p Fo(;)17 b Ft(0\))1185 2016 y Fu(2)1251 2052 y Fi(6)p Ft(=)28 b(0)p Fo(;)114 b(\013)1607 2067 y Fu(2)1646 2052 y Ft(\(0)p Fo(;)17 b Ft(0\))1864 2016 y Fu(2)1925 2052 y Ft(+)22 b Fo(\013)2085 2067 y Fu(4)2125 2052 y Ft(\(0)p Fo(;)17 b Ft(0\))2343 2016 y Fu(2)2409 2052 y Fi(6)p Ft(=)27 b(0.)-118 2222 y(Then,)36 b(the)e(system)i(has)e(four)g (solutions)f Fo(p)1499 2186 y Fm(k)1573 2222 y Ft(=)d(\()p Fo(\022)1765 2186 y Fm(k)1762 2247 y Fu(1)1808 2222 y Fo(;)17 b(\022)1900 2186 y Fm(k)1897 2247 y Fu(2)1942 2222 y Ft(\),)35 b Fo(k)f Ft(=)p 2233 2148 142 4 v 30 w(1)p Fo(;)17 b Ft(4)33 b(\(w)m(e)j(stress)f(that,)g(in)e(this)h(case,) i(the)-118 2343 y(solutions)h(do)g(not)h(dep)s(end)g(on)g(the)g(ra)m(y) g(to)g(whic)m(h)g(\()p Fo(I)1942 2307 y Fu(0)1934 2367 y(1)1981 2343 y Fo(;)17 b(I)2076 2307 y Fu(0)2068 2367 y(2)2115 2343 y Ft(\))38 b(b)s(elongs\).)58 b(Due)37 b(to)h(\()p Fk(LLL)p Ft(\))g(the)g(implicit)-118 2463 y(function)c(theorem)h(is)g(applicable)e(to)i(\(24\),)g(and)g(giv)m(es) h(four)e(solutions)g Fo(p)2668 2427 y Fm(k)2743 2463 y Ft(=)e(\()p Fo(\022)2937 2427 y Fm(k)2934 2488 y Fu(1)2980 2463 y Ft(\()p Fo(I)3069 2427 y Fu(0)3061 2488 y(1)3108 2463 y Fo(;)17 b(I)3203 2427 y Fu(0)3195 2488 y(2)3242 2463 y Ft(\))p Fo(;)g(\022)3372 2427 y Fm(k)3369 2488 y Fu(2)3414 2463 y Ft(\()p Fo(I)3503 2427 y Fu(0)3495 2488 y(1)3543 2463 y Fo(;)g(I)3638 2427 y Fu(0)3630 2488 y(2)3677 2463 y Ft(\)\),)-118 2584 y Fo(k)31 b Ft(=)p 67 2509 V 27 w(1)p Fo(;)17 b Ft(4)o(.)42 b(In)27 b(fact,)h(only)e(a)h (p)s(ositiv)m(e)f(measure)i(Can)m(torian)e(set)h(of)g(tori)e(p)s (ersists,)k(but)e(the)g(implicit)c(func-)-118 2704 y(tion)31 b(theorem)h(w)m(orks)i(here)f(uniformly)e(in)g Fo(I)41 b Ft(out)32 b(of)g(some)g(neigh)m(b)s(orho)s(o)s(d)f(of)h(the)h(former) f(homo)s(clinic)-118 2824 y(orbit.)p 158 2824 42 42 v -118 3157 a Fp(5)161 b(The)53 b(Melnik)l(o)l(v)h(p)t(oten)l(tial)-118 3376 y Ft(The)31 b(metho)s(d)f(of)g(detecting)g(homo)s(clinic)e(orbits) h(to)h(in)m(v)-5 b(arian)m(t)29 b(tori)g(presen)m(ted)k(in)d(previous)g (section)h(is,)-118 3497 y(in)j(a)h(sense,)i(blind:)47 b(it)34 b(is)h(rather)g(hard)g(to)f(apply)h(it)f(to)h(concrete)h (Hamiltonians)c(and)j(p)s(erturbations.)-118 3617 y(No)m(w,)42 b(w)m(e)f(are)f(in)m(terested)h(in)f(detecting)g(the)g(e\013ectiv)m(e)h (existence)h(of)d(transv)m(erse)j(homo)s(clinic)37 b(orbits)-118 3737 y(for)32 b(concrete)j(p)s(erturbations)d Fo("H)1154 3752 y Fu(1)1193 3737 y Ft(,)h(as)g(w)m(ell)g(as)g(in)f(obtaining)f(a)i (\014rst)g(appro)m(ximation)e(for)h(\(the)i(traces)-118 3858 y(in)i Fo(N)88 3822 y Fm(s)163 3858 y Ft(of)7 b(\))37 b(these)h(orbits.)57 b(The)38 b(standard)g(pro)s(cedure)g(for)f(this)g (purp)s(ose)h(is)e(the)i(P)m(oincar)m(\023)-46 b(e{Melnik)m(o)m(v)-118 3978 y(metho)s(d,)29 b(whic)m(h)g(pro)m(vides)h(the)f(Melnik)m(o)m(v)g (function)g(as)g(an)f(appro)m(ximation)f(to)h(the)i(splitting)c (distance)-118 4098 y(\(i.e.)32 b(the)h(distance)g(b)s(et)m(w)m(een)i (the)e(stable)f(and)h(unstable)f(manifolds)f(of)h(a)g(giv)m(en)h (torus\).)28 4219 y(The)24 b(transv)m(erse)g(homo)s(clinic)19 b(orbits)j(are)h(usually)e(appro)m(ximated)h(as)h(simple)d(zero)s(es)k (of)e(the)h(\(v)m(ector\))-118 4339 y(Melnik)m(o)m(v)k(function)f(but,) i(as)e(in)g([4],)h(the)g(Hamiltonian)c(c)m(haracter)k(of)f(the)h (equations)f(allo)m(ws)g(us)h(to)f(lo)s(ok)-118 4460 y(for)39 b(nondegenerate)j(critical)37 b(p)s(oin)m(ts)j(of)f(the)i (\(scalar\))e(Melnik)m(o)m(v)i(p)s(oten)m(tial,)f(whose)h(gradien)m(t)f (is)f(the)-118 4580 y(Melnik)m(o)m(v)33 b(function.)28 4700 y(F)-8 b(or)24 b(the)i(sak)m(e)g(of)f(simplicit)m(y)-8 b(,)24 b(w)m(e)i(restrict)f(ourselv)m(es)h(to)f(the)h(case)g(of)e(an)h (unp)s(erturb)s(ed)h(Hamiltonian)-118 4821 y Fo(H)-37 4836 y Fu(0)35 4821 y Ft(as)32 b(in)g(the)h(examples)g(considered)g(in) f(Section)g(1.1,)g(plus)h(a)f(p)s(erturbation)g Fo("H)2960 4836 y Fu(1)2999 4821 y Ft(\()p Fo(x)3092 4836 y Fu(1)3131 4821 y Fo(;)17 b(y)3223 4836 y Fu(1)3262 4821 y Fo(;)g(x)3361 4836 y Fu(2)3401 4821 y Fo(;)g(y)3493 4836 y Fu(2)3531 4821 y Fo(;)g(p;)g(q)t Ft(\).)28 4941 y(W)-8 b(e)39 b(kno)m(w)g(from)f (Section)g(3)g(that)g(the)h(p)s(ersisten)m(t)g(tori)e(and)i(their)f(in) m(v)-5 b(arian)m(t)36 b(manifolds)g(b)s(ecome)-118 5062 y(transparen)m(t)k(in)e(the)h(lo)s(cal)e(KAM)i(co)s(ordinates)f(in)m (tro)s(duced)h(there.)64 b(Ho)m(w)m(ev)m(er,)42 b(since)e(the)f(Melnik) m(o)m(v)-118 5182 y(function)22 b(will)e(b)s(e)j(de\014ned)i(with)d (the)h(help)f(of)h(in)m(tegrals)e(along)h(tra)5 b(jectories)22 b(on)h(the)g(global)d(unp)s(erturb)s(ed)-118 5302 y(homo)s(clinic)f (manifold,)j(it)f(is)h(natural)g(to)g(use)h(the)g(global)d(original)g (co)s(ordinates)i Fo(z)32 b Ft(=)c(\()p Fo(x)3119 5317 y Fu(1)3158 5302 y Fo(;)17 b(y)3250 5317 y Fu(1)3289 5302 y Fo(;)g(x)3388 5317 y Fu(2)3428 5302 y Fo(;)g(y)3520 5317 y Fu(2)3558 5302 y Fo(;)g(p;)g(q)t Ft(\))-118 5423 y(in)42 b(order)h(to)g(deduce)i(an)e(expression)h(for)f(the)g(Melnik)m (o)m(v)h(function.)74 b(T)-8 b(o)43 b(b)s(e)g(more)g(precise,)j(w)m(e)e (are)1782 5753 y(18)p eop %%Page: 19 19 19 18 bop -118 407 a Ft(going)32 b(to)h(use)h(the)g(co)s(ordinates)f Fo(\020)k Ft(=)29 b(\()p Fo(\030)1386 422 y Fu(1)1425 407 y Fo(;)17 b(\030)1512 422 y Fu(2)1550 407 y Fo(;)g(')1658 422 y Fu(1)1697 407 y Fo(;)g(')1805 422 y Fu(2)1844 407 y Fo(;)g(p;)g(q)t Ft(\),)33 b(i.e.)g(w)m(e)i(c)m(hange)f(to)f(the)h(p)s (olar)e(co)s(ordinates)-118 527 y(in)m(tro)s(duced)h(in)e(\(5\).)28 648 y(The)k(reason)e(for)g(using)g(p)s(olar)f(co)s(ordinates)h(is)g (that)g(the)h(Melnik)m(o)m(v)g(function)f(will)e(pro)m(vide)j(a)f (\014rst)-118 768 y(appro)m(ximation)20 b(for)i(the)h(splitting)d (distance)j(along)e(the)i(actions)f Fo(\030)32 b Ft(=)27 b(\()p Fo(\030)2552 783 y Fu(1)2591 768 y Fo(;)17 b(\030)2678 783 y Fu(2)2717 768 y Ft(\),)24 b(whic)m(h)f(are)g(\014rst)g(in)m (tegrals)-118 888 y(of)g Fo(H)65 903 y Fu(0)104 888 y Ft(.)40 b(Notice)23 b(that)g(the)g(unp)s(erturb)s(ed)i(homo)s(clinic)20 b(manifold)g(of)j(a)g(torus)g(is)g(giv)m(en)g(b)m(y)h Fi(f)p Fo(\030)32 b Ft(=)c(const)p Fo(;)17 b(p)28 b Ft(=)-118 1009 y(0)p Fi(g)p Ft(,)39 b(and)f(it)f(will)e(b)s(e)j(enough)h(to)e (giv)m(e)h(a)g(measure)g(for)f(the)i(splitting)c(along)i(the)h Fo(\030)5 b Ft(-directions)36 b(due)j(to)-118 1129 y(the)33 b(conserv)-5 b(ation)32 b(of)g(energy)-8 b(.)28 1249 y(First,)30 b(w)m(e)i(describ)s(e)g(the)f(parameterizations)e(to)h(b)s (e)h(used.)44 b(W)-8 b(e)31 b(denote)h Fi(T)56 b Ft(a)31 b(p)s(ersisten)m(t)g(Diophan-)-118 1370 y(tine)22 b(torus)h(in)e Fo(W)516 1334 y Fm(c)551 1370 y Ft(,)j(with)e(actions)g Fo(I)1186 1334 y Fu(0)1253 1370 y Ft(=)28 b(\()p Fo(I)1446 1334 y Fu(0)1438 1394 y(1)1485 1370 y Fo(;)17 b(I)1580 1334 y Fu(0)1572 1394 y(2)1619 1370 y Ft(\).)40 b(This)22 b(torus)h(can)g(b)s(e)f(parameterized)g(b)m(y)i Fo(\022)31 b Ft(=)c(\()p Fo(\022)3479 1385 y Fu(1)3519 1370 y Fo(;)17 b(\022)3608 1385 y Fu(2)3648 1370 y Ft(\))27 b Fi(2)-118 1490 y Fe(T)-62 1454 y Fu(2)4 1490 y Ft(in)g(the)i(co)s(ordinates)e (pro)m(vided)h(b)m(y)h(KAM)f(theorem)f(in)g(Section)h(3.)41 b(Let)28 b Fi(Z)2770 1454 y Fn(\003)2810 1490 y Ft(\()p Fo(I)2899 1454 y Fu(0)2938 1490 y Fo(;)17 b(\022)s(;)g(")p Ft(\))27 b(b)s(e)h(the)g(param-)-118 1611 y(eterization)d(of)i(the)g (torus)g Fi(T)53 b Ft(in)26 b(the)h(co)s(ordinates)f Fo(\020)8 b Ft(,)27 b(and)g(let)f Fi(I)2244 1574 y Fn(\003)2284 1611 y Ft(\()p Fo(I)2373 1574 y Fu(0)2413 1611 y Fo(;)17 b(\022)s(;)g(")p Ft(\))27 b(=)g(\()p Fi(I)2862 1574 y Fn(\003)2855 1635 y Fu(1)2902 1611 y Ft(\()p Fo(I)2991 1574 y Fu(0)3030 1611 y Fo(;)17 b(\022)s(;)g(")p Ft(\))p Fo(;)g Fi(I)3355 1574 y Fn(\003)3348 1635 y Fu(2)3394 1611 y Ft(\()p Fo(I)3483 1574 y Fu(0)3522 1611 y Fo(;)g(\022)s(;)g(")p Ft(\)\))-118 1731 y(denote)33 b(its)f Fo(\030)5 b Ft(-comp)s(onen)m (ts:)43 b Fi(Z)1067 1695 y Fn(\003)1134 1731 y Ft(=)28 b(\()p Fi(I)1337 1695 y Fn(\003)1330 1756 y Fu(1)1377 1731 y Fo(;)17 b Fi(I)1482 1695 y Fn(\003)1475 1756 y Fu(2)1522 1731 y Fo(;)g(:)g(:)g(:)o Ft(\).)28 1851 y(Let)49 b(us)g(denote)h Fi(A)771 1815 y Fm(s)807 1851 y Ft(,)j Fi(A)967 1815 y Fm(u)1060 1851 y Ft(the)c(traces)g(of)f(the)h(torus)g Fi(T)74 b Ft(on)49 b(the)g(transv)m(erse)i(sections)e Fo(N)3529 1815 y Fm(s)3566 1851 y Ft(,)k Fo(N)3734 1815 y Fm(u)-118 1972 y Ft(\(i.e.)36 b(the)h(in)m(tersection)g(of)f(the)h (stable)f(and)h(unstable)g(manifolds)d(of)i Fi(T)62 b Ft(with)36 b Fo(N)2934 1936 y Fm(s)2972 1972 y Ft(,)h Fo(N)3124 1936 y Fm(u)3170 1972 y Ft(\).)55 b(Notice)37 b(that)-118 2092 y(these)k(traces)g(are)f(2-dimensional)d(tori.)64 b(Recall)39 b(that)h(the)g(transv)m(erse)i(sections)f(ha)m(v)m(e)g(b)s (een)g(de\014ned)-118 2213 y(in)e(\(2\))g(b)m(y)h(a)f(n)m(um)m(b)s(er)h Fo(d)p Ft(,)h(not)f(dep)s(ending)f(on)h Fo(")p Ft(.)64 b(This)39 b Fo(d)g Ft(can)h(b)s(e)g(c)m(hosen)h(small)c(enough)j(in)f (suc)m(h)i(a)-118 2333 y(w)m(a)m(y)31 b(that)f(the)g(traces)h(are)f (con)m(tained)g(in)f(the)i(domain)d(of)i(v)-5 b(alidit)m(y)28 b(of)h(the)i(KAM)f(results)g(of)g(Section)g(3.)-118 2453 y(W)-8 b(e)40 b(use)h(for)e(the)i(traces)f Fi(A)934 2417 y Fm(s;u)1071 2453 y Ft(the)g(same)g(parameters)g(as)g(for)g(the)g (torus)g Fi(T)26 b Ft(:)58 b(as)40 b(b)s(efore,)i(w)m(e)f(denote)-118 2574 y Fi(Z)-38 2538 y Fm(s;u)59 2574 y Ft(\()p Fo(I)148 2538 y Fu(0)187 2574 y Fo(;)17 b(\022)s(;)g(")p Ft(\))30 b(the)h(whole)g(parameterization,)e(and)i Fi(I)1906 2538 y Fm(s;u)2031 2574 y Ft(=)d(\()p Fi(I)2234 2526 y Fm(s;u)2227 2595 y Fu(1)2332 2574 y Fo(;)17 b Fi(I)2437 2526 y Fm(s;u)2430 2595 y Fu(2)2534 2574 y Ft(\))31 b(their)f Fo(\030)5 b Ft(-comp)s(onen)m(ts.)42 b(W)-8 b(e)31 b(see)-118 2694 y(from)k(KAM)h(Hamiltonian)c(equations)k(\(9\))g(that)f(the)i(angles)e Fo(\022)i Fi(2)d Fe(T)2475 2658 y Fu(2)2549 2694 y Ft(in)h(the)i (parameterizations)d Fi(Z)3713 2658 y Fn(\003)3752 2694 y Ft(,)-118 2814 y Fi(Z)-38 2778 y Fm(s)-1 2814 y Ft(,)k Fi(Z)144 2778 y Fm(u)225 2814 y Ft(can)f(b)s(e)g(tak)m(en)h(in)e(suc)m (h)i(a)e(w)m(a)m(y)i(that)f(the)g(tra)5 b(jectory)37 b(starting)f(at)g(the)h(p)s(oin)m(t)f Fi(Z)3293 2778 y Fm(s)3330 2814 y Ft(\()p 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Fo(I)96 3260 y Fu(0)135 3296 y Fo(;)17 b(\022)s(;)g(")p Ft(\))41 b(the)h (parameterization)d(of)i Fo(S)1515 3311 y Fm(")1552 3296 y Ft(\()p Fi(A)1670 3260 y Fm(u)1715 3296 y Ft(\))g(inherited)g(from)f Fi(A)2534 3260 y Fm(u)2579 3296 y Ft(,)k(i.e.)2845 3271 y(\026)2816 3296 y Fi(Z)2896 3260 y Fm(u)2982 3296 y Ft(is)e(the)g(p)s(oin)m(t)e(where)-118 3416 y(the)45 b(tra)5 b(jectory)46 b(starting)e(at)h Fi(Z)1118 3380 y Fm(u)1208 3416 y Ft(in)m(tersects)i Fo(N)1741 3380 y Fm(s)1778 3416 y Ft(.)81 b(As)46 b(b)s(efore,)i(w)m(e)e(denote)2870 3391 y(\026)2861 3416 y Fi(I)2922 3380 y Fm(u)3016 3416 y Ft(=)j(\()3188 3391 y(\026)3179 3416 y Fi(I)3240 3380 y Fm(u)3233 3441 y Fu(1)3286 3416 y Fo(;)3339 3391 y Ft(\026)3330 3416 y Fi(I)3391 3380 y Fm(u)3384 3441 y Fu(2)3436 3416 y Ft(\))c(the)h Fo(\030)5 b Ft(-)-118 3537 y(comp)s(onen)m(ts)36 b(of)f(the)h(p)s(oin)m(t)994 3512 y(\026)964 3537 y Fi(Z)1044 3501 y Fm(u)1089 3537 y Ft(.)52 b(As)36 b(a)f(measure)h(for)f(the)h(splitting)d(distance,)k (w)m(e)f(are)g(going)e(to)h(tak)m(e)-118 3657 y(the)i(distance)g(b)s (et)m(w)m(een)i(the)f(tori)d Fi(A)1260 3621 y Fm(s)1333 3657 y Ft(and)i Fo(S)1587 3672 y Fm(")1624 3657 y Ft(\()p Fi(A)1742 3621 y Fm(u)1787 3657 y Ft(\),)h(restricted)f(to)g(the)g Fo(\030)2667 3672 y Fm(i)2695 3657 y Ft(-directions.)55 b(Nev)m(ertheless,)-118 3778 y(the)40 b(phase)i(drift)d(that)h(o)s (ccurs)h(along)d(an)m(y)j(tra)5 b(jectory)41 b(has)f(to)g(b)s(e)h(tak)m (en)g(in)m(to)e(accoun)m(t)i(in)f(order)g(to)-118 3898 y(measure)33 b(this)f(distance)h(correctly)-8 b(.)28 4018 y(Since)37 b(the)g(tra)5 b(jectories)36 b(on)g(the)h(in)m(v)-5 b(arian)m(t)35 b(manifolds)f(of)i(the)h(torus)g Fi(T)62 b Ft(are)36 b(close)h(to)f(the)g(unp)s(er-)-118 4139 y(turb)s(ed)43 b(homo)s(clinic)c(tra)5 b(jectories,)45 b(w)m(e)f(lo)s(ok)d(at)h(the)h(case)g Fo(")h Ft(=)g(0)f(in)e(order)i (to)f(c)m(ho)s(ose)h(the)g(phases.)-118 4259 y(Recalling)29 b(the)j(function)f Fo(q)896 4274 y Fu(0)936 4259 y Ft(\()p Fo(t)p Ft(\))h(in)e(the)i(examples)g(giv)m(en)g(in)f(Section)g(1.1,)h (a)f(homo)s(clinic)e(tra)5 b(jectory)32 b(to)-118 4379 y Fi(T)58 b Ft(for)32 b Fo(H)224 4394 y Fu(0)296 4379 y Ft(is)g(giv)m(en)h(b)m(y)904 4560 y Fo(\020)947 4575 y Fu(0)986 4560 y Ft(\()p Fo(t)p Ft(\))27 b(=)h Fo(\020)1271 4575 y Fu(0)1310 4560 y Ft(\()p Fo(t;)17 b(I)1478 4519 y Fu(0)1517 4560 y Fo(;)g(\022)s Ft(\))28 b(=)f(\()p Fo(I)1867 4519 y Fu(0)1906 4560 y Fo(;)17 b(\022)25 b Ft(+)d Fo(t)8 b Ft(~)-57 b Fo(!)t(;)35 b Ft(_)-45 b Fo(q)2305 4575 y Fu(0)2344 4560 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(q)2542 4575 y Fu(0)2582 4560 y Ft(\()p Fo(t)p Ft(\)\))p Fo(;)848 b Ft(\(25\))-118 4741 y(where)34 b(w)m(e)f(ha)m(v)m(e)h (denoted)42 b(~)-57 b Fo(!)31 b Ft(=)c(\()8 b(~)-57 b Fo(!)1195 4756 y Fu(1)1234 4741 y Fo(;)25 b Ft(~)-57 b Fo(!)1339 4756 y Fu(2)1378 4741 y Ft(\),)33 b(with)-85 4972 y(~)-57 b Fo(!)-32 4987 y Fu(1)35 4972 y Ft(=)149 4904 y Fo(@)5 b(h)261 4919 y Fu(0)p 149 4948 153 4 v 156 5040 a Fo(@)g(\030)255 5055 y Fu(1)311 4972 y Ft(\()p Fo(I)400 4930 y Fu(0)392 4996 y(1)439 4972 y Fo(;)17 b(I)534 4930 y Fu(0)526 4996 y(2)573 4972 y Ft(\))28 b(=)g Fo(!)804 4987 y Fu(1)865 4972 y Ft(+)22 b(2)p Fo(AI)1136 4930 y Fu(0)1128 4996 y(1)1197 4972 y Ft(+)g(2)p Fo(B)5 b(I)1474 4930 y Fu(0)1466 4996 y(2)1536 4972 y Ft(+)22 b Fi(\001)17 b(\001)g(\001)d Fo(;)122 b Ft(~)-56 b Fo(!)1969 4987 y Fu(2)2036 4972 y Ft(=)2149 4904 y Fo(@)5 b(h)2261 4919 y Fu(0)p 2149 4948 V 2156 5040 a Fo(@)g(\030)2255 5055 y Fu(2)2312 4972 y Ft(\()p Fo(I)2401 4930 y Fu(0)2393 4996 y(1)2440 4972 y Fo(;)17 b(I)2535 4930 y Fu(0)2527 4996 y(2)2574 4972 y Ft(\))27 b(=)h Fo(!)2804 4987 y Fu(2)2865 4972 y Ft(+)22 b(2)p Fo(B)5 b(I)3142 4930 y Fu(0)3134 4996 y(1)3204 4972 y Ft(+)22 b(2)p Fo(C)7 b(I)3479 4930 y Fu(0)3471 4996 y(2)3540 4972 y Ft(+)22 b Fi(\001)17 b(\001)g(\001)-118 5203 y Ft(It)29 b(is)g(clear)g(that)g(tra)5 b(jectory)30 b(\(25\))f(is)g(asymptotic)f(for)h Fo(t)f Fi(!)f(\0061)p Ft(,)k(with)e(exp)s(onen)m(tial)f(estimates,)i(to)f(the) -118 5323 y(follo)m(wing)h(tra)5 b(jectory)33 b(on)f(the)h(unp)s (erturb)s(ed)h(torus:)1041 5504 y Fo(\020)1092 5463 y Fn(\003)1084 5529 y Fu(0)1131 5504 y Ft(\()p Fo(t)p Ft(\))27 b(=)h Fo(\020)1424 5463 y Fn(\003)1416 5529 y Fu(0)1462 5504 y Ft(\()p Fo(t;)17 b(I)1630 5463 y Fu(0)1670 5504 y Fo(;)g(\022)s Ft(\))27 b(=)h(\()p Fo(I)2020 5463 y Fu(0)2059 5504 y Fo(;)17 b(\022)25 b Ft(+)d Fo(t)8 b Ft(~)-57 b Fo(!)s(;)17 b Ft(0)p Fo(;)g Ft(0\))p Fo(:)1782 5753 y Ft(19)p eop %%Page: 20 20 20 19 bop -118 407 a Ft(Let)37 b Fo(T)118 422 y Fu(1)157 407 y Fo(;)17 b(T)258 422 y Fu(2)332 407 y Fo(>)34 b Ft(0)i(constan)m(ts)i(suc)m(h)g(that)e Fo(\020)1445 422 y Fu(0)1484 407 y Ft(\()p Fi(\000)p Fo(T)1656 422 y Fu(1)1696 407 y Ft(\))e Fi(2)h Fo(N)1957 371 y Fm(u)2002 407 y Ft(,)j Fo(\020)2110 422 y Fu(0)2149 407 y Ft(\()p Fo(T)2244 422 y Fu(2)2283 407 y Ft(\))d Fi(2)f Fo(N)2544 371 y Fm(s)2581 407 y Ft(.)56 b(W)-8 b(e)36 b(ha)m(v)m(e)i Fo(T)3121 422 y Fu(1)3161 407 y Fo(;)17 b(T)3262 422 y Fu(2)3335 407 y Fi(\030)35 b Ft(ln)o(\(1)p Fo(=d)p Ft(\),)-118 527 y(where)f Fo(d)e Ft(is)g(the)h(n)m(um)m(b)s(er)g(\(c)m (hosen)h(small)c(enough\))j(de\014ning)g(the)g(sections)g Fo(N)2836 491 y Fm(s)2873 527 y Ft(,)g Fo(N)3021 491 y Fm(u)3066 527 y Ft(.)-118 756 y Fk(Remark)k(3)49 b Fj(It)34 b(is)f(worth)h(str)-5 b(essing)34 b(the)g(fact)f(that)i(at)f Fo(")27 b Ft(=)h(0)p Fj(,)34 b(the)g(Hamiltonian)f Fo(H)3048 771 y Fu(0)3121 756 y Fj(in)h(the)g(examples)-118 876 y(c)-5 b(onsider)g(e)g(d)40 b(is)h(the)g(sum)g(of)g(two)h(indep)-5 b(endent)40 b(subHamiltonians)g Fo(h)2509 891 y Fu(0)2590 876 y Fj(and)g Fo(p)2834 840 y Fu(2)2874 876 y Fo(=)p Ft(2)26 b(+)h Fo(V)22 b Ft(\()p Fo(q)t Ft(\))41 b Fj(\(de)-5 b(c)g(ouple)g(d)-118 996 y(subsystems\).)59 b(In)40 b(this)f(c)-5 b(ase)40 b(al)5 b(l)39 b(homo)-5 b(clinic)39 b(orbits)g(to)h(a)g (\014xe)-5 b(d)39 b(torus)i(in)e Fo(W)2853 960 y Fm(c)2928 996 y Fj(p)-5 b(ossess)38 b(the)i(pr)-5 b(op)g(erty)-118 1117 y(of)39 b(the)g(same)f(asymptotics)g(\(the)h(same)g(orbit)g(on)f (the)h(torus\))h(as)e Fo(t)e Fi(!)f Ft(+)p Fi(1)j Fj(and)h Fo(t)c Fi(!)g(\0001)p Fj(.)57 b(F)-7 b(or)39 b(the)-118 1237 y(c)-5 b(ase)30 b(of)h(c)-5 b(ouple)g(d)30 b(inte)-5 b(gr)g(able)31 b(Hamiltonians)f(these)g(asymptotic)h(orbits)g(ar)-5 b(e)31 b(di\013er)-5 b(ent)30 b(generic)-5 b(al)5 b(ly,)31 b(and)-118 1357 y(this)38 b(has)f(to)h(b)-5 b(e)37 b(taken)h(into)g(ac) -5 b(c)g(ount)37 b(for)h(the)g(Melnikov)f(p)-5 b(otential,)38 b(which)f(b)-5 b(e)g(c)g(omes)37 b(the)g(sum)h(of)g(two)-118 1478 y(distinct)45 b(inte)-5 b(gr)g(als)44 b(for)h Fo(t)i(<)g Ft(0)d Fj(and)h Fo(t)i(>)f Ft(0)p Fj(.)75 b(In)45 b(fact,)i(the)e (Poinc)-5 b(ar)n(\023)-47 b(e{Melnikov)43 b(metho)-5 b(d)44 b(for)h(mor)-5 b(e)-118 1598 y(gener)g(al)36 b(unp)-5 b(erturb)g(e)g(d)37 b(Hamiltonians)f Fo(H)1435 1613 y Fu(0)1511 1598 y Fj(was)g(develop)-5 b(e)g(d)36 b(in)h([4)o(])g(intr)-5 b(o)g(ducing)36 b(an)h(alternative)f(single)-118 1719 y(inte)-5 b(gr)g(al)34 b(expr)-5 b(ession)34 b(for)h(the)f(Melnikov)h (p)-5 b(otential)34 b(\(a)h(pr)-5 b(evious)34 b(work)g(in)h(this)g(dir) -5 b(e)g(ction)34 b(was)g([15]\).)28 1947 y Ft(No)m(w,)g(w)m(e)g(can)f (de\014ne)h(the)g(follo)m(wing)c(\(v)m(ector\))k(function)e(as)h(an)g (exact)h(measure)f(for)f(the)i(splitting)-118 2067 y(distance:)687 2188 y Fi(M)p Ft(\()p Fo(I)896 2146 y Fu(0)935 2188 y Fo(;)17 b(\022)s(;)g(")p Ft(\))27 b(=)h(\()1333 2162 y(\026)1324 2188 y Fi(I)1385 2146 y Fm(u)1430 2188 y Ft(\()p Fo(I)1519 2146 y Fu(0)1558 2188 y Fo(;)17 b(\022)25 b Fi(\000)e Fo(T)1829 2203 y Fu(1)1876 2188 y Ft(~)-56 b Fo(!)s(;)17 b(")p Ft(\))k Fi(\000)i(I)2243 2146 y Fm(s)2280 2188 y Ft(\()p Fo(I)2369 2146 y Fu(0)2409 2188 y Fo(;)17 b(\022)25 b Ft(+)d Fo(T)2678 2203 y Fu(2)2725 2188 y Ft(~)-57 b Fo(!)t(;)17 b(")p Ft(\)\))p Fo(:)-118 2362 y Ft(The)36 b(next)h(theorem)e(pro)m(vides)h(a)f(\014rst)h(order)f (appro)m(ximation)f(for)g(the)i(function)f Fi(M)g Ft(in)g(terms)g(of)g (the)-118 2482 y(gradien)m(t)d(of)g(a)g(scalar)g(function,)g(called)g (the)h(Melnik)m(o)m(v)g(p)s(oten)m(tial,)e(de\014ned)j(as)f(follo)m (ws:)902 2729 y Fo(L)p Ft(\()p Fo(I)1057 2688 y Fu(0)1096 2729 y Fo(;)17 b(\022)s Ft(\))28 b(=)f Fi(\000)1451 2612 y Fh(Z)1534 2638 y Fn(1)1498 2801 y(\0001)1644 2633 y Fh(h)1708 2704 y Ft(^)1683 2729 y Fo(H)1764 2744 y Fu(1)1803 2729 y Ft(\()p Fo(\020)1884 2744 y Fu(0)1923 2729 y Ft(\()p Fo(t)p Ft(\)\))22 b Fi(\000)2219 2704 y Ft(^)2194 2729 y Fo(H)2275 2744 y Fu(1)2314 2729 y Ft(\()p Fo(\020)2403 2688 y Fn(\003)2395 2754 y Fu(0)2442 2729 y Ft(\()p Fo(t)p Ft(\)\))2591 2633 y Fh(i)2647 2729 y Fo(dt;)846 b Ft(\(26\))-118 3010 y(where)204 2985 y(^)179 3010 y Fo(H)260 3025 y Fu(1)299 3010 y Ft(\()p Fo(\020)8 b Ft(\))47 b(denotes)h(the)h(p)s (erturbation)d Fo(H)1691 3025 y Fu(1)1778 3010 y Ft(expressed)51 b(in)46 b(the)j(co)s(ordinates)e Fo(\020)8 b Ft(.)87 b(The)49 b(in)m(tegral)-118 3130 y(is)41 b(absolutely)f(con)m(v)m (ergen)m(t)j(since)f Fo(\020)1248 3145 y Fu(0)1287 3130 y Ft(\()p Fo(t)p Ft(\))f(is)g(asymptotic)f(for)h Fo(t)h Fi(!)g(\0061)f Ft(to)g Fo(\020)2830 3094 y Fn(\003)2822 3155 y Fu(0)2868 3130 y Ft(\()p Fo(t)p Ft(\))g(\(with)g(exp)s(onen)m (tial)-118 3251 y(estimates\).)-118 3479 y Fk(Theorem)c(3)49 b Fj(F)-7 b(or)39 b(the)h(splitting)f(distanc)-5 b(e)39 b(along)g(the)h Fo(\030)2049 3494 y Fm(i)2077 3479 y Fj(-dir)-5 b(e)g(ctions,)40 b(the)g(fol)5 b(lowing)38 b(\014rst)i(or)-5 b(der)40 b(ap-)-118 3599 y(pr)-5 b(oximation)34 b(holds:)1393 3720 y Fi(M)27 b Ft(=)h Fo(")p Fi(r)1773 3735 y Fm(\022)1812 3720 y Fo(L)22 b Ft(+)g Fi(O)s Ft(\()p Fo(")2164 3679 y Fu(2)2203 3720 y Ft(\))p Fo(:)1338 b Ft(\(27\))-118 3948 y Fk(Pro)s(of)p Ft(.)68 b(F)-8 b(or)40 b(an)m(y)i Fo(\022)j Ft(=)c(\()p Fo(\022)915 3963 y Fu(1)955 3948 y Fo(;)17 b(\022)1044 3963 y Fu(2)1084 3948 y Ft(\),)43 b(let)d(us)h(consider)g(the)h(tra)5 b(jectories)41 b Fo(\020)2611 3912 y Fm(s;u)2707 3948 y Ft(\()p Fo(t)p Ft(\))h(=)g Fo(\020)3029 3912 y Fm(s;u)3125 3948 y Ft(\()p Fo(t;)17 b(I)3293 3912 y Fu(0)3332 3948 y Fo(;)g(\022)s(;)g(")p Ft(\))40 b(suc)m(h)-118 4068 y(that)f Fo(\020)151 4032 y Fm(u)195 4068 y Ft(\()p Fi(\000)p Fo(T)367 4083 y Fu(1)407 4068 y Ft(\))g(=)f Fi(Z)678 4032 y Fm(u)724 4068 y Ft(\()p Fo(I)813 4032 y Fu(0)852 4068 y Fo(;)17 b(\022)29 b Fi(\000)e Fo(T)1131 4083 y Fu(1)1179 4068 y Ft(~)-57 b Fo(!)s(;)17 b(")p Ft(\))39 b(and)g Fo(\020)1649 4032 y Fm(s)1685 4068 y Ft(\()p Fo(T)1780 4083 y Fu(2)1819 4068 y Ft(\))g(=)g Fi(Z)2091 4032 y Fm(s)2128 4068 y Ft(\()p Fo(I)2217 4032 y Fu(0)2256 4068 y Fo(;)17 b(\022)29 b Ft(+)e Fo(T)2534 4083 y Fu(2)2581 4068 y Ft(~)-57 b Fo(!)t(;)17 b(")p Ft(\).)62 b(The)40 b(tra)5 b(jectories)40 b Fo(\020)3633 4032 y Fm(s)3668 4068 y Ft(\()p Fo(t)p Ft(\))-118 4189 y(and)i Fo(\020)132 4153 y Fm(u)176 4189 y Ft(\()p Fo(t)p Ft(\))g(are)g Fi(O)s Ft(\()p Fo(")p Ft(\)-close)f(to)h Fo(\020)1152 4204 y Fu(0)1191 4189 y Ft(\()p Fo(t)p Ft(\))g(for)g Fo(t)i Fi(\025)g Fo(T)1760 4204 y Fu(2)1841 4189 y Ft(and)e Fo(t)i Fi(\024)h(\000)p Fo(T)2375 4204 y Fu(1)2457 4189 y Ft(resp)s(ectiv)m(ely)-8 b(,)45 b(due)e(to)e(that)h(these)-118 4309 y(parts)e(of)g(the)h(tra)5 b(jectories)41 b(are)f(con)m(tained)g (in)g(the)h(domain)d(of)i(v)-5 b(alidit)m(y)38 b(of)i(the)h(KAM)g (results,)h(and)-118 4430 y(they)37 b(are)e(asymptotic)h(to)f(the)h (torus)g(as)h Fo(t)c Fi(!)g(\0061)p Ft(.)53 b(Extending)37 b(these)g(estimates)e(to)h Fi(\000)p Fo(T)3320 4445 y Fu(1)3393 4430 y Fi(\024)e Fo(t)f Fi(\024)h Fo(T)3740 4445 y Fu(2)-118 4550 y Ft(\(an)j(in)m(terv)-5 b(al)36 b(whose)j(length)e(is)g Fi(\030)f Ft(ln)o(\(1)p Fo(=d)p Ft(\),)i(outside)f(of)g(the)h(scop)s(e)g(of)f(KAM)h(results\),)h(w)m(e) f(see)h(from)-118 4670 y(Gron)m(w)m(all)k(inequalit)m(y)h(that)h(the)g (tra)5 b(jectory)45 b Fo(\020)1699 4634 y Fm(u)1744 4670 y Ft(\()p Fo(t)p Ft(\))g(is)f Fi(O)s Ft(\()p Fo("=d)2276 4634 y Fm(\013)2325 4670 y Ft(\)-close)g(to)g Fo(\020)2815 4685 y Fu(0)2854 4670 y Ft(\()p Fo(t)p Ft(\))h(for)g(some)f Fo(\013)50 b(>)e Ft(0.)3740 4634 y Fu(3)-118 4791 y Ft(Ho)m(w)m(ev)m (er,)d(since)d(the)f(n)m(um)m(b)s(er)g Fo(d)g Ft(do)s(es)g(not)g(dep)s (end)h(on)f Fo(")p Ft(,)h(it)e(follo)m(ws)g(that)g Fo(\020)2912 4754 y Fm(u)2956 4791 y Ft(\()p Fo(t)p Ft(\))h(is)g Fi(O)s Ft(\()p Fo(")p Ft(\)-close)f(to)-118 4911 y Fo(\020)-75 4926 y Fu(0)-36 4911 y Ft(\()p Fo(t)p Ft(\))32 b(for)f(all)f Fo(t)e Fi(\024)g Fo(T)615 4926 y Fu(2)654 4911 y Ft(.)44 b(Using)31 b(this)h(estimate,)f(and)h(the)g(fact)g(that)f(the)i(tra)5 b(jectory)32 b Fo(\020)3036 4875 y Fm(u)3080 4911 y Ft(\()p Fo(t)p Ft(\))g(in)m(tersects)h Fo(N)3742 4875 y Fm(s)-118 5031 y Ft(transv)m(ersely)-8 b(,)35 b(w)m(e)g(deduce)g(that)1155 5006 y(\026)1126 5031 y Fi(Z)1206 4995 y Fm(u)1280 5031 y Ft(=)30 b Fo(\020)1437 4995 y Fm(u)1481 5031 y Ft(\()p Fo(T)1576 5046 y Fu(2)1638 5031 y Ft(+)23 b Fi(O)s Ft(\()p Fo(")p Ft(\)\))33 b(and,)h(for)f(the)h(actions,)2918 5006 y(\026)2909 5031 y Fi(I)2970 4995 y Fm(u)2963 5056 y(i)3045 5031 y Ft(=)29 b Fo(\030)3198 4995 y Fm(u)3193 5056 y(i)3242 5031 y Ft(\()p Fo(T)3337 5046 y Fu(2)3377 5031 y Ft(\))23 b(+)f Fi(O)s Ft(\()p Fo(")3702 4995 y Fu(2)3742 5031 y Ft(\))-118 5152 y(since)142 5126 y(_)121 5152 y Fo(\030)164 5167 y Fm(i)219 5152 y Ft(=)28 b Fi(O)s Ft(\()p Fo(")p Ft(\).)p -118 5245 1560 4 v -6 5306 a Fd(3)31 5336 y Fv(This)k Fb(\013)h Fv(is)e(giv)n(en)g(b)n(y)h(the)g (Lipsc)n(hitz)g(constan)n(t)f(for)g(the)h(Hamiltonian)g(v)n(ector)f (\014eld)h(asso)r(ciated)e(to)i Fb(H)3356 5348 y Fd(0)3393 5336 y Fv(;)i(it)f(can)e(b)r(e)-118 5436 y(obtained)c(from)g(a)h(b)r (ound)g(\(on)f(a)g(compact)g(set\))h(for)f(the)h(second)f(deriv)-5 b(ativ)n(es)27 b(of)g Fb(H)2570 5448 y Fd(0)2608 5436 y Fv(.)1782 5753 y Ft(20)p eop %%Page: 21 21 21 20 bop 28 407 a Ft(On)23 b(the)h(torus)f Fi(T)i Ft(,)g(let)d Fo(\020)893 371 y Fn(\003)932 407 y Ft(\()p Fo(t)p Ft(\))28 b(=)f Fo(\020)1225 371 y Fn(\003)1264 407 y Ft(\()p Fo(t;)17 b(I)1432 371 y Fu(0)1471 407 y Fo(;)g(\022)s(;)g(")p Ft(\))22 b(denote)i(the)f(tra)5 b(jectory)24 b(to)e(whic)m(h)i Fo(\020)3052 371 y Fm(s)3088 407 y Ft(\()p Fo(t)p Ft(\))f(is)f (asymptotic)-118 527 y(for)33 b Fo(t)c Fi(!)g(1)p Ft(,)k(with)g Fo(\020)659 491 y Fn(\003)698 527 y Ft(\()p Fo(T)793 542 y Fu(2)832 527 y Ft(\))c(=)g Fi(Z)1084 491 y Fn(\003)1124 527 y Ft(\()p Fo(I)1213 491 y Fu(0)1252 527 y Fo(;)17 b(\022)25 b Ft(+)e Fo(T)1522 542 y Fu(2)1569 527 y Ft(~)-57 b Fo(!)t(;)17 b(")p Ft(\).)45 b(This)34 b(tra)5 b(jectory)34 b Fo(\020)2557 491 y Fn(\003)2595 527 y Ft(\()p Fo(t)p Ft(\))g(is)e Fi(O)s Ft(\()p Fo(")p Ft(\)-close)h(to)g Fo(\020)3480 491 y Fn(\003)3472 552 y Fu(0)3519 527 y Ft(\()p Fo(t)p Ft(\))g(for)-118 648 y(all)d Fo(t)e Fi(2)g Fe(R)p Ft(.)50 b(The)33 b(tra)5 b(jectory)34 b Fo(\020)1012 611 y Fm(u)1056 648 y Ft(\()p Fo(t)p Ft(\))e(is)g(also)g(asymptotic)g (for)g Fo(t)c Fi(!)f(\0001)33 b Ft(to)f(the)h(tra)5 b(jectory)33 b Fo(\020)3337 611 y Fn(\003)3376 648 y Ft(\()p Fo(t)p Ft(\).)28 768 y(F)-8 b(or)32 b(the)h(comp)s(onen)m(ts)g(of)f(the)h (function)f Fi(M)p Ft(,)g(w)m(e)i(write)23 963 y(\026)14 988 y Fi(I)75 947 y Fm(u)68 1013 y(i)142 988 y Fi(\000)23 b(I)303 947 y Fm(s)296 1013 y(i)368 988 y Ft(=)28 b(\()519 963 y(\026)510 988 y Fi(I)571 947 y Fm(u)564 1013 y(i)638 988 y Fi(\000)23 b(I)799 947 y Fn(\003)792 1013 y Fm(i)839 988 y Ft(\))f Fi(\000)h Ft(\()p Fi(I)1098 947 y Fm(s)1091 1013 y(i)1157 988 y Fi(\000)g(I)1318 947 y Fn(\003)1311 1013 y Fm(i)1358 988 y Ft(\))k(=)h(\()p Fo(\030)1613 947 y Fm(u)1608 1013 y(i)1657 988 y Ft(\()p Fo(T)1752 1003 y Fu(2)1792 988 y Ft(\))22 b Fi(\000)g Fo(\030)1999 947 y Fn(\003)1994 1013 y Fm(i)2038 988 y Ft(\()p Fo(T)2133 1003 y Fu(2)2173 988 y Ft(\)\))g Fi(\000)h Ft(\()p Fo(\030)2457 947 y Fm(s)2452 1013 y(i)2493 988 y Ft(\()p Fo(T)2588 1003 y Fu(2)2627 988 y Ft(\))f Fi(\000)h Fo(\030)2835 947 y Fn(\003)2830 1013 y Fm(i)2874 988 y Ft(\()p Fo(T)2969 1003 y Fu(2)3008 988 y Ft(\)\))f(+)g Fi(O)s Ft(\()p Fo(")3370 947 y Fu(2)3409 988 y Ft(\))p Fo(:)132 b Ft(\(28\))-118 1208 y(T)-8 b(o)43 b(obtain)f(the)i(\014rst)g(order)f(appro)m (ximation,)h(w)m(e)g(use)g(that)f Fo(\030)2295 1223 y Fu(1)2334 1208 y Ft(,)j Fo(\030)2450 1223 y Fu(2)2532 1208 y Ft(are)d(\014rst)h(in)m(tegrals)e(of)h Fo(H)3524 1223 y Fu(0)3606 1208 y Ft(in)g(a)-118 1328 y(neigh)m(b)s(orho)s(o)s(d) 31 b Fo(U)43 b Ft(of)33 b(the)g(singular)e(p)s(oin)m(t)g Fo(O)s Ft(:)813 1591 y(_)792 1617 y Fo(\030)835 1632 y Fm(i)890 1617 y Ft(=)d Fo(")p Fi(f)p Fo(\030)1133 1632 y Fm(i)1160 1617 y Fo(;)1229 1592 y Ft(^)1204 1617 y Fo(H)1285 1632 y Fu(1)1324 1617 y Fi(g)g Ft(=)f Fi(\000)p Fo(")1638 1550 y(@)1721 1525 y Ft(^)1694 1550 y Fo(H)1775 1565 y Fu(1)p 1638 1594 178 4 v 1652 1686 a Fo(@)5 b(')1772 1701 y Fm(i)1853 1617 y Ft(=)28 b Fo(")2020 1471 y Fh( )2085 1617 y Fo(y)2133 1632 y Fm(i)2170 1550 y Fo(@)5 b(H)2307 1565 y Fu(1)p 2170 1594 V 2189 1686 a Fo(@)g(x)2300 1701 y Fm(i)2380 1617 y Fi(\000)22 b Fo(x)2534 1632 y Fm(i)2573 1550 y Fo(@)5 b(H)2710 1565 y Fu(1)p 2573 1594 V 2595 1686 a Fo(@)g(y)2699 1701 y Fm(i)2760 1471 y Fh(!)2842 1617 y Fo(:)737 b Ft(\(29\))-118 1900 y(F)-8 b(or)32 b(the)h(\\stable")f(term)g(in)f(\(28\),)h(w)m(e)i(obtain:)162 2194 y Fi(\000)p Ft(\()p Fo(\030)325 2153 y Fm(s)320 2219 y(i)361 2194 y Ft(\()p Fo(T)456 2209 y Fu(2)496 2194 y Ft(\))22 b Fi(\000)h Fo(\030)704 2153 y Fn(\003)699 2219 y Fm(i)743 2194 y Ft(\()p Fo(T)838 2209 y Fu(2)877 2194 y Ft(\)\))28 b(=)1084 2077 y Fh(Z)1167 2103 y Fn(1)1130 2266 y Fm(T)1171 2275 y Fl(2)1259 2098 y Fh(h)1319 2168 y Ft(_)1298 2194 y Fo(\030)1346 2153 y Fm(s)1341 2219 y(i)1382 2194 y Ft(\()p Fo(t)p Ft(\))22 b Fi(\000)1636 2168 y Ft(_)1615 2194 y Fo(\030)1663 2153 y Fn(\003)1658 2219 y Fm(i)1702 2194 y Ft(\()p Fo(t)p Ft(\))1813 2098 y Fh(i)1869 2194 y Fo(dt)27 b Ft(=)h Fi(\000)p Fo(")2226 2077 y Fh(Z)2309 2103 y Fn(1)2272 2266 y Fm(T)2313 2275 y Fl(2)2400 2048 y Fh(")2458 2127 y Fo(@)2540 2102 y Ft(^)2514 2127 y Fo(H)2595 2142 y Fu(1)p 2458 2171 V 2472 2263 a Fo(@)5 b(')2592 2278 y Fm(i)2645 2194 y Ft(\()p Fo(\020)2734 2153 y Fm(s)2770 2194 y Ft(\()p Fo(t)p Ft(\)\))22 b Fi(\000)3051 2127 y Fo(@)3133 2102 y Ft(^)3107 2127 y Fo(H)3188 2142 y Fu(1)p 3051 2171 V 3065 2263 a Fo(@)5 b(')3185 2278 y Fm(i)3238 2194 y Ft(\()p Fo(\020)3327 2153 y Fn(\003)3366 2194 y Ft(\()p Fo(t)p Ft(\)\))3515 2048 y Fh(#)3580 2194 y Fo(dt)357 2489 y Ft(=)28 b Fi(\000)p Fo(")601 2372 y Fh(Z)683 2398 y Fn(1)646 2560 y Fm(T)687 2569 y Fl(2)775 2343 y Fh(")833 2422 y Fo(@)915 2396 y Ft(^)889 2422 y Fo(H)970 2437 y Fu(1)p 833 2466 V 847 2557 a Fo(@)5 b(')967 2572 y Fm(i)1020 2489 y Ft(\()p Fo(\020)1101 2504 y Fu(0)1140 2489 y Ft(\()p Fo(t)p Ft(\)\))22 b Fi(\000)1421 2422 y Fo(@)1503 2396 y Ft(^)1477 2422 y Fo(H)1558 2437 y Fu(1)p 1421 2466 V 1435 2557 a Fo(@)5 b(')1555 2572 y Fm(i)1608 2489 y Ft(\()p Fo(\020)1697 2448 y Fn(\003)1689 2514 y Fu(0)1736 2489 y Ft(\()p Fo(t)p Ft(\)\))1885 2343 y Fh(#)1950 2489 y Fo(dt)22 b Ft(+)g Fi(O)s Ft(\()p Fo(")2322 2448 y Fu(2)2361 2489 y Ft(\))p Fo(;)-118 2771 y Ft(where)46 b(w)m(e)g(ha)m(v)m(e)g(used)g(the)f Fi(O)s Ft(\()p Fo(")p Ft(\)-closeness)h(of)e(the)h(p)s(erturb)s(ed)h (tra)5 b(jectories)45 b(to)f(the)h(unp)s(erturb)s(ed)-118 2892 y(ones,)31 b(for)f Fo(t)e Fi(\025)g Fo(T)496 2907 y Fu(2)535 2892 y Ft(.)43 b(The)31 b(in)m(tegral)d(in)m(v)m(olv)m(ed)j (is)e(absolutely)g(con)m(v)m(ergen)m(t,)k(since)d Fo(\020)2891 2855 y Fm(s)2927 2892 y Ft(\()p Fo(t)p Ft(\))g(is)g(asymptotic)f(for) -118 3012 y Fo(t)f Fi(!)f(1)h Ft(to)h Fo(\020)367 2976 y Fn(\003)405 3012 y Ft(\()p Fo(t)p Ft(\))g(with)f(exp)s(onen)m(tial)h (estimates.)41 b(F)-8 b(or)28 b(the)h(\\unstable")g(term)f(in)g (\(28\),)h(w)m(e)h(can)f(pro)s(ceed)-118 3132 y(analogously)i (\(although)h(w)m(e)i(in)m(tegrate)e(in)g(a)h(\\longer")e(in)m(terv)-5 b(al,)32 b(this)h(do)s(es)g(not)g(mak)m(e)g(the)g(estimates)-118 3253 y(w)m(orse,)40 b(b)s(ecause)f(the)f(length)f(of)g(the)h (additional)d(in)m(terv)-5 b(al)36 b Fi(\000)p Fo(T)2272 3268 y Fu(1)2348 3253 y Fi(\024)h Fo(t)f Fi(\024)h Fo(T)2704 3268 y Fu(2)2781 3253 y Ft(do)s(es)h(not)g(dep)s(end)g(on)g Fo(")p Ft(\),)-118 3373 y(and)33 b(w)m(e)g(obtain:)208 3643 y Fo(\030)256 3602 y Fm(u)251 3667 y(i)301 3643 y Ft(\()p Fo(T)396 3658 y Fu(2)435 3643 y Ft(\))22 b Fi(\000)h Fo(\030)643 3602 y Fn(\003)638 3667 y Fm(i)682 3643 y Ft(\()p Fo(T)777 3658 y Fu(2)816 3643 y Ft(\))28 b(=)985 3526 y Fh(Z)1068 3552 y Fm(T)1109 3561 y Fl(2)1032 3714 y Fn(\0001)1178 3546 y Fh(h)1238 3616 y Ft(_)1217 3643 y Fo(\030)1265 3602 y Fm(u)1260 3667 y(i)1309 3643 y Ft(\()p Fo(t)p Ft(\))23 b Fi(\000)1563 3616 y Ft(_)1542 3643 y Fo(\030)1590 3602 y Fn(\003)1585 3667 y Fm(i)1629 3643 y Ft(\()p Fo(t)p Ft(\))1740 3546 y Fh(i)1796 3643 y Fo(dt)28 b Ft(=)f Fi(\000)p Fo(")2153 3526 y Fh(Z)2236 3552 y Fm(T)2277 3561 y Fl(2)2199 3714 y Fn(\0001)2345 3497 y Fh(")2404 3575 y Fo(@)2486 3550 y Ft(^)2460 3575 y Fo(H)2541 3590 y Fu(1)p 2404 3620 V 2418 3711 a Fo(@)5 b(')2538 3726 y Fm(i)2591 3643 y Ft(\()p Fo(\020)2680 3602 y Fm(u)2724 3643 y Ft(\()p Fo(t)p Ft(\)\))22 b Fi(\000)3005 3575 y Fo(@)3087 3550 y Ft(^)3061 3575 y Fo(H)3142 3590 y Fu(1)p 3005 3620 V 3019 3711 a Fo(@)5 b(')3139 3726 y Fm(i)3192 3643 y Ft(\()p Fo(\020)3281 3602 y Fn(\003)3319 3643 y Ft(\()p Fo(t)p Ft(\)\))3468 3497 y Fh(#)3533 3643 y Fo(dt)403 3937 y Ft(=)28 b Fi(\000)p Fo(")647 3820 y Fh(Z)730 3847 y Fm(T)771 3856 y Fl(2)693 4009 y Fn(\0001)839 3791 y Fh(")897 3870 y Fo(@)980 3845 y Ft(^)953 3870 y Fo(H)1034 3885 y Fu(1)p 897 3914 V 911 4006 a Fo(@)5 b(')1031 4021 y Fm(i)1084 3937 y Ft(\()p Fo(\020)1165 3952 y Fu(0)1204 3937 y Ft(\()p Fo(t)p Ft(\)\))23 b Fi(\000)1485 3870 y Fo(@)1567 3845 y Ft(^)1541 3870 y Fo(H)1622 3885 y Fu(1)p 1485 3914 V 1499 4006 a Fo(@)5 b(')1619 4021 y Fm(i)1672 3937 y Ft(\()p Fo(\020)1761 3896 y Fn(\003)1753 3962 y Fu(0)1800 3937 y Ft(\()p Fo(t)p Ft(\)\))1949 3791 y Fh(#)2014 3937 y Fo(dt)22 b Ft(+)g Fi(O)s Ft(\()p Fo(")2386 3896 y Fu(2)2425 3937 y Ft(\))p Fo(:)-118 4220 y Ft(Replacing)40 b(in)h(\(28\))g(the)h(appro)m(ximations)d(obtained,)44 b(w)m(e)e(obtain)f(an)g(in)m(tegral)2967 4149 y Fh(R)3022 4175 y Fn(1)3006 4245 y(\0001)3177 4220 y Ft(\(the)h(Melnik)m(o)m(v) -118 4340 y(function\))k(that)g(giv)m(es)g(a)g(\014rst)h(order)f(appro) m(ximation)e(to)i(the)h(function)f Fi(M)p Ft(,)j(with)d(an)g(error)g (term)-118 4460 y Fi(O)s Ft(\()p Fo(")48 4424 y Fu(2)87 4460 y Ft(\).)d(This)33 b(in)m(tegral)e(is)h(the)h(gradien)m(t)f(in)g (\()p Fo(\022)1618 4475 y Fu(1)1658 4460 y Fo(;)17 b(\022)1747 4475 y Fu(2)1786 4460 y Ft(\))33 b(of)f(the)h(Melnik)m(o)m(v)g(p)s (oten)m(tial)e(de\014ned)j(in)e(\(26\).)p 3662 4460 42 42 v 28 4631 a(Since)e(the)f(splitting)e(distance)j(is)f(appro)m (ximated)f(b)m(y)j(the)e(gradien)m(t)g(of)g(the)h(Melnik)m(o)m(v)f(p)s (oten)m(tial)f Fo(L)p Ft(,)-118 4751 y(the)35 b(nondegenerate)h (critical)d(p)s(oin)m(ts)h(in)h Fo(\022)f Ft(=)e(\()p Fo(\022)1709 4766 y Fu(1)1749 4751 y Fo(;)17 b(\022)1838 4766 y Fu(2)1877 4751 y Ft(\))35 b(of)g Fo(L)g Ft(will)d(giv)m(e)j (rise)g(to)g(transv)m(erse)i(homo)s(clinic)-118 4871 y(orbits.)73 b(W)-8 b(e)43 b(p)s(oin)m(t)f(out)h(that,)i(in)d(related)g (cases,)47 b(it)42 b(can)h(b)s(e)g(sho)m(wn)h([4])e(that)h(the)g(whole) g(splitting)-118 4992 y(distance)28 b(\(and)g(not)g(only)g(its)g (\014rst)g(order)g(appro)m(ximation\))e(is)i(also)f(the)h(gradien)m(t)g (of)g(a)f(scalar)h(function,)-118 5112 y(called)45 b(the)i(splitting)d (p)s(oten)m(tial)h(\(due)i(to)g(the)f(Hamiltonian)d(c)m(haracter)48 b(of)e(the)h(equations,)j(whic)m(h)-118 5233 y(implies)31 b(Lagrangian)g(prop)s(erties)i(for)g(the)h(in)m(v)-5 b(arian)m(t)31 b(manifolds\).)44 b(This)33 b(fact)g(is)g(v)m(ery)i (useful)e(in)f(order)-118 5353 y(to)f(ensure)i(the)f(existence)h(of)e (homo)s(clinic)d(orbits)j(in)g(degenerate)i(\(non-transv)m(erse\))g (cases,)g(but)f(w)m(e)h(do)-118 5473 y(not)f(need)i(it)d(here.)1782 5753 y(21)p eop %%Page: 22 22 22 21 bop 28 407 a Ft(No)m(w,)31 b(w)m(e)f(are)f(going)f(to)h(study)h (the)f(nondegenerate)i(critical)c(p)s(oin)m(ts)h(in)h Fo(\022)j Ft(of)d(the)g(Melnik)m(o)m(v)h(p)s(oten-)-118 527 y(tial.)56 b(This)37 b(requires)h(explicit)e(computations)h(and,)h (to)f(giv)m(e)h(an)f(illustration,)e(w)m(e)j(only)f(consider)h(the)-118 648 y(case)h(of)e(the)i(p)s(endulum)e(\(example)g(1,)i(among)e(the)i (three)f(ones)h(considered)g(in)e(Section)h(1.1\),)h(and)f(a)-118 768 y(more)32 b(concrete)i(p)s(erturbation:)1154 974 y Fo(H)1235 989 y Fu(1)1302 974 y Ft(=)27 b(\(cos)17 b Fo(q)26 b Fi(\000)d Fo(\027)6 b Ft(\))p Fo(f)11 b Ft(\()p Fo(x)2003 989 y Fu(1)2043 974 y Fo(;)17 b(y)2135 989 y Fu(1)2173 974 y Fo(;)g(x)2272 989 y Fu(2)2312 974 y Fo(;)g(y)2404 989 y Fu(2)2443 974 y Ft(\))p Fo(;)1098 b Ft(\(30\))-118 1181 y(where)36 b Fo(\027)41 b Ft(is)34 b(a)g(parameter.)50 b(It)34 b(is)h(not)f(hard)h(to)f(see)i(that)e(for)g Fo(\027)k Ft(=)31 b(1)k(all)d(tori)i(p)s(ersist)g(with)h(their)f(lo)s (cal)-118 1301 y(in)m(v)-5 b(arian)m(t)39 b(manifolds,)h(whereas)i(for) e Fo(\027)48 b Fi(6)p Ft(=)41 b(1)f(w)m(e)h(ha)m(v)m(e)h(a)e(ma)5 b(jorit)m(y)39 b(of)h(p)s(ersisten)m(t)i(Diophan)m(tine)d(tori)-118 1421 y(from)c(KAM)i(theory)-8 b(.)56 b(In)37 b(b)s(oth)g(cases,)i(the)e (global)d(in)m(v)-5 b(arian)m(t)36 b(manifolds)e(ma)m(y)i(not)h (coincide,)g(and)g(to)-118 1542 y(study)32 b(their)e(transv)m(erse)j (in)m(tersections)f(w)m(e)g(use)g(the)f(Melnik)m(o)m(v)g(p)s(oten)m (tial.)41 b(Applying)31 b(\(26\))f(and)h(using)-118 1662 y(that)1309 1802 y(cos)17 b Fo(q)1499 1817 y Fu(0)1539 1802 y Ft(\()p Fo(t)p Ft(\))22 b Fi(\000)h Ft(1)k(=)h Fi(\000)2153 1735 y Ft(2)p 2039 1779 277 4 v 2039 1874 a(cosh)2224 1832 y Fu(2)2280 1874 y Fo(t)2325 1802 y(;)1254 b Ft(\(31\))-118 2000 y(w)m(e)29 b(obtain)f(the)h(follo)m(wing)c (expression)30 b(for)e(the)h(Melnik)m(o)m(v)f(p)s(oten)m(tial,)g(whic)m (h)h(do)s(es)g(not)f(dep)s(end)i(on)e(the)-118 2120 y(v)-5 b(alue)32 b(of)g Fo(\027)6 b Ft(:)1139 2292 y Fo(L)p Ft(\()p Fo(I)1294 2250 y Fu(0)1334 2292 y Fo(;)17 b(\022)s Ft(\))27 b(=)h(2)1661 2174 y Fh(Z)1743 2201 y Fn(1)1706 2363 y(\0001)1884 2198 y Ft(^)1863 2224 y Fo(f)10 b Ft(\()p Fo(I)2010 2188 y Fu(0)2050 2224 y Fo(;)17 b(\022)25 b Ft(+)d Fo(t)8 b Ft(~)-57 b Fo(!)s Ft(\))p 1863 2268 537 4 v 1993 2364 a(cosh)2178 2321 y Fu(2)2234 2364 y Fo(t)2409 2292 y(dt:)28 2496 y Ft(In)38 b(the)f(next)h(t)m(w)m(o)g(theorems,)h(t) m(w)m(o)f(di\013eren)m(t)f(t)m(yp)s(es)i(of)e(functions)g Fo(f)48 b Ft(are)37 b(considered)h(in)e(the)i(p)s(er-)-118 2616 y(turbation)43 b(\(30\),)k(and)d(the)h(existence)h(and)e(n)m(um)m (b)s(er)h(of)f(transv)m(erse)i(homo)s(clinic)c(orbits)h(for)h(suc)m(h)i (a)-118 2736 y(p)s(erturbation)31 b(is)g(studied.)44 b(First,)31 b(in)g(Theorem)i(4,)e(w)m(e)i(consider)g Fo(f)38 b Ft(=)28 b Fi(O)2618 2751 y Fu(2)2657 2736 y Ft(\()p Fo(x)2750 2751 y Fu(1)2790 2736 y Fo(;)17 b(y)2882 2751 y Fu(1)2921 2736 y Fo(;)g(x)3020 2751 y Fu(2)3059 2736 y Fo(;)g(y)3151 2751 y Fu(2)3190 2736 y Ft(\),)32 b(and)g(writing)-118 2857 y(do)m(wn)39 b(the)f(Hamiltonian)d(equations) j(one)h(easily)e(c)m(hec)m(ks)k(that)d(the)g(homo)s(clinic)d(lo)s(op)i (\000)h(p)s(ersists)h(\(to)-118 2977 y(b)s(e)e(more)e(precise,)k(it)c (is)h(k)m(ept)i(unc)m(hanged\).)57 b(Instead,)39 b(in)c(Theorem)i(5)f (w)m(e)i(consider)f(a)f(more)g(general)-118 3097 y(function)f Fo(f)47 b Ft(and,)37 b(in)e(general,)i(the)f(lo)s(op)e(will)g(not)i(b)s (e)g(preserv)m(ed.)56 b(Their)36 b(statemen)m(ts)h(are)f(similar)d(to) -118 3218 y(those)c(of)g(Theorems)g(1)g(and)g(2,)g(but)h(no)m(w)f(w)m (e)h(pro)m(vide)f(a)g(condition)e(that)i(can)g(b)s(e)g(c)m(hec)m(k)m (ed)j(for)c(concrete)-118 3338 y(examples.)28 3459 y(W)-8 b(e)32 b(recall)e(that)h(a)h(scalar)e(function)h(on)h(a)f(manifold)e (is)i(called)f(a)h(Morse)i(function)e(pro)m(vided)h(all)d(its)-118 3579 y(critical)h(p)s(oin)m(ts)i(are)h(nondegenerate.)-118 3792 y Fk(Theorem)k(4)49 b Fj(L)-5 b(et)35 b Fo(f)j Ft(=)28 b Fi(O)899 3807 y Fu(2)939 3792 y Ft(\()p Fo(x)1032 3807 y Fu(1)1071 3792 y Fo(;)17 b(y)1163 3807 y Fu(1)1202 3792 y Fo(;)g(x)1301 3807 y Fu(2)1341 3792 y Fo(;)g(y)1433 3807 y Fu(2)1471 3792 y Ft(\))35 b Fj(in)f(the)h(p)-5 b(erturb)g(ation)35 b(\(30\),)f(denote)g Fo(f)2976 3807 y Fu(2)3050 3792 y Fj(its)h(quadr)-5 b(atic)34 b(p)-5 b(art)-118 3912 y(\(i.e.)44 b Fo(f)58 b Ft(=)46 b Fo(f)377 3927 y Fu(2)446 3912 y Ft(+)30 b Fi(O)631 3927 y Fu(3)670 3912 y Fj(\).)76 b(F)-7 b(or)44 b(any)h(action)f Fo(I)1562 3876 y Fu(0)1648 3912 y Ft(=)j(\()p Fo(I)1860 3876 y Fu(0)1852 3937 y(1)1899 3912 y Fo(;)17 b(I)1994 3876 y Fu(0)1986 3937 y(2)2033 3912 y Ft(\))p Fj(,)47 b(let)e(us)h(denote)e Fo(L)2820 3927 y Fu(2)2860 3912 y Ft(\()p Fo(I)2949 3876 y Fu(0)2988 3912 y Fo(;)17 b Fi(\001)p Ft(\))44 b Fj(the)i(p)-5 b(art)45 b(of)g(the)-118 4032 y(Melnikov)33 b(p)-5 b(otential)33 b(c)-5 b(oming)32 b(fr)-5 b(om)33 b Fo(f)1303 4047 y Fu(2)1342 4032 y Fj(,)h(and)f(let)g Fi(B)k Fj(b)-5 b(e)33 b(the)h(set)f(of)g(actions)g Fo(I)2759 3996 y Fu(0)2832 4032 y Fj(such)g(that)h Fo(L)3316 4047 y Fu(2)3356 4032 y Ft(\()p Fo(I)3445 3996 y Fu(0)3484 4032 y Fo(;)17 b Fi(\001)p Ft(\))33 b Fj(is)g(a)-118 4153 y(Morse)38 b(function)g(of)f Fo(\022)g Fi(2)d Fe(T)925 4117 y Fu(2)964 4153 y Fj(.)54 b(Then,)38 b(for)g Fo(I)1546 4117 y Fu(0)1623 4153 y Fj(in)g(an)g(op)-5 b(en)37 b(subset)h(of)g Fi(B)s Fj(,)h(and)f Fo(")f Fj(smal)5 b(l)37 b(enough,)i(ther)-5 b(e)-118 4273 y(exist)35 b(either)f(8,)h(12)f(or)h(16)g(tr)-5 b(ansverse)34 b(homo)-5 b(clinic)33 b(orbits.)-118 4486 y Fk(Pro)s(of.)48 b Ft(Let)34 b Fo(f)41 b Ft(=)30 b Fo(b)645 4501 y Fu(1)685 4486 y Fo(x)740 4450 y Fu(2)740 4510 y(1)803 4486 y Ft(+)23 b Fo(b)943 4501 y Fu(2)983 4486 y Fo(x)1038 4501 y Fu(1)1077 4486 y Fo(y)1125 4501 y Fu(1)1188 4486 y Ft(+)g Fo(b)1328 4501 y Fu(3)1367 4486 y Fo(x)1422 4501 y Fu(1)1462 4486 y Fo(x)1517 4501 y Fu(2)1580 4486 y Ft(+)g Fo(b)1720 4501 y Fu(4)1760 4486 y Fo(x)1815 4501 y Fu(1)1855 4486 y Fo(y)1903 4501 y Fu(2)1965 4486 y Ft(+)g Fo(b)2105 4501 y Fu(5)2145 4486 y Fo(y)2197 4450 y Fu(2)2193 4510 y(1)2259 4486 y Ft(+)g Fo(b)2399 4501 y Fu(6)2439 4486 y Fo(y)2487 4501 y Fu(1)2526 4486 y Fo(x)2581 4501 y Fu(2)2644 4486 y Ft(+)g Fo(b)2784 4501 y Fu(7)2824 4486 y Fo(y)2872 4501 y Fu(1)2911 4486 y Fo(y)2959 4501 y Fu(2)3021 4486 y Ft(+)g Fo(b)3161 4501 y Fu(8)3201 4486 y Fo(x)3256 4450 y Fu(2)3256 4510 y(2)3319 4486 y Ft(+)g Fo(b)3459 4501 y Fu(9)3499 4486 y Fo(x)3554 4501 y Fu(2)3594 4486 y Fo(y)3642 4501 y Fu(2)3704 4486 y Ft(+)-118 4606 y Fo(b)-77 4621 y Fu(10)-2 4606 y Fo(y)50 4570 y Fu(2)46 4631 y(2)111 4606 y Ft(+)f Fi(O)288 4621 y Fu(3)328 4606 y Ft(\()p Fo(x)421 4621 y Fu(1)460 4606 y Fo(;)17 b(y)552 4621 y Fu(1)591 4606 y Fo(;)g(x)690 4621 y Fu(2)730 4606 y Fo(;)g(y)822 4621 y Fu(2)860 4606 y Ft(\).)44 b(In)33 b(p)s(olar)e(co)s(ordinates,)h (this)g(function)g(b)s(ecomes)450 4786 y(^)429 4812 y Fo(f)11 b Ft(\()p Fo(\030)569 4827 y Fu(1)608 4812 y Fo(;)17 b(\030)695 4827 y Fu(2)734 4812 y Fo(;)g(')842 4827 y Fu(1)881 4812 y Fo(;)g(')989 4827 y Fu(2)1028 4812 y Ft(\))512 4958 y(=)83 b Fo(\030)714 4973 y Fu(1)753 4958 y Ft([\()p Fo(b)859 4973 y Fu(1)921 4958 y Ft(+)22 b Fo(b)1060 4973 y Fu(5)1100 4958 y Ft(\))g(+)g(\()p Fo(b)1337 4973 y Fu(1)1399 4958 y Fi(\000)h Fo(b)1540 4973 y Fu(5)1580 4958 y Ft(\))17 b(cos)g(2)p Fo(')1895 4973 y Fu(1)1956 4958 y Ft(+)22 b Fo(b)2095 4973 y Fu(2)2151 4958 y Ft(sin)17 b(2)p Fo(')2401 4973 y Fu(1)2440 4958 y Ft(])671 5127 y(+)747 5022 y Fh(q)p 830 5022 165 4 v 105 x Fo(\030)873 5142 y Fu(1)912 5127 y Fo(\030)955 5142 y Fu(2)994 5127 y Ft([\()p Fo(b)1100 5142 y Fu(3)1162 5127 y Fi(\000)23 b Fo(b)1303 5142 y Fu(7)1343 5127 y Ft(\))17 b(cos\()p Fo(')1630 5142 y Fu(1)1691 5127 y Ft(+)22 b Fo(')1853 5142 y Fu(2)1893 5127 y Ft(\))g(+)g(\()p Fo(b)2130 5142 y Fu(4)2192 5127 y Ft(+)g Fo(b)2331 5142 y Fu(6)2371 5127 y Ft(\))17 b(sin)o(\()p Fo(')2647 5142 y Fu(1)2708 5127 y Ft(+)22 b Fo(')2870 5142 y Fu(2)2910 5127 y Ft(\))1043 5313 y(+)g(\()p Fo(b)1220 5328 y Fu(3)1282 5313 y Ft(+)g Fo(b)1421 5328 y Fu(7)1461 5313 y Ft(\))17 b(cos\()p Fo(')1748 5328 y Fu(1)1810 5313 y Fi(\000)23 b Fo(')1974 5328 y Fu(2)2013 5313 y Ft(\))f Fi(\000)h Ft(\()p Fo(b)2252 5328 y Fu(4)2314 5313 y Fi(\000)f Fo(b)2454 5328 y Fu(6)2494 5313 y Ft(\))17 b(sin)o(\()p Fo(')2770 5328 y Fu(1)2832 5313 y Fi(\000)22 b Fo(')2995 5328 y Fu(2)3035 5313 y Ft(\)])671 5487 y(+)p Fo(\030)790 5502 y Fu(2)829 5487 y Ft([\()p Fo(b)935 5502 y Fu(8)997 5487 y Ft(+)g Fo(b)1136 5502 y Fu(10)1211 5487 y Ft(\))g(+)g(\()p Fo(b)1448 5502 y Fu(8)1510 5487 y Fi(\000)h Fo(b)1651 5502 y Fu(10)1726 5487 y Ft(\))17 b(cos)g(2)p Fo(')2041 5502 y Fu(2)2102 5487 y Ft(+)22 b Fo(b)2241 5502 y Fu(9)2298 5487 y Ft(sin)16 b(2)p Fo(')2547 5502 y Fu(2)2586 5487 y Ft(])23 b(+)f Fi(O)2832 5391 y Fh(\020)2882 5487 y Fi(k)p Fo(\030)5 b Fi(k)3030 5446 y Fu(3)p Fm(=)p Fu(2)3139 5391 y Fh(\021)3205 5487 y Fo(:)1782 5753 y Ft(22)p eop %%Page: 23 23 23 22 bop -118 407 a Ft(Inserting)48 b(this)f(expansion)i(in)m(to)e (the)h(in)m(tegral)f(and)h(taking)f(in)m(to)g(accoun)m(t)i(that)e(in)m (tegrals)g(of)h(o)s(dd)-118 527 y(functions)32 b(v)-5 b(anish,)33 b(and)g(using)f(also)f(that)41 b(~)-57 b Fo(!)1544 542 y Fm(i)1599 527 y Ft(=)28 b Fo(!)1764 542 y Fm(i)1814 527 y Ft(+)22 b Fi(O)s Ft(\()p Fi(k)p Fo(I)2133 491 y Fu(0)2172 527 y Fi(k)p Ft(\),)32 b(w)m(e)i(get)243 747 y Fo(L)p Ft(\()p Fo(I)398 706 y Fu(0)390 772 y(1)437 747 y Fo(;)17 b(I)532 706 y Fu(0)524 772 y(2)571 747 y Fo(;)g(\022)660 762 y Fu(1)700 747 y Fo(;)g(\022)789 762 y Fu(2)829 747 y Ft(\))326 892 y(=)83 b(2)p Fo(I)585 851 y Fu(0)577 917 y(1)624 892 y Ft([2\()p Fo(b)779 907 y Fu(1)841 892 y Ft(+)22 b Fo(b)980 907 y Fu(5)1019 892 y Ft(\))h(+)f Fi(J)1246 907 y Fu(2)p Fm(!)1325 916 y Fl(1)1363 892 y Ft(\(\()p Fo(b)1480 907 y Fu(1)1542 892 y Fi(\000)h Fo(b)1683 907 y Fu(5)1722 892 y Ft(\))17 b(cos)g(2)p Fo(\022)2018 907 y Fu(1)2080 892 y Ft(+)22 b Fo(b)2219 907 y Fu(2)2276 892 y Ft(sin)16 b(2)p Fo(\022)2506 907 y Fu(1)2546 892 y Ft(\)])485 1065 y(+2)610 957 y Fh(q)p 692 957 181 4 v 692 1065 a Fo(I)743 1031 y Fu(0)735 1087 y(1)783 1065 y Fo(I)834 1031 y Fu(0)826 1087 y(2)873 1065 y Ft([)p Fi(J)968 1080 y Fm(!)1012 1089 y Fl(1)1046 1080 y Fu(+)p Fm(!)1145 1089 y Fl(2)1183 1065 y Ft(\(\()p Fo(b)1300 1080 y Fu(3)1362 1065 y Fi(\000)23 b Fo(b)1503 1080 y Fu(7)1543 1065 y Ft(\))17 b(cos\()p Fo(\022)1811 1080 y Fu(1)1873 1065 y Ft(+)22 b Fo(\022)2016 1080 y Fu(2)2056 1065 y Ft(\))g(+)g(\()p Fo(b)2293 1080 y Fu(4)2355 1065 y Ft(+)g Fo(b)2494 1080 y Fu(6)2534 1065 y Ft(\))17 b(sin)o(\()p Fo(\022)2791 1080 y Fu(1)2853 1065 y Ft(+)22 b Fo(\022)2996 1080 y Fu(2)3036 1065 y Ft(\)\))922 1251 y(+)g Fi(J)1088 1266 y Fm(!)1132 1275 y Fl(1)1166 1266 y Fn(\000)p Fm(!)1265 1275 y Fl(2)1304 1251 y Ft(\(\()p Fo(b)1421 1266 y Fu(3)1483 1251 y Ft(+)g Fo(b)1622 1266 y Fu(7)1661 1251 y Ft(\))17 b(cos)q(\()p Fo(\022)1930 1266 y Fu(1)1992 1251 y Fi(\000)22 b Fo(\022)2136 1266 y Fu(2)2176 1251 y Ft(\))g Fi(\000)h Ft(\()p Fo(b)2415 1266 y Fu(4)2477 1251 y Fi(\000)f Fo(b)2617 1266 y Fu(6)2657 1251 y Ft(\))17 b(sin\()p Fo(\022)2915 1266 y Fu(1)2977 1251 y Fi(\000)22 b Fo(\022)3121 1266 y Fu(2)3161 1251 y Ft(\)\)])485 1422 y(+2)p Fo(I)661 1381 y Fu(0)653 1447 y(2)700 1422 y Ft([2\()p Fo(b)855 1437 y Fu(8)916 1422 y Ft(+)g Fo(b)1055 1437 y Fu(10)1131 1422 y Ft(\))g(+)g Fi(J)1357 1437 y Fu(2)p Fm(!)1436 1446 y Fl(2)1474 1422 y Ft(\(\()p Fo(b)1591 1437 y Fu(8)1653 1422 y Fi(\000)h Fo(b)1794 1437 y Fu(10)1869 1422 y Ft(\))17 b(cos)g(2)p Fo(\022)2165 1437 y Fu(2)2226 1422 y Ft(+)22 b Fo(b)2365 1437 y Fu(9)2422 1422 y Ft(sin)16 b(2)p Fo(\022)2652 1437 y Fu(2)2692 1422 y Ft(\)])22 b(+)g Fi(O)2976 1326 y Fh(\020)3025 1422 y Fi(k)p Fo(I)3126 1381 y Fu(0)3166 1422 y Fi(k)3216 1381 y Fu(3)p Fm(=)p Fu(2)3325 1326 y Fh(\021)3392 1422 y Fo(;)187 b Ft(\(32\))-118 1642 y(where)34 b(w)m(e)f(de\014ne)919 1779 y Fi(J)987 1794 y Fm(a)1056 1779 y Ft(=)1160 1662 y Fh(Z)1243 1688 y Fn(1)1206 1851 y(\0001)1383 1712 y Ft(cos)17 b Fo(at)p 1362 1756 277 4 v 1362 1852 a Ft(cosh)1547 1809 y Fu(2)1603 1852 y Fo(t)1648 1779 y(dt)28 b Ft(=)2049 1712 y Fo(\031)t(a)p 1875 1756 458 4 v 1875 1847 a Ft(sinh\()p Fo(\031)t(a=)p Ft(2\))2343 1779 y Fo(;)114 b(a)28 b Fi(6)p Ft(=)f(0)p Fo(;)864 b Ft(\(33\))-118 2009 y(computed)36 b(b)m(y)g(residue)g (theory)-8 b(.)53 b(W)-8 b(e)36 b(ha)m(v)m(e)g(also)f(used)i(that)e Fi(J)2257 2024 y Fu(0)2329 2009 y Ft(=)d(2.)52 b(The)36 b(Melnik)m(o)m(v)g(p)s(oten)m(tial)e(\(32\))-118 2130 y(b)s(ecomes)f(a)f(generic)h(function)f(of)g(the)h(t)m(yp)s(e)881 2350 y Fo(L)28 b Ft(=)g Fo(L)1145 2365 y Fu(2)1207 2350 y Ft(+)22 b Fi(O)1403 2253 y Fh(\020)1453 2350 y Fi(k)p Fo(I)1554 2309 y Fu(0)1593 2350 y Fi(k)1643 2309 y Fu(3)p Fm(=)p Fu(2)1753 2253 y Fh(\021)1819 2350 y Fo(;)1760 b Ft(\(34\))881 2508 y Fo(L)947 2523 y Fu(2)1015 2508 y Ft(=)27 b Fo(s)1164 2523 y Fu(0)1226 2508 y Ft(+)22 b Fo(s)1370 2523 y Fu(1)1426 2508 y Ft(cos\(2)p Fo(\022)1688 2523 y Fu(1)1750 2508 y Fi(\000)h Fo(\033)1905 2523 y Fu(1)1945 2508 y Ft(\))f(+)g Fo(s)2149 2523 y Fu(2)2205 2508 y Ft(cos\()p Fo(\022)2418 2523 y Fu(1)2480 2508 y Ft(+)g Fo(\022)2623 2523 y Fu(2)2685 2508 y Fi(\000)h Fo(\033)2840 2523 y Fu(2)2880 2508 y Ft(\))1226 2653 y(+)f Fo(s)1370 2668 y Fu(3)1426 2653 y Ft(cos\()p Fo(\022)1639 2668 y Fu(1)1701 2653 y Fi(\000)h Fo(\022)1846 2668 y Fu(2)1908 2653 y Fi(\000)g Fo(\033)2063 2668 y Fu(3)2102 2653 y Ft(\))f(+)g Fo(s)2306 2668 y Fu(4)2362 2653 y Ft(cos)q(\(2)p Fo(\022)2625 2668 y Fu(2)2687 2653 y Fi(\000)g Fo(\033)2841 2668 y Fu(4)2881 2653 y Ft(\))p Fo(;)-118 2895 y Ft(where)40 b(the)f(\(non-negativ)m(e\))g(co)s(e\016cien)m(ts)h Fo(s)1547 2910 y Fm(i)1614 2895 y Ft(are)e(homogeneous)h(linear)e(p)s (olynomials)f(in)i Fo(I)3382 2859 y Fu(0)3374 2920 y(1)3421 2895 y Ft(,)3489 2792 y Fh(q)p 3572 2792 181 4 v 103 x Fo(I)3623 2861 y Fu(0)3615 2917 y(1)3662 2895 y Fo(I)3713 2861 y Fu(0)3705 2917 y(2)3752 2895 y Ft(,)-118 3026 y Fo(I)-67 2990 y Fu(0)-75 3050 y(2)-28 3026 y Ft(,)d(and)f(the)h (phases)g Fo(\033)761 3041 y Fm(i)824 3026 y Ft(do)f(not)g(dep)s(end)i (on)e Fo(I)1665 2990 y Fu(0)1704 3026 y Ft(.)48 b(Since)35 b(w)m(e)g(are)f(in)m(terested)h(in)f(the)g(critical)e(p)s(oin)m(ts)i (of)-118 3146 y Fo(L)f Ft(with)f(resp)s(ect)i(to)e Fo(\022)701 3161 y Fu(1)741 3146 y Ft(,)h Fo(\022)846 3161 y Fu(2)885 3146 y Ft(,)g(w)m(e)h(can)f(ob)m(viously)f(assume)h Fo(s)2082 3161 y Fu(0)2149 3146 y Ft(=)28 b(0.)28 3266 y(By)f(Lemma)d(2,)j(if)e Fo(L)768 3281 y Fu(2)833 3266 y Ft(is)h(a)f(Morse)i(function)f(it)e (has)j(8,)g(12)e(or)h(16)f(critical)e(p)s(oin)m(ts.)41 b(This)26 b(result)g(can)g(b)s(e)-118 3387 y(extended)j(to)e(the)g (whole)g(Melnik)m(o)m(v)h(p)s(oten)m(tial)d Fo(L)j Ft(in)e(\(34\))h (pro)m(vided)g Fi(k)p Fo(I)2558 3351 y Fu(0)2598 3387 y Fi(k)f Ft(is)h(small)e(enough,)k(excluding)-118 3507 y(a)35 b(neigh)m(b)s(orho)s(o)s(d)f(of)g(the)i(ra)m(ys)g Fo(I)1125 3471 y Fu(0)1117 3532 y(1)1164 3507 y Fo(=I)1264 3471 y Fu(0)1256 3532 y(2)1335 3507 y Ft(=)c(const)k(where)g(the)g (Morse)g(condition)e(is)g(not)h(ful\014lled.)50 b(As)35 b(in)-118 3628 y(Theorem)e(1,)f(the)h(critical)d(p)s(oin)m(ts)j(tend)g (to)f(a)g(constan)m(t)i(along)d(eac)m(h)i(ra)m(y)-8 b(,)33 b(as)g(\()p Fo(I)2880 3591 y Fu(0)2872 3652 y(1)2919 3628 y Fo(;)17 b(I)3014 3591 y Fu(0)3006 3652 y(2)3053 3628 y Ft(\))28 b Fi(!)f Ft(\(0)p Fo(;)17 b Ft(0\).)p 3534 3628 42 42 v 28 3798 a(Although)35 b(this)g(theorem)g(do)s(es)h (not)g(giv)m(e)f(explicit)f(conditions)g(determining)g(the)i(n)m(um)m (b)s(er)g(of)f(ho-)-118 3918 y(mo)s(clinic)41 b(orbits,)46 b(it)e(is)f(p)s(ossible)h(to)g(study)h(this)f(in)f(concrete)j (examples.)78 b(As)45 b(an)f(illustration,)f(w)m(e)-118 4039 y(consider)33 b(the)g(p)s(erturbation)f(\(30\))f(with)i(the)g (function)-8 4308 y Fo(f)38 b Ft(=)285 4241 y(1)p 192 4285 235 4 v 192 4376 a(8)p Fi(J)309 4391 y Fu(2)p Fm(!)388 4400 y Fl(1)436 4308 y Ft(\()p Fo(x)529 4267 y Fu(2)529 4333 y(1)591 4308 y Fi(\000)23 b Fo(y)743 4267 y Fu(2)739 4333 y(1)781 4308 y Ft(\))f(+)1085 4241 y Fo(\014)p 949 4285 333 4 v 949 4376 a Ft(4)p Fi(J)1066 4391 y Fm(!)1110 4400 y Fl(1)1144 4391 y Fu(+)p Fm(!)1243 4400 y Fl(2)1291 4308 y Ft(\()p Fo(x)1384 4323 y Fu(1)1424 4308 y Fo(x)1479 4323 y Fu(2)1541 4308 y Fi(\000)h Fo(y)1689 4323 y Fu(1)1728 4308 y Fo(y)1776 4323 y Fu(2)1815 4308 y Ft(\))f(+)2119 4241 y Fo(\014)p 1983 4285 V 1983 4376 a Ft(4)p Fi(J)2100 4391 y Fm(!)2144 4400 y Fl(1)2178 4391 y Fn(\000)p Fm(!)2277 4400 y Fl(2)2325 4308 y Ft(\()p Fo(x)2418 4323 y Fu(1)2458 4308 y Fo(x)2513 4323 y Fu(2)2575 4308 y Ft(+)g Fo(y)2721 4323 y Fu(1)2760 4308 y Fo(y)2808 4323 y Fu(2)2847 4308 y Ft(\))g(+)3108 4241 y(1)p 3015 4285 235 4 v 3015 4376 a(8)p Fi(J)3132 4391 y Fu(2)p Fm(!)3211 4400 y Fl(2)3259 4308 y Ft(\()p Fo(x)3352 4267 y Fu(2)3352 4333 y(2)3414 4308 y Fi(\000)h Fo(y)3566 4267 y Fu(2)3562 4333 y(2)3604 4308 y Ft(\))p Fo(;)-118 4583 y Ft(with)34 b Fo(!)167 4598 y Fu(1)206 4583 y Ft(,)g Fo(!)328 4598 y Fu(2)402 4583 y Ft(\014xed,)h(and)g Fo(\014)g(>)c Ft(0)i(as)i(a)f(parameter)f (\(recall)g(that)h Fi(J)2390 4598 y Fm(a)2465 4583 y Ft(has)g(b)s(een)h(de\014ned)h(in)d(\(33\)\).)48 b(The)-118 4703 y(Melnik)m(o)m(v)33 b(p)s(oten)m(tial)e(in)h(\(32\))f(b)s(ecomes) 246 4968 y Fo(L)d Ft(=)g Fo(L)510 4983 y Fu(2)577 4968 y Ft(=)681 4860 y Fh(q)p 764 4860 181 4 v 108 x Fo(I)815 4934 y Fu(0)807 4990 y(1)854 4968 y Fo(I)905 4934 y Fu(0)897 4990 y(2)961 4847 y Fh(\022)1033 4901 y Fo(r)p 1032 4945 49 4 v 1032 5036 a Ft(2)1107 4968 y(cos)18 b(2)p Fo(\022)1349 4983 y Fu(1)1410 4968 y Ft(+)k Fo(\014)h Ft(cos\()p Fo(\022)1799 4983 y Fu(1)1861 4968 y Ft(+)f Fo(\022)2004 4983 y Fu(2)2044 4968 y Ft(\))g(+)g Fo(\014)g Ft(cos)q(\()p Fo(\022)2493 4983 y Fu(1)2555 4968 y Fi(\000)h Fo(\022)2700 4983 y Fu(2)2740 4968 y Ft(\))f(+)2931 4901 y(1)p 2908 4945 96 4 v 2908 5036 a(2)p Fo(r)3030 4968 y Ft(cos)17 b(2)p Fo(\022)3271 4983 y Fu(2)3310 4847 y Fh(\023)3388 4968 y Fo(;)191 b Ft(\(35\))-118 5260 y(with)35 b Fo(r)f Ft(=)293 5159 y Fh(q)p 376 5159 230 4 v 101 x Fo(I)427 5226 y Fu(0)419 5282 y(1)467 5260 y Fo(=I)567 5226 y Fu(0)559 5282 y(2)606 5260 y Ft(.)51 b(Remo)m(ving)34 b(the)h(term)1549 5157 y Fh(q)p 1632 5157 181 4 v 103 x Fo(I)1683 5226 y Fu(0)1675 5282 y(1)1722 5260 y Fo(I)1773 5226 y Fu(0)1765 5282 y(2)1848 5260 y Ft(\(that)f(do)s(es)i(not)f(in\015uence)h(the)f (critical)e(p)s(oin)m(ts)i(of)-118 5380 y Fo(L)p Ft(,)d(unless)g(it)e (v)-5 b(anishes\),)33 b(the)e(co)s(e\016cien)m(ts)i(considered)f(in)f (the)h(pro)s(of)e(of)h(Lemma)f(2)h(b)s(ecome)h Fo(s)3438 5395 y Fu(1)3505 5380 y Ft(=)27 b Fo(r)s(=)p Ft(2,)1782 5753 y(23)p eop %%Page: 24 24 24 23 bop -118 407 a Fo(s)-72 422 y Fu(2)7 407 y Ft(=)40 b Fo(s)169 422 y Fu(3)249 407 y Ft(=)g Fo(\014)6 b Ft(,)41 b Fo(s)540 422 y Fu(4)620 407 y Ft(=)f(1)p Fo(=)p Ft(2)p Fo(r)s Ft(,)h(and)f(the)g(phases)h(are)f(all)e Fo(\033)2055 422 y Fm(i)2124 407 y Ft(=)i(0.)65 b(After)40 b(the)g(linear)e(c)m (hange)j(\(16\),)g(w)m(e)-118 527 y(obtain)198 495 y Fh(e)185 527 y Fo(L)251 542 y Fu(2)291 527 y Ft(\()p Fo( )392 542 y Fu(1)432 527 y Fo(;)17 b( )539 542 y Fu(2)579 527 y Ft(\))32 b(as)h(in)f(\(17\),)g(with)1155 747 y Fo(\025)1212 762 y Fu(0)1252 747 y Ft(\()p Fo( )1353 762 y Fu(2)1392 747 y Ft(\))c(=)f Fo(\014)c Ft(cos)17 b Fo( )1849 762 y Fu(2)1889 747 y Fo(;)1155 937 y(\025)1212 952 y Fu(1)1252 937 y Ft(\()p Fo( )1353 952 y Fu(2)1392 937 y Ft(\))28 b(=)1572 870 y Fo(r)p 1571 914 49 4 v 1571 1005 a Ft(2)1652 937 y(+)22 b Fo(\014)h Ft(cos)17 b Fo( )2038 952 y Fu(2)2100 937 y Ft(+)2231 870 y(1)p 2208 914 96 4 v 2208 1005 a(2)p Fo(r)2330 937 y Ft(cos)g(2)p Fo( )2589 952 y Fu(2)2629 937 y Fo(;)1155 1170 y(\025)1212 1185 y Fu(2)1252 1170 y Ft(\()p Fo( )1353 1185 y Fu(2)1392 1170 y Ft(\))28 b(=)f Fo(\014)c Ft(sin)16 b Fo( )1838 1185 y Fu(2)1900 1170 y Ft(+)2031 1103 y(1)p 2008 1147 V 2008 1238 a(2)p Fo(r)2130 1170 y Ft(sin)g(2)p Fo( )2378 1185 y Fu(2)2418 1170 y Fo(:)-118 1421 y Ft(After)35 b(some)g(computations,)f(w)m(e)i(see)h(that)d(the)i(trigonometric)c(p)s (olynomial)g(in)i(\(23\))g(can)h(b)s(e)h(written)-118 1541 y(as)573 1681 y Fo(g)t Ft(\()p Fo( )725 1696 y Fu(2)764 1681 y Ft(\))27 b(=)h(\(1)22 b Fi(\000)g Fo(\014)1202 1640 y Fu(2)1241 1681 y Ft(\))17 b(sin)1416 1640 y Fu(2)1472 1681 y Fo( )1535 1696 y Fu(2)1591 1560 y Fh(\022)1652 1681 y Ft(cos)1783 1640 y Fu(2)1839 1681 y Fo( )1902 1696 y Fu(2)1964 1681 y Ft(+)22 b Fo(\014)2139 1560 y Fh(\022)2200 1681 y Fo(r)j Ft(+)2377 1614 y(1)p 2377 1658 49 4 v 2378 1750 a Fo(r)2436 1560 y Fh(\023)2514 1681 y Ft(cos)17 b Fo( )2724 1696 y Fu(2)2786 1681 y Ft(+)22 b Fo(\014)2945 1640 y Fu(2)2984 1560 y Fh(\023)3062 1681 y Fo(:)-118 1893 y Ft(It)27 b(is)g(clear)g(that)f Fo(g)t Ft(\()p Fo( )660 1908 y Fu(2)700 1893 y Ft(\))h(has)g Fo( )996 1908 y Fu(2)1064 1893 y Ft(=)g(0)p Fo(;)17 b(\031)31 b Ft(as)c(double)g(zero)s(es,)i(whic)m(h)f(giv)m(e)f(rise)g(from)f (\(20{21\))g(to)h(4)g(critical)-118 2013 y(p)s(oin)m(ts:)53 b(\(0)p Fo(;)17 b Ft(0\),)37 b(\(0)p Fo(;)17 b(\031)t Ft(\),)38 b(\()p Fo(\031)t(;)17 b Ft(0\),)38 b(\()p Fo(\031)t(;)17 b(\031)t Ft(\),)38 b(for)f(the)h(function)2120 1981 y Fh(e)2108 2013 y Fo(L)2174 2028 y Fu(2)2213 2013 y Ft(.)58 b(Besides,)40 b(there)e(can)g(exist)f(additional)-118 2134 y(critical)25 b(p)s(oin)m(ts)j(for)f(this)h(function,)g(asso)s (ciated)g(to)g(zero)s(es)h Fo( )2150 2149 y Fu(2)2217 2134 y Ft(giv)m(en)f(b)m(y)h(cos)17 b Fo( )2808 2149 y Fu(2)2876 2134 y Ft(=)27 b Fi(\000)p Fo(\014)6 b(=r)31 b Ft(and)d(cos)17 b Fo( )3636 2149 y Fu(2)3704 2134 y Ft(=)-118 2254 y Fi(\000)p Fo(\014)6 b(r)s Ft(.)74 b(The)44 b(n)m(um)m(b)s(er)f(of)g(suc)m(h)h(zero)s(es)g(can)f(b)s(e)g(0,)i(2)e (or)f(4)h(\(excluding)f(degenerate)j(cases\),)h(and)d(w)m(e)-118 2375 y(deduce)30 b(that)e(the)g(Melnik)m(o)m(v)h(p)s(oten)m(tial)d (\(35\))i(ma)m(y)g(ha)m(v)m(e)h(8,)g(12)f(or)g(16)f(critical)f(p)s(oin) m(ts.)42 b(More)28 b(precisely)-8 b(,)-118 2495 y(w)m(e)34 b(ha)m(v)m(e:)27 2698 y Fi(\017)49 b Ft(F)-8 b(or)29 b(0)f Fo(<)f(\014)33 b(<)28 b Ft(1,)i(there)h(are)f(16)g(critical)e(p)s (oin)m(ts)h(if)g Fo(\014)34 b(<)27 b(r)k(<)c Ft(1)p Fo(=\014)6 b Ft(,)30 b Fo(r)g Fi(6)p Ft(=)e(1,)i(and)g(12)g(critical)e(p)s(oin)m (ts)126 2819 y(if)j Fo(r)g(<)c(\014)38 b Ft(or)32 b Fo(r)f(>)c Ft(1)p Fo(=\014)6 b Ft(.)27 3022 y Fi(\017)49 b Ft(F)-8 b(or)39 b Fo(\014)46 b(>)41 b Ft(1,)h(there)f(are)f(12)g(critical)e(p)s (oin)m(ts)i(if)f Fo(r)k(<)e Ft(1)p Fo(=\014)k Ft(or)40 b Fo(r)j(>)e(\014)6 b Ft(,)42 b(and)e(8)g(critical)e(p)s(oin)m(ts)i(if) 126 3143 y(1)p Fo(=\014)33 b(<)27 b(r)j(<)e(\014)6 b Ft(,)32 b Fo(r)f Fi(6)p Ft(=)c(1.)-118 3346 y(Assuming)46 b Fo(")g Ft(small)e(enough,)50 b(eac)m(h)d(critical)d(p)s(oin)m(t)i (giv)m(es)g(rise)g(to)g(a)g(homo)s(clinic)e(orbit,)k(though)f(a)-118 3481 y(neigh)m(b)s(orho)s(o)s(d)31 b(of)h(the)h(ra)m(ys)980 3380 y Fh(q)p 1063 3380 230 4 v 101 x Fo(I)1114 3447 y Fu(0)1106 3503 y(1)1153 3481 y Fo(=I)1253 3447 y Fu(0)1245 3503 y(2)1320 3481 y Ft(=)27 b(0)p Fo(;)17 b Ft(1)p Fo(=\014)6 b(;)17 b Ft(1)p Fo(;)g(\014)6 b(;)17 b Fi(1)30 b Ft(ha)m(v)m(e)k(to)e (b)s(e)h(excluded.)28 3652 y(In)46 b(the)g(next)g(theorem)f(w)m(e)i (study)f(a)g(more)e(general)h(case)h(\(i.e.)f(not)h(assuming)e Fo(f)61 b Ft(=)49 b Fi(O)3469 3667 y Fu(2)3509 3652 y Ft(\),)f(and)-118 3772 y(establish)32 b(the)h(n)m(um)m(b)s(er)g(of)f (transv)m(erse)j(homo)s(clinic)29 b(orbits,)j(in)g(a)g(m)m(uc)m(h)i (simpler)d(w)m(a)m(y)-8 b(.)-118 4000 y Fk(Theorem)37 b(5)49 b Fj(Consider)29 b(an)g(arbitr)-5 b(ary)30 b Fo(f)41 b Fj(in)30 b(the)f(p)-5 b(erturb)g(ation)30 b(\(30\).)43 b(Under)30 b(generic)f(c)-5 b(onditions)29 b(the)-118 4121 y(c)-5 b(o)g(e\016cients)34 b(of)g(or)-5 b(der)35 b(1)g(of)f Fo(f)11 b Fj(,)35 b(for)g(an)f(op)-5 b(en)35 b(subset)g(of)f(actions)g Fo(I)2388 4085 y Fu(0)2456 4121 y Ft(=)27 b(\()p Fo(I)2648 4085 y Fu(0)2640 4145 y(1)2687 4121 y Fo(;)17 b(I)2782 4085 y Fu(0)2774 4145 y(2)2821 4121 y Ft(\))p Fj(,)35 b(and)g Fo(")f Fj(smal)5 b(l)34 b(enough,)-118 4241 y(ther)-5 b(e)35 b(exist)f(exactly)h(4)g(tr) -5 b(ansverse)34 b(homo)-5 b(clinic)33 b(orbits.)-118 4469 y Fk(Pro)s(of.)43 b Ft(Let)598 4590 y Fo(f)38 b Ft(=)28 b Fo(a)839 4605 y Fu(0)901 4590 y Ft(+)22 b Fo(a)1050 4605 y Fu(11)1125 4590 y Fo(x)1180 4605 y Fu(1)1242 4590 y Ft(+)g Fo(a)1391 4605 y Fu(12)1466 4590 y Fo(y)1514 4605 y Fu(1)1575 4590 y Ft(+)g Fo(a)1724 4605 y Fu(21)1799 4590 y Fo(x)1854 4605 y Fu(2)1916 4590 y Ft(+)g Fo(a)2065 4605 y Fu(22)2140 4590 y Fo(y)2188 4605 y Fu(2)2249 4590 y Ft(+)g Fi(O)2426 4605 y Fu(2)2466 4590 y Ft(\()p Fo(x)2559 4605 y Fu(1)2598 4590 y Fo(;)17 b(y)2690 4605 y Fu(1)2729 4590 y Fo(;)g(x)2828 4605 y Fu(2)2868 4590 y Fo(;)g(y)2960 4605 y Fu(2)2998 4590 y Ft(\))p Fo(:)543 b Ft(\(36\))-118 4764 y(Computing)31 b(the)i(Melnik)m(o)m(v)g(p)s(oten)m(tial)e(as)i(in) f(the)h(pro)s(of)e(of)i(Theorem)f(4,)h(w)m(e)g(obtain)-61 5006 y Fo(L)28 b Ft(=)g(4)p Fo(a)237 5021 y Fu(0)298 5006 y Ft(+)22 b(2)445 4898 y Fh(q)p 528 4898 140 4 v 108 x Ft(2)p Fo(I)628 4972 y Fu(0)620 5028 y(1)667 5006 y Fi(J)735 5021 y Fm(!)779 5030 y Fl(1)817 5006 y Ft(\()p Fo(a)906 5021 y Fu(11)998 5006 y Ft(cos)17 b Fo(\022)1190 5021 y Fu(1)1252 5006 y Ft(+)22 b Fo(a)1401 5021 y Fu(12)1492 5006 y Ft(sin)17 b Fo(\022)1674 5021 y Fu(1)1714 5006 y Ft(\))22 b(+)g(2)1921 4898 y Fh(q)p 2004 4898 V 108 x Ft(2)p Fo(I)2104 4972 y Fu(0)2096 5028 y(2)2143 5006 y Fi(J)2211 5021 y Fm(!)2255 5030 y Fl(2)2293 5006 y Ft(\()p Fo(a)2382 5021 y Fu(21)2473 5006 y Ft(cos)17 b Fo(\022)2665 5021 y Fu(2)2728 5006 y Ft(+)22 b Fo(a)2877 5021 y Fu(22)2968 5006 y Ft(sin)16 b Fo(\022)3149 5021 y Fu(2)3189 5006 y Ft(\))22 b(+)g Fi(O)s Ft(\()p Fi(k)p Fo(I)3568 4965 y Fu(0)3608 5006 y Fi(k)p Ft(\))p Fo(;)-118 5226 y Ft(and)36 b(it)e(has)i(4)g(critical)d(p)s(oin)m(ts)i(pro)m (vided)h Fo(a)1516 5190 y Fu(2)1516 5251 y(11)1616 5226 y Ft(+)24 b Fo(a)1767 5190 y Fu(2)1767 5251 y(12)1875 5226 y Fi(6)p Ft(=)32 b(0)k(and)g Fo(a)2312 5190 y Fu(2)2312 5251 y(21)2411 5226 y Ft(+)24 b Fo(a)2562 5190 y Fu(2)2562 5251 y(22)2670 5226 y Fi(6)p Ft(=)32 b(0)k(\(compare)f(with)g(condi-) -118 5346 y(tion)f(\()p Fk(LLL)p Ft(\))i(in)f(Section)g(4\).)51 b(Suc)m(h)36 b(critical)d(p)s(oin)m(ts)i(tend)h(to)e(constan)m(t)j(as)e (\()p Fo(I)2840 5310 y Fu(0)2832 5371 y(1)2879 5346 y Fo(;)17 b(I)2974 5310 y Fu(0)2966 5371 y(2)3013 5346 y Ft(\))32 b Fi(!)g Ft(\(0)p Fo(;)17 b Ft(0\))34 b(\(in)h(this)-118 5467 y(case,)e(the)g(constan)m(t)h(is)e(the)h(same)f(for)g(all)f(ra)m (ys)i Fo(I)1729 5430 y Fu(0)1721 5491 y(1)1769 5467 y Fo(=I)1869 5430 y Fu(0)1861 5491 y(2)1935 5467 y Ft(=)28 b(const)q(\).)p 2370 5467 42 42 v 1782 5753 a(24)p eop %%Page: 25 25 25 24 bop 28 407 a Ft(Although)37 b(the)g Fo(\022)681 422 y Fm(i)710 407 y Ft(-deriv)-5 b(ativ)m(es)37 b(of)g Fo(L)g Ft(tend)h(to)f(0)g(as)h(\()p Fo(I)2103 371 y Fu(0)2095 431 y(1)2142 407 y Fo(;)17 b(I)2237 371 y Fu(0)2229 431 y(2)2276 407 y Ft(\))36 b Fi(!)f Ft(\(0)p Fo(;)17 b Ft(0\),)38 b(this)f(do)s(es)g(not)h(imply)d(the)-118 527 y(p)s(ersistence)j(of)f (the)g(homo)s(clinic)d(lo)s(op)h(\000)i(for)f Fo(")f Fi(6)p Ft(=)g(0,)i(since)h(the)f(splitting)e(of)h(this)h(lo)s(op)e(ma)m (y)i(app)s(ear)-118 648 y(in)h(the)h(higher)f(order)h(terms)f(in)g (\(27\).)61 b(In)39 b(the)g(same)f(w)m(a)m(y)-8 b(,)42 b(the)d(4)f(critical)e(p)s(oin)m(ts)i(predicted)i(b)m(y)f(the)-118 768 y(Melnik)m(o)m(v)27 b(p)s(oten)m(tial)e(ma)m(y)i(not)f(giv)m(e)h (true)g(homo)s(clinic)c(in)m(tersections)28 b(if)d(\()p Fo(I)2683 732 y Fu(0)2675 793 y(1)2722 768 y Fo(;)17 b(I)2817 732 y Fu(0)2809 793 y(2)2856 768 y Ft(\))27 b(is)f(v)m(ery)i(close)f(to)f(\(0)p Fo(;)17 b Ft(0\))-118 888 y(\(to)36 b(b)s(e)h(more)e(precise,)k(a)d(neigh)m(b)s(orho)s(o)s(d) f(of)h(the)h(lines)f Fo(I)2038 903 y Fu(1)2112 888 y Ft(=)e(0,)j Fo(I)2378 903 y Fu(2)2452 888 y Ft(=)d(0,)j(of)f(width)h Fi(O)s Ft(\()p Fo(")3237 852 y Fu(2)3276 888 y Ft(\),)g(has)g(to)f(b)s (e)-118 1009 y(excluded)45 b(from)d(the)i(op)s(en)g(set)g(in)f(the)h (statemen)m(t)g(of)f(Theorem)h(5\).)76 b(Th)m(us,)48 b(the)c(usual)g(P)m(oincar)m(\023)-46 b(e{)-118 1129 y(Melnik)m(o)m(v)34 b(metho)s(d,)g(as)g(in)m(tro)s(duced)g(at)f(the)i (b)s(eginning)d(of)i(this)f(section,)i(do)s(es)f(not)g(apply)f(in)g(a)h (small)-118 1249 y(neigh)m(b)s(orho)s(o)s(d)d(of)h(the)h(lo)s(op.)-118 1579 y Fp(6)161 b(A)54 b(\014rst)e(appro)l(ximation)i(for)g(the)f (splitting)h(of)g(the)f(lo)t(op)-118 1798 y Ft(Finally)-8 b(,)22 b(w)m(e)i(sho)m(w)g(that)f(the)g(splitting)e(of)h(the)i(lo)s(op) d(\000)i(can)g(b)s(e)h(detected)g(with)f(a)f(mo)s(di\014ed)g(v)m (ersion)i(of)e(the)-118 1918 y(P)m(oincar)m(\023)-46 b(e{Melnik)m(o)m(v)34 b(metho)s(d,)f(that)h(can)f(b)s(e)h(applied)f(to) g(the)h(examples)g(in)m(tro)s(duced)f(in)g(Section)h(1.1.)-118 2038 y(Due)24 b(to)g(the)h(degeneracy)i(of)d(the)g(p)s(olar)f(co)s (ordinates)h(de\014ned)i(in)e(\(5\),)i(w)m(e)f(dev)m(elop)g(the)g (metho)s(d)f(directly)-118 2159 y(in)32 b(the)h(original)c(co)s (ordinates)j Fo(z)g Ft(=)c(\()p Fo(x)1308 2174 y Fu(1)1348 2159 y Fo(;)17 b(y)1440 2174 y Fu(1)1479 2159 y Fo(;)g(x)1578 2174 y Fu(2)1617 2159 y Fo(;)g(y)1709 2174 y Fu(2)1748 2159 y Fo(;)g(p;)g(q)t Ft(\).)28 2279 y(W)-8 b(e)49 b(are)g(going)f(to) g(obtain)g(a)g(\014rst)h(appro)m(ximation)e(for)h(the)i(splitting)c (distance)j(b)s(et)m(w)m(een)i(the)-118 2400 y(\(global\))33 b(in)m(v)-5 b(arian)m(t)35 b(manifolds)f Fo(W)1213 2363 y Fm(u)1257 2400 y Ft(,)j Fo(W)1427 2363 y Fm(s)1500 2400 y Ft(inherited)e(from)g(the)h(initial)c(lo)s(op)j(\000.)54 b(The)37 b(unstable)f(man-)-118 2520 y(ifold)k Fo(W)217 2484 y Fm(u)304 2520 y Ft(is)i(giv)m(en)g(b)m(y)i(a)e(tra)5 b(jectory)43 b Fo(z)1427 2484 y Fm(u)1472 2520 y Ft(\()p Fo(t)p Ft(\))i(=)f(\()p Fo(x)1841 2484 y Fm(u)1841 2545 y Fu(1)1886 2520 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(y)2093 2484 y Fm(u)2089 2545 y Fu(1)2137 2520 y Ft(\()p Fo(t)p Ft(\))p Fo(;)g(x)2347 2484 y Fm(u)2347 2545 y Fu(2)2393 2520 y Ft(\()p Fo(t)p Ft(\))p Fo(;)g(y)2600 2484 y Fm(u)2596 2545 y Fu(2)2644 2520 y Ft(\()p Fo(t)p Ft(\))p Fo(;)g(p)2848 2484 y Fm(u)2892 2520 y Ft(\()p Fo(t)p Ft(\))p Fo(;)g(q)3094 2484 y Fm(u)3139 2520 y Ft(\()p Fo(t)p Ft(\)\))42 b(that)g(tends)-118 2640 y(to)i(the)h(singular)f(p)s(oin)m(t)f Fo(O)k Ft(for)e Fo(t)j Fi(!)g(\0001)p Ft(,)g(and)d(analogously)e(with)h(the)i(stable)e (manifold)e Fo(W)3582 2604 y Fm(s)3663 2640 y Ft(for)-118 2761 y Fo(t)57 b Fi(!)f(1)p Ft(.)95 b(Both)49 b(tra)5 b(jectories)50 b Fo(z)1190 2725 y Fm(u)1236 2761 y Ft(\()p Fo(t)p Ft(\),)k Fo(z)1477 2725 y Fm(s)1514 2761 y Ft(\()p Fo(t)p Ft(\))c(are)g Fi(O)s Ft(\()p Fo(")p Ft(\)-p)s(erturbations)e(of) i(the)g(lo)s(op)e(\000,)54 b(giv)m(en)c(b)m(y)-118 2881 y Fo(z)-73 2896 y Fu(0)-33 2881 y Ft(\()p Fo(t)p Ft(\))32 b(=)g(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)33 b Ft(_)-43 b Fo(q)671 2896 y Fu(0)708 2881 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(q)906 2896 y Fu(0)946 2881 y Ft(\()p Fo(t)p Ft(\)\))35 b(\(notice)g(that)f(this)h(is)g(the)g (limit)d(of)j(the)g(homo)s(clinic)d(tra)5 b(jectory)36 b(\(25\))-118 3001 y(as)d(\()p Fo(I)91 2965 y Fu(0)83 3026 y(1)130 3001 y Fo(;)17 b(I)225 2965 y Fu(0)217 3026 y(2)264 3001 y Ft(\))27 b Fi(!)h Ft(\(0)p Fo(;)17 b Ft(0\),)32 b(for)g(an)m(y)h(\()p Fo(\022)1150 3016 y Fu(1)1190 3001 y Fo(;)17 b(\022)1279 3016 y Fu(2)1318 3001 y Ft(\)\).)28 3122 y(T)-8 b(o)24 b(giv)m(e)g(a)g(measure)g(for)g(the)g(splitting)e (distance)i(b)s(et)m(w)m(een)i Fo(W)2321 3086 y Fm(u)2390 3122 y Ft(and)e Fo(W)2677 3086 y Fm(s)2713 3122 y Ft(,)i(w)m(e)f(can)f (restrict)g(ourselv)m(es)-118 3242 y(to)43 b(the)h(co)s(ordinates)f(\() p Fo(x)813 3257 y Fu(1)853 3242 y Fo(;)17 b(y)945 3257 y Fu(1)983 3242 y Fo(;)g(x)1082 3257 y Fu(2)1122 3242 y Fo(;)g(y)1214 3257 y Fu(2)1253 3242 y Ft(\).)76 b(Indeed,)47 b(the)d(distance)g(along)e Fo(p)h Ft(is)g(then)h(determined)f(\(on)h(a) -118 3363 y(section)33 b Fo(q)e Ft(=)d(const\))33 b(b)m(y)g(the)g (conserv)-5 b(ation)33 b(of)f(energy)-8 b(.)-118 3562 y Fk(Theorem)37 b(6)49 b Fj(Consider)29 b(in)i(the)g(p)-5 b(erturb)g(ation)31 b(\(30\))f(an)g(arbitr)-5 b(ary)31 b Fo(f)11 b Fj(,)31 b(written)g(as)g(in)f(\(36\).)43 b(Then,)31 b(for)-118 3683 y Fo(")j Fj(smal)5 b(l)34 b(enough)h(the)g(fol)5 b(lowing)33 b(appr)-5 b(oximations)33 b(hold:)512 3878 y Fo(x)567 3837 y Fm(u)567 3903 y(i)613 3878 y Ft(\()p Fo(t)p Ft(\))22 b Fi(\000)h Fo(x)901 3837 y Fm(s)901 3903 y(i)938 3878 y Ft(\()p Fo(t)p Ft(\))28 b(=)f Fi(\000)p Ft(2)p Fo(")p Fi(J)1420 3893 y Fm(!)1464 3903 y Fg(i)1494 3878 y Ft(\()p Fo(a)1583 3893 y Fm(i)p Fu(1)1663 3878 y Ft(cos)17 b Fo(!)1871 3893 y Fm(i)1899 3878 y Fo(t)23 b Fi(\000)f Fo(a)2107 3893 y Fm(i)p Fu(2)2188 3878 y Ft(sin)16 b Fo(!)2385 3893 y Fm(i)2413 3878 y Fo(t)p Ft(\))22 b(+)g Fi(O)s Ft(\()p Fo(")2772 3837 y Fu(2)2811 3878 y Ft(\))p Fo(;)730 b Ft(\(37\))512 4023 y Fo(y)564 3982 y Fm(u)560 4048 y(i)608 4023 y Ft(\()p Fo(t)p Ft(\))23 b Fi(\000)f Fo(y)893 3982 y Fm(s)889 4048 y(i)929 4023 y Ft(\()p Fo(t)p Ft(\))28 b(=)g Fi(\000)p Ft(2)p Fo(")p Fi(J)1412 4038 y Fm(!)1456 4048 y Fg(i)1485 4023 y Ft(\()p Fo(a)1574 4038 y Fm(i)p Fu(1)1655 4023 y Ft(sin)16 b Fo(!)1852 4038 y Fm(i)1880 4023 y Fo(t)22 b Ft(+)g Fo(a)2086 4038 y Fm(i)p Fu(2)2167 4023 y Ft(cos)17 b Fo(!)2375 4038 y Fm(i)2403 4023 y Fo(t)p Ft(\))22 b(+)g Fi(O)s Ft(\()p Fo(")2762 3982 y Fu(2)2801 4023 y Ft(\))p Fo(;)116 b(i)28 b Ft(=)g(1)p Fo(;)17 b Ft(2)p Fo(:)290 b Ft(\(38\))-118 4223 y Fk(Pro)s(of.)68 b Ft(W)-8 b(e)41 b(\014rst)h(consider)f(the)g(unstable)h(manifold)c Fo(W)2121 4187 y Fm(u)2165 4223 y Ft(.)69 b(W)-8 b(e)41 b(write)g Fo(x)2750 4187 y Fm(u)2750 4248 y(i)2795 4223 y Ft(\()p Fo(t)p Ft(\))h(=)g Fo("x)3167 4176 y Fm(u;)p Fn(\003)3167 4246 y Fm(i)3267 4223 y Ft(\()p Fo(t)p Ft(\))28 b(+)f Fi(O)s Ft(\()p Fo(")3675 4187 y Fu(2)3714 4223 y Ft(\),)-118 4344 y Fo(y)-66 4307 y Fm(u)-70 4368 y(i)-22 4344 y Ft(\()p Fo(t)p Ft(\))h(=)f Fo("y)318 4296 y Fm(u;)p Fn(\003)314 4366 y Fm(i)417 4344 y Ft(\()p Fo(t)p Ft(\))22 b(+)g Fi(O)s Ft(\()p Fo(")814 4307 y Fu(2)853 4344 y Ft(\),)33 b(whic)m(h)g(satisfy)g(the)g(Hamiltonian)28 b(equations)623 4588 y(_)-44 b Fo(x)661 4547 y Fm(u)661 4613 y(i)734 4588 y Ft(=)28 b Fi(\000)p Fo(!)976 4603 y Fm(i)1004 4588 y Fo(y)1056 4547 y Fm(u)1052 4613 y(i)1123 4588 y Ft(+)22 b Fi(O)1300 4603 y Fu(3)1339 4588 y Ft(\()p Fo(x)1432 4547 y Fm(u)1432 4613 y Fu(1)1478 4588 y Fo(;)17 b(y)1574 4547 y Fm(u)1570 4613 y Fu(1)1618 4588 y Fo(;)g(x)1717 4547 y Fm(u)1717 4613 y Fu(2)1762 4588 y Fo(;)g(y)1858 4547 y Fm(u)1854 4613 y Fu(2)1902 4588 y Ft(\))22 b Fi(\000)g Fo(")2117 4521 y(@)5 b(H)2254 4536 y Fu(1)p 2117 4565 178 4 v 2135 4656 a Fo(@)g(x)2246 4671 y Fm(i)2304 4588 y Ft(\()p Fo(x)2397 4547 y Fm(u)2397 4613 y Fu(1)2442 4588 y Fo(;)17 b(y)2538 4547 y Fm(u)2534 4613 y Fu(1)2582 4588 y Fo(;)g(x)2681 4547 y Fm(u)2681 4613 y Fu(2)2726 4588 y Fo(;)g(y)2822 4547 y Fm(u)2818 4613 y Fu(2)2866 4588 y Fo(;)g(p)2959 4547 y Fm(u)3004 4588 y Fo(;)g(q)3095 4547 y Fm(u)3140 4588 y Ft(\))p Fo(;)624 4841 y Ft(_)-45 b Fo(y)658 4800 y Fm(u)654 4866 y(i)730 4841 y Ft(=)28 b Fo(!)895 4856 y Fm(i)923 4841 y Fo(x)978 4800 y Fm(u)978 4866 y(i)1045 4841 y Ft(+)22 b Fi(O)1222 4856 y Fu(3)1262 4841 y Ft(\()p Fo(x)1355 4800 y Fm(u)1355 4866 y 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Fm(u;)p Fn(\003)-8 5472 y Fm(i)120 5450 y Ft(=)28 b Fi(\000)p Fo(!)362 5465 y Fm(i)390 5450 y Fo(y)442 5402 y Fm(u;)p Fn(\003)438 5472 y Fm(i)564 5450 y Fi(\000)673 5382 y Fo(@)5 b(H)810 5397 y Fu(1)p 673 5426 V 691 5518 a Fo(@)g(x)802 5533 y Fm(i)860 5450 y Ft(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)33 b Ft(_)-43 b Fo(q)1313 5465 y Fu(0)1351 5450 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(q)1549 5465 y Fu(0)1588 5450 y Ft(\()p Fo(t)p Ft(\)\))p Fo(;)230 b Ft(_)-45 b Fo(y)2028 5402 y Fm(u;)p Fn(\003)2024 5472 y Fm(i)2155 5450 y Ft(=)27 b Fo(!)2319 5465 y Fm(i)2347 5450 y Fo(x)2402 5402 y Fm(u;)p Fn(\003)2402 5472 y Fm(i)2525 5450 y Ft(+)2633 5382 y Fo(@)5 b(H)2770 5397 y Fu(1)p 2633 5426 V 2655 5518 a Fo(@)g(y)2759 5533 y Fm(i)2820 5450 y Ft(\(0)p Fo(;)17 b Ft(0)p Fo(;)g Ft(0)p Fo(;)g Ft(0)p Fo(;)33 b Ft(_)-43 b Fo(q)3273 5465 y Fu(0)3310 5450 y Ft(\()p Fo(t)p Ft(\))p Fo(;)17 b(q)3508 5465 y Fu(0)3548 5450 y Ft(\()p Fo(t)p Ft(\)\))p Fo(:)1782 5753 y Ft(25)p eop %%Page: 26 26 26 25 bop -118 407 a Ft(Then,)34 b(from)d(the)i(v)-5 b(ariation-of-constan)m(ts)30 b(form)m(ula)h(w)m(e)j(get)e(the)h (expressions)87 588 y Fo(x)142 541 y Fm(u;)p Fn(\003)142 611 y Fm(i)242 588 y Ft(\()p Fo(t)p Ft(\))28 b(=)g Fo(g)536 547 y Fm(u)532 613 y(i)p Fu(1)595 588 y Ft(\()p Fo(t)p Ft(\))17 b(cos)g Fo(!)931 603 y Fm(i)959 588 y Fo(t)22 b Fi(\000)h Fo(g)1167 547 y Fm(u)1163 613 y(i)p Fu(2)1226 588 y Ft(\()p Fo(t)p Ft(\))17 b(sin)f Fo(!)1551 603 y Fm(i)1579 588 y Fo(t;)212 b(y)1905 541 y Fm(u;)p Fn(\003)1901 611 y Fm(i)2004 588 y Ft(\()p Fo(t)p Ft(\))28 b(=)f Fo(g)2297 547 y Fm(u)2293 613 y(i)p Fu(1)2356 588 y Ft(\()p Fo(t)p Ft(\))17 b(sin)f Fo(!)2681 603 y Fm(i)2709 588 y Fo(t)23 b Ft(+)f Fo(g)2916 547 y Fm(u)2912 613 y(i)p Fu(2)2975 588 y Ft(\()p Fo(t)p Ft(\))17 b(cos)g Fo(!)3311 603 y Fm(i)3339 588 y Fo(t;)205 b Ft(\(39\))-118 770 y(where)562 989 y Fo(g)613 948 y Fm(u)609 1014 y(i)p Fu(1)672 989 y Ft(\()p Fo(t)p Ft(\))28 b(=)914 872 y Fh(Z)997 898 y Fm(t)960 1061 y Fn(\0001)1107 843 y Fh( )1172 989 y Fi(\000)17 b Ft(cos)h Fo(!)1475 1004 y Fm(i)1503 989 y Fo(s)k Fi(\001)1630 922 y Fo(@)5 b(H)1767 937 y Fu(1)p 1630 966 178 4 v 1649 1058 a Fo(@)g(x)1760 1073 y Fm(i)1818 989 y Ft(\()p Fo(z)1901 1004 y Fu(0)1940 989 y Ft(\()p Fo(s)p Ft(\)\))22 b(+)g(sin)17 b Fo(!)2418 1004 y Fm(i)2446 989 y Fo(s)22 b Fi(\001)2573 922 y Fo(@)5 b(H)2710 937 y Fu(1)p 2573 966 V 2596 1058 a Fo(@)g(y)2700 1073 y Fm(i)2761 989 y Ft(\()p Fo(z)2844 1004 y Fu(0)2883 989 y Ft(\()p Fo(s)p Ft(\)\))3043 843 y Fh(!)3125 989 y Fo(ds;)357 b Ft(\(40\))562 1272 y Fo(g)613 1230 y Fm(u)609 1296 y(i)p Fu(2)672 1272 y Ft(\()p Fo(t)p Ft(\))28 b(=)914 1154 y Fh(Z)997 1181 y Fm(t)960 1343 y Fn(\0001)1107 1125 y Fh( )1172 1272 y Ft(sin)17 b Fo(!)1370 1287 y Fm(i)1398 1272 y Fo(s)22 b Fi(\001)1526 1204 y Fo(@)5 b(H)1663 1219 y Fu(1)p 1526 1248 V 1544 1340 a Fo(@)g(x)1655 1355 y Fm(i)1713 1272 y Ft(\()p Fo(z)1796 1287 y Fu(0)1835 1272 y Ft(\()p Fo(s)p Ft(\)\))22 b(+)g(cos)17 b Fo(!)2323 1287 y Fm(i)2351 1272 y Fo(s)23 b Fi(\001)2479 1204 y Fo(@)5 b(H)2616 1219 y Fu(1)p 2479 1248 V 2502 1340 a Fo(@)g(y)2606 1355 y Fm(i)2666 1272 y Ft(\()p Fo(z)2749 1287 y Fu(0)2789 1272 y Ft(\()p Fo(s)p Ft(\)\))2949 1125 y Fh(!)3031 1272 y Fo(ds:)451 b Ft(\(41\))-118 1515 y(Using)32 b(equalit)m(y)g(\(31\),)g(w)m(e)i(see)g(that)723 1679 y Fo(@)5 b(H)860 1694 y Fu(1)p 723 1723 V 742 1815 a Fo(@)g(x)853 1830 y Fm(i)911 1747 y Ft(\()p Fo(z)994 1762 y Fu(0)1033 1747 y Ft(\()p Fo(s)p Ft(\)\))28 b(=)f Fi(\000)1468 1679 y Ft(2)p Fo(a)1568 1694 y Fm(i)p Fu(1)p 1411 1723 277 4 v 1411 1819 a Ft(cosh)1597 1776 y Fu(2)1653 1819 y Fo(t)1698 1747 y(;)1947 1679 y(@)5 b(H)2084 1694 y Fu(1)p 1947 1723 178 4 v 1969 1815 a Fo(@)g(y)2073 1830 y Fm(i)2134 1747 y Ft(\()p Fo(z)2217 1762 y Fu(0)2256 1747 y Ft(\()p Fo(s)p Ft(\)\))28 b(=)f Fi(\000)2691 1679 y Ft(2)p Fo(a)2791 1694 y Fm(i)p Fu(2)p 2634 1723 277 4 v 2634 1819 a Ft(cosh)2820 1776 y Fu(2)2876 1819 y Fo(t)2921 1747 y(;)-118 1979 y Ft(and)33 b(this)f(implies)e(that)i(the) h(in)m(tegrals)f(\(40{41\))f(are)i(absolutely)e(con)m(v)m(ergen)m(t.)28 2099 y(W)-8 b(e)44 b(can)g(pro)s(ceed)g(in)f(a)h(similar)c(w)m(a)m(y)45 b(for)e(the)h(stable)f(manifold)e Fo(W)2679 2063 y Fm(s)2715 2099 y Ft(,)47 b(and)c(obtain)g(expressions)-118 2219 y(analogous)38 b(to)g(\(39{41\),)i(but)f(no)m(w)1246 2148 y Fh(R)1302 2175 y Fm(t)1286 2245 y Fn(\0001)1454 2219 y Ft(is)g(replaced)g(b)m(y)h Fi(\000)2186 2148 y Fh(R)2241 2175 y Fn(1)2225 2245 y Fm(t)2316 2219 y Ft(.)62 b(Their)39 b(di\013erence)h(is)e(giv)m(en)h(at)g(\014rst)-118 2340 y(order)33 b(b)m(y)-118 2521 y Fo(x)-63 2474 y Fm(u;)p Fn(\003)-63 2544 y Fm(i)37 2521 y Ft(\()p Fo(t)p Ft(\))12 b Fi(\000)g Fo(x)304 2474 y Fm(s;)p Fn(\003)304 2544 y Fm(i)397 2521 y Ft(\()p Fo(t)p Ft(\))27 b(=)h Fi(4)p Fo(g)775 2536 y Fm(i)p Fu(1)854 2521 y Ft(cos)18 b Fo(!)1063 2536 y Fm(i)1090 2521 y Fo(t)12 b Fi(\000)g(4)p Fo(g)1362 2536 y Fm(i)p Fu(2)1442 2521 y Ft(sin)17 b Fo(!)1640 2536 y Fm(i)1668 2521 y Fo(t;)212 b(y)1994 2474 y Fm(u;)p 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b(we)f(have)g(me)-5 b(asur)g(e)g(d)44 b(the)i(splitting)f(along)f(the)i(c)-5 b(o)g(or)g(dinates)126 3128 y Ft(\()p Fo(x)219 3143 y Fu(1)259 3128 y Fo(;)17 b(y)351 3143 y Fu(1)389 3128 y Fo(;)g(x)488 3143 y Fu(2)528 3128 y Fo(;)g(y)620 3143 y Fu(2)659 3128 y Ft(\))p Fj(,)35 b(which)f(ar)-5 b(e)35 b(not)g(\014rst)g(inte)-5 b(gr)g(als)34 b(of)h(the)g(unp)-5 b(erturb)g(e)g(d)35 b(Hamiltonian)g Fo(H)3414 3143 y Fu(0)3488 3128 y Fj(\(unlike)126 3249 y(the)45 b(appr)-5 b(o)g(ach)44 b(fol)5 b(lowe)-5 b(d)44 b(in)h(\(29\)\).)74 b(But)46 b(the)f(fact)g(that,)j(in)d Fo(H)2582 3264 y Fu(0)2621 3249 y Fj(,)j(the)d(variation)f(of)h(such)g(c)-5 b(o-)126 3369 y(or)g(dinates)38 b(c)-5 b(an)39 b(b)-5 b(e)39 b(describ)-5 b(e)g(d)39 b(by)g(line)-5 b(ar)39 b(di\013er)-5 b(ential)38 b(e)-5 b(quations)39 b(al)5 b(lowe)-5 b(d)38 b(us)i(to)f(obtain)g(a)g(\014rst)126 3490 y(appr)-5 b(oximation)33 b(analo)-5 b(gous)34 b(to)h(the)g (Melnikov)f(function)h(also)f(in)g(this)h(c)-5 b(ase.)-3 3682 y(2.)49 b(As)h(a)f(c)-5 b(onse)g(quenc)g(e)49 b(of)g(The)-5 b(or)g(em)49 b(6,)k(the)d(distanc)-5 b(e)49 b(b)-5 b(etwe)g(en)49 b Fo(W)2670 3646 y Fm(s)2756 3682 y Fj(and)g Fo(W)3066 3646 y Fm(u)3161 3682 y Fj(along)g(the)g(\014rst)126 3802 y(inte)-5 b(gr)g(als)39 b Fo(\030)561 3817 y Fm(i)625 3802 y Ft(=)d(\()p Fo(x)830 3766 y Fu(2)830 3827 y Fm(i)895 3802 y Ft(+)25 b Fo(y)1048 3766 y Fu(2)1044 3827 y Fm(i)1087 3802 y Ft(\))p Fo(=)p Ft(2)39 b Fj(is)g Fi(O)s Ft(\()p Fo(")1537 3766 y Fu(2)1576 3802 y Ft(\))p Fj(.)58 b(This)39 b(explains)f(why)i(the)f(splitting)g(of)g(the)h(lo)-5 b(op)39 b Ft(\000)g Fj(\(as)126 3923 y(wel)5 b(l)46 b(as)h(for)f(tori)h (close)f(to)h Ft(\000)p Fj(\))g(c)-5 b(ould)47 b(not)f(b)-5 b(e)47 b(dete)-5 b(cte)g(d)47 b(fr)-5 b(om)46 b(the)h(Melnikov)f(appr) -5 b(oximation)126 4043 y(c)g(onsider)g(e)g(d)33 b(in)i(The)-5 b(or)g(em)34 b(5.)-3 4236 y(3.)49 b(The)39 b(lo)-5 b(op)39 b(is)h(not)g(pr)-5 b(eserve)g(d)39 b(if)h(at)g(le)-5 b(ast)40 b(one)f(of)h(the)f(c)-5 b(o)g(e\016cients)39 b Fo(a)2734 4251 y Fm(ij)2835 4236 y Fj(do)-5 b(es)39 b(not)h(vanish.)59 b(This)126 4356 y(agr)-5 b(e)g(es)32 b(with)g(the)g(fact)h(that)g(the)f(set)h Fi(L)f Fj(of)g(Hamiltonians)g (with)g(homo)-5 b(clinic)31 b(lo)-5 b(ops)32 b(has)g(c)-5 b(o)g(dimen-)126 4476 y(sion)34 b(4,)h(as)f(explaine)-5 b(d)34 b(at)h(the)f(b)-5 b(e)g(ginning)34 b(of)g(Se)-5 b(ction)35 b(4.)-118 4803 y Fp(Ac)l(kno)l(wledgmen)l(ts)-118 5023 y Ft(This)45 b(w)m(ork)h(has)f(b)s(een)h(supp)s(orted)g(in)e(part) h(b)m(y)h(the)f(INT)-8 b(AS)46 b(gran)m(t)f(00-221.)79 b(O.K.)44 b(and)h(L.L.)g(also)-118 5143 y(ac)m(kno)m(wledge)33 b(a)e(partial)e(supp)s(ort)j(of)f(CRDF)g(\(a)m(w)m(ard)h(R)m (U-M1-2583-MO-04\))c(and)k(Russian)f(F)-8 b(ounda-)-118 5263 y(tion)32 b(of)i(Basic)f(Researc)m(h)i(\(gran)m(t)e (04-01-00483\),)e(L.L.)j(ac)m(kno)m(wledges)h(a)f(supp)s(ort)f(from)g (the)h(program)-118 5384 y(\\Univ)m(ersities)29 b(of)f(Russia")h(\(pro) 5 b(ject)30 b(No.)42 b(1905\).)g(A.D.)29 b(and)g(P)-8 b(.G.)29 b(w)m(ere)i(also)d(partially)e(supp)s(orted)31 b(b)m(y)-118 5504 y(the)i(Catalan)e(gran)m(t)i(2001SGR-70,)d(and)j(the) g(MCyT-FEDER)g(gran)m(t)g(BFM2003-9504.)1782 5753 y(26)p eop %%Page: 27 27 27 26 bop -118 407 a Fp(References)-73 610 y Fq([1])50 b(R.)33 b(Abraham)f(and)g(J.)g(E.)h(Marsden.)46 b Fa(F)-7 b(oundations)37 b(of)e(me)-5 b(chanics)p Fq(.)48 b(Benjamin/Cummings,) 31 b(Reading,)72 723 y(Mass.,)g(1978.)43 b(Second)30 b(edition.)-73 900 y([2])50 b(V.)34 b(I.)f(Arnol)516 867 y Fn(0)538 900 y Fq(d.)48 b Fa(Mathematic)-5 b(al)37 b(metho)-5 b(ds)38 b(of)d(classic)-5 b(al)37 b(me)-5 b(chanics)p Fq(,)35 b(v)m(olume)e(60)h(of)f Fa(Gr)-5 b(aduate)37 b(T)-7 b(exts)36 b(in)72 1013 y(Mathematics)p Fq(.)42 b(Springer-V)-8 b(erlag,)29 b(New)i(Y)-8 b(ork,)31 b(second)g(edition,)e(1989.)-73 1191 y([3])50 b(J.)30 b(Carr.)41 b Fa(Applic)-5 b(ations)34 b(of)f(c)-5 b(entr)g(e)34 b(manifold)g(the)-5 b(ory)p Fq(,)33 b(v)m(olume)d(35)h(of)g Fa(Applie)-5 b(d)34 b(Mathematic)-5 b(al)34 b(Scienc)-5 b(es)p Fq(.)72 1303 y(Springer-V)d(erlag,)30 b(New)g(Y)-8 b(ork,)31 b(1981.)-73 1481 y([4])50 b(A.)42 b(Delshams)g(and)f(P)-8 b(.)43 b(Guti)m(\023)-43 b(errez.)76 b(Splitting)39 b(p)s(oten)m(tial)i (and)h(the)g(Poincar)m(\023)-43 b(e-Melnik)m(o)m(v)43 b(metho)s(d)f(for)72 1594 y(whisk)m(ered)29 b(tori)h(in)f(Hamiltonian)g (systems.)40 b Fa(J.)33 b(Nonline)-5 b(ar)33 b(Sci.)p Fq(,)e(10\(4\):433{476,)36 b(2000.)-73 1771 y([5])50 b(L.)36 b(H.)f(Eliasson.)55 b(Normal)35 b(forms)g(for)g(Hamiltonian)f (systems)i(with)e(Poisson)h(comm)m(uting)g(in)m(tegrals|)72 1884 y(elliptic)28 b(case.)42 b Fa(Comment.)33 b(Math.)h(Helv.)p Fq(,)29 b(65\(1\):4{35,)35 b(1990.)-73 2062 y([6])50 b(S.)39 b(M.)g(Gra\013.)67 b(On)39 b(the)g(conserv)-5 b(ation)39 b(of)h(h)m(yp)s(erb)s(olic)c(in)m(v)-5 b(arian)m(t)38 b(tori)h(for)g(Hamiltonian)e(systems.)67 b Fa(J.)72 2175 y(Di\013er)-5 b(ential)33 b(Equations)p Fq(,)e(15:1{69,)j(1974.)-73 2352 y([7])50 b(M.)31 b(W.)g(Hirsc)m(h,)f(C.)g(C.)g(Pugh,)g(and)g(M.)h (Sh)m(ub.)39 b Fa(Invariant)34 b(manifolds)p Fq(.)42 b(Springer-V)-8 b(erlag,)30 b(Berlin,)e(1977.)72 2465 y(Lecture)j(Notes)g(in)e(Mathematics,)j(V)-8 b(ol.)31 b(583.)-73 2642 y([8])50 b(L.)42 b(M.)h(Lerman.)74 b(Iso)s(energetical) 42 b(structure)g(of)g(in)m(tegrable)g(Hamiltonian)e(systems)i(in)f(an)g (extended)72 2755 y(neigh)m(b)s(orho)s(o)s(d)34 b(of)i(a)h(simple)d (singular)g(p)s(oin)m(t:)52 b(three)36 b(degrees)h(of)g(freedom.)58 b(In)35 b(L.)h(Lerman,)i(G.)f(P)m(olo-)72 2868 y(to)m(vski)-10 b(\025)-35 b(\020,)37 b(and)f(L.)g(Shilnik)m(o)m(v,)g(editors,)h Fa(Metho)-5 b(ds)39 b(of)f(qualitative)h(the)-5 b(ory)39 b(of)f(di\013er)-5 b(ential)40 b(e)-5 b(quations)38 b(and)72 2981 y(r)-5 b(elate)g(d)41 b(topics,)h(Supplement)p Fq(,)f(v)m(olume)c (200)i(of)f Fa(A)n(mer.)h(Math.)h(So)-5 b(c.)40 b(T)-7 b(r)i(ansl.)40 b(Ser.)g(2)p Fq(,)g(pages)e(219{242.)72 3094 y(Amer.)31 b(Math.)g(So)s(c.,)f(Pro)m(vidence,)h(RI,)f(2000.)43 b(Adv)-5 b(ances)30 b(in)f(the)i(Mathematical)g(Sciences,)f(48.)-73 3272 y([9])50 b(L.)35 b(M.)h(Lerman)e(and)g(Y)-8 b(a.)37 b(L.)e(Umanskiy)-8 b(.)53 b Fa(F)-7 b(our-dimensional)39 b(inte)-5 b(gr)g(able)38 b(Hamiltonian)h(systems)f(with)72 3384 y(simple)28 b(singular)f(p)-5 b(oints)28 b(\(top)-5 b(olo)g(gic)g(al)31 b(asp)-5 b(e)g(cts\))p Fq(,)27 b(v)m(olume)d(176)h (of)f Fa(T)-7 b(r)i(anslations)30 b(of)d(Mathematic)-5 b(al)28 b(Mono-)72 3497 y(gr)-5 b(aphs)p Fq(.)43 b(American)30 b(Mathematical)h(So)s(ciet)m(y)-8 b(,)31 b(Pro)m(vidence,)g(RI,)f (1998.)-118 3675 y([10])50 b(D.)28 b(McDu\013)g(and)f(D.)h(Salamon.)35 b Fa(Intr)-5 b(o)g(duction)32 b(to)e(symple)-5 b(ctic)31 b(top)-5 b(olo)g(gy)p Fq(.)39 b(Oxford)26 b(Mathematical)i(Mono-)72 3788 y(graphs.)i(Oxford)f(Univ)m(ersit)m(y)h(Press,)g(New)h(Y)-8 b(ork,)31 b(second)f(edition,)g(1998.)-118 3965 y([11])50 b(J.)29 b(Milnor.)38 b Fa(Morse)32 b(the)-5 b(ory)p Fq(.)40 b(Annals)28 b(of)i(Mathematics)g(Studies,)f(No.)h(51.)g(Princeton)f (Univ)m(ersit)m(y)g(Press,)72 4078 y(Princeton,)h(N.J.,)h(1963.)-118 4256 y([12])50 b(J.)26 b(Moser.)33 b(The)25 b(analytic)g(in)m(v)-5 b(arian)m(ts)25 b(of)h(an)f(area-preserving)h(mapping)e(near)h(a)h(h)m (yp)s(erb)s(olic)d(\014xed)i(p)s(oin)m(t.)72 4369 y Fa(Comm.)34 b(Pur)-5 b(e)32 b(Appl.)h(Math.)p Fq(,)e(9:673{692,)k(1956.)-118 4546 y([13])50 b(J.)35 b(P\177)-45 b(osc)m(hel.)54 b(In)m(tegrabilit)m (y)34 b(of)i(Hamiltonian)d(systems)i(on)g(Can)m(tor)g(sets.)55 b Fa(Comm.)38 b(Pur)-5 b(e)36 b(Appl.)i(Math.)p Fq(,)72 4659 y(35\(5\):653{696,)e(1982.)-118 4836 y([14])50 b(I.)32 b(R.)g(Shafarevic)m(h.)44 b Fa(Basic)34 b(algebr)-5 b(aic)35 b(ge)-5 b(ometry.)35 b(1)p Fq(.)45 b(Springer-V)-8 b(erlag,)31 b(Berlin,)g(second)g(edition,)g(1994.)72 4949 y(V)-8 b(arieties)31 b(in)e(pro)5 b(jectiv)m(e)31 b(space.)-118 5127 y([15])50 b(D.)34 b(V.)f(T)-8 b(reshev.)47 b(Hyp)s(erb)s(olic)31 b(tori)h(and)g(asymptotic)h(surfaces)g(in)f(Hamiltonian)f(systems.)48 b Fa(R)n(ussian)35 b(J.)72 5240 y(Math.)e(Phys.)p Fq(,)e (2\(1\):93{110,)k(1994.)-118 5417 y([16])50 b(J.)30 b(V)-8 b(ey)g(.)42 b(Sur)29 b(certains)h(syst)m(\022)-43 b(emes)32 b(dynamiques)d(s)m(\023)-43 b(eparables.)41 b Fa(A)n(mer.)32 b(J.)g(Math.)p Fq(,)f(100\(3\):591{614,)36 b(1978.)1782 5753 y Ft(27)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF ---------------0405070533445--