%\magnification=1200
\parskip=0pt
\def\R{{\bf R}}
\def \d{{\rm d}}
\def \ep{\epsilon}
\def \( {\big( }
\def \) {\big) }
\def \gs {\lower3pt \hbox {${\buildrel > \over \sim}$}}
\def \nhy {nonhyperbolic}
\def \q {\quad}
\def \qq {\qquad}
\def \pd{\partial}
\def \pn{\par\noindent}
\def \bs{\bigskip}
\def \en{\eqno}
\def \Ref{\pn{\bf References}\bs\pn\parskip=5 pt\parindent=0 pt}
\baselineskip 0.55 cm
\def\.#1{\dot #1}
{\nopagenumbers
%\font\petit = cmr8
~ \vskip 2 truecm
{\bf \centerline {Nonhyperbolic homoclinic chaos}}
\vskip 1.5 truecm
\centerline {{\bf G. Cicogna}\footnote{$^{(*)}$}{E-mail :
cicogna@difi.unipi.it} and {\bf M. Santoprete}}
\centerline{Dipartimento di Fisica, Universit\`a di Pisa, }
\centerline{Via Buonarroti 2, Ed. B}
\centerline{I-56127, Pisa, Italy}
\vskip 2 truecm
\pn
{\bf Abstract.}
\pn
Homoclinic chaos is usually examined with the hypothesis of hyperbolicity
of the critical point. We consider here, following a (suitably adjusted)
classical analytic method, the case of non-hyperbolic points
and show that, under a Melnikov-type condition plus an additional
assumption, the negatively and positively asymptotic sets persist
under periodic perturbations, together with their infinitely many
intersections on the Poincar\'e section.
We also examine, by means of essentially the same procedure, the case of
(heteroclinic) orbits tending to the infinity; this case includes in
particular the classical Sitnikov 3--body problem.
\bs\bs\pn
PACS. no: 03.20, 05.45
\pn
{\it Keywords}: homoclinic chaos, nonhyperbolic critical point, Melnikov
theory, Sitnikov problem.
\vskip 2 truecm\pn
{\hfill To appear in Phys. Letters A}
\vfill\eject }
\pageno=2
~ \vskip 3 truecm
It is well known that perturbing homoclinic (or heteroclinic) orbits of
dynamical systems may lead to the phenomenon of transversal
intersections of stable and unstable manifolds of the
critical point, and that this is one of the routes for the
appearance of chaos; it is also known that Melnikov method is a very
efficient analytical criterion to determine the occurrence of the intersection
of stable and unstable manifolds. It is impossible to give here even a partial
account of the vast literature on this subject, and we will
quote here only those papers [1-7] which are more strictly related to the
present approach (for a more complete list of references, see also [8]).
It can be noticed, however, that almost the totality of the papers
dealing with homoclinic chaos and Melnikov method considers only the case
of {\it hyperbolic} critical points (some exceptions will be quoted
below). We want to show here that a suitable extension of a classical and
purely analytical method, used in [5] for the hyperbolic case, can also
cover -- under suitable hypotheses -- the \nhy\ case.
For definiteness, even if more general situations could be considered,
we will deal here only with planar systems originated from unperturbed
Hamiltonians of the form
$$H={1\over 2} p^2 + V(x)\en (1)$$
We point out that this method can (and will) be applied not only to the
case of homoclinic orbits approaching \nhy\ points corresponding to some
``superquadratic'' unstable equilibrium points $x_0\in\R$ of the
unperturbed potential $V(x)$, but also to the quite different case of
(heteroclinic) orbits tending to $\pm\infty$ for
$t\to\pm\infty$ respectively, appearing in the case of potentials having
an equilibrium point at the infinity.
For the sake of concreteness, we shall provide the construction and the
proof only for the case of superquadratic stationary points, but
it will appear clear that the argument can be repeated equally well for
the other one. For what concerns the first case, we refer also to [9],
where however a completely different approach is used; the second situation
includes in particular the classical Sitnikov restricted $3-$body problem
in celestial mechanics, which has been considered in detail by Moser [4],
by means of the introduction of a singular change of coordinates
(see also [10,11]).
More specifically, we will show that the negatively
and positively asymptotic sets of the unstable equilibrium point persist
even in the \nhy\ case and under periodic perturbation. We also obtain that
the occurrence of an intersection of these sets on the Poincar\'e section
(together with the chain of their infinitely many subsequent intersections)
can be detected by a Melnikov criterion, plus an additional condition,
which must be introduced in this case to compensate the lack of the
``exponential dichotomy'' peculiar of the hyperbolic case.
\vfill\eject\pn
{\bf 1.} We consider a Hamiltonian dynamical problem with Hamiltonian
of the form (1), plus a smooth perturbation $g$ depending on one or more
real (small) parameters $\ep$:
$$\eqalign { \.x= & \ p \cr \.p=& -{\d V\over {\d x}} + g(\ep,x,\.x,t)
\qq {\rm with}\q g(0,x,\.x,t)=0 } \en(2)$$
where the potential $V(x)$ of the unperturbed problem is assumed to be
analytical and to have a ``superquadratic'' unstable equilibrium point at
$x=x_0$ (say, $x_0=0$), which may be either a ``cubic-like''
stationary point, or a local maximum for $V(x)$ (in this case, clearly,
$m<0$ in (3) below):
$$V(0)={\d V(0)\over{\d x}}=\ldots{\d^{\nu-1} V(0)\over{\d x^{\nu-1}}}=0
\q ; \q {\d^{\nu} V(0)\over{\d x^{\nu}}}=m\not=0 \q {\rm for\ some \ integer}
\ \nu>2 \en(3)$$
We assume that the unperturbed problem possesses a homoclinic orbit
doubly asymptotic to $x_0=0$:
$$\chi=\chi(t) \qq {\rm with} \qq \lim_{t\to\pm\infty}\chi(t)=0\en(4)$$
this happens, e.g., if $V(x)$ also admits at least a stable equilibrium point
$x_1$ (e.g., with $x_1>x_0$), and there is another point $x_2>x_1$ such
that $V(x_2)=0$ and $\d V(x)/\d x<0$ for $x_00$ for
$x_10$, of course), it is easy to check,
recalling also (12), that if
$$ n_1>\nu-2 \qq {\rm and}\qq n_2>{2(\nu-2)\over \nu} \en(20)$$
then condition (16) is satisfied.
\bs
Considering e.g. the following explicit example, with $V$ given by (5),
$$\ddot x=2x^3-3x^5+\ep_1x^3\sin\omega t+\ep_2\.x^3\en(21)$$
there are two homoclinic orbits of the unperturbed Hamiltonian, given by
$$\chi(t)={\pm 1\over{\sqrt{1+t^2}}}$$
conditions (20) are satisfied (here $\nu=4$) and the Melnikov condition
(17) can be explicitly evaluated, giving transversal homoclinic intersections
if
$$\Big|{\ep_1\over{\ep_2}}\Big|>{3\exp(\omega)\over{32\ \omega(1+\omega)}}
\en(22)$$
A numerical integration of this problem, performed along the same lines
as in [6, Chapt.2] (see [8] for details), shows that, choosing e.g.
$\omega=1,\ \ep_2=.05$, transversal intersections appear when
$\ep_1\gs .006$, in quite good agreement with the value $\ep_1=.00637$
given by (22).
\bs
Summarizing, we can state the above results in following form:
\bs\pn
{\bf Theorem.} Consider a planar dynamical system of the form (2) where
the potential $V(x)$ admits a superquadratic unstable equilibrium in some
point, say $x_0=0$, producing then a \nhy\ unstable
equilibrium point for the unperturbed system (given by $\ep=0$).
Assume that the unperturbed system admits a homoclinic orbit $\chi(t)$ doubly
asymptotic to $x_0$. Assume for simplicity the perturbation $g$ of the
form (19) with the conditions (20) satisfied. If in addition Melnikov
conditions (17) and (18) are satisfied, then the perturbed problem admits
a negatively and positively asymptotic sets of the unstable equilibrium
point, which admit an infinite sequence of homoclinic intersections on
the Poincar\'e section, giving rise to a (\nhy\ homoclinic) chaotic
dynamical flow.
\bs\bs\pn
{\bf 2.} Instead of discussing some of the possible extensions of the
above results to more general dynamical systems or to problems in greater
dimension, we prefer here to examine a quite different situation which
in fact can be discussed with a similar procedure. Let us consider a problem
with Hamiltonian (1), where now the potential is such that
$$V(x)<0 \q , \q \forall x\in\R \qq {\rm with} \qq
\lim_{x\to\pm\infty}V(x)=0 \en(23)$$
and there is only a stationary point $x_0$ (a minimum) for $V(x)$:
$${\d V(x_0)\over{\d x}}=0 \en(23')$$
There is then a (heteroclinic) orbit approaching $-\infty$ and $+\infty$:
$$\chi=\chi(t) \qq{\rm with}\qq \lim_{t\to\mp\infty}\chi(t)=\mp\infty
\en(24)$$
A well known example is given by the classical Sitnikov restricted
$3-$body problem in celestial mechanics, which -- at the limit of
zero eccentricity -- is described by the potential~[4]
$$V(x)={-1\over{\sqrt{x^2+{1\over 4}}}}\en(25)$$
More in general, we can consider a potential such that
$$V(x)\sim {1\over{|x|^\mu}}\q {\rm for}\q |x|\to\infty\qq{\rm with}\q
\mu>0,\ {\rm real} \en(26)$$
Clearly, these problems are, {\it per se}, \nhy , the equilibrium point
being at the infinity; actually, Moser [4] examined the above problem (25),
in the presence of the periodic perturbation produced by nonzero eccentricity,
using the McGehee [16] singular coordinate transformation defined by
$x=2/y^2$ and $\d t=(4/y^3)\ \d s$, which in fact
transforms the problem into a hyperbolic one near the point $y=0$, and he
was able to prove the existence of Smale horseshoe dynamics (see also
[10,11]).
Our above procedure can be equally well used in this new case, i.e.
for problems with potential of the form (23,27): indeed, writing the problem
as in (8), we can again introduce the variation equation (10-11).
In this case, one finds that two independent solutions of the homogeneous
part of (10) have the following behaviour, for $|t|\to\infty$
$$\eqalign{ \.\chi(t)\sim &\ |t|^{-\mu/(2+\mu)}\to 0 \cr
\psi(t)\sim &\ |t|^{(2+2\mu)/(2+\mu)}\to\infty } \en (27)$$
This allows us to repeat exactly the same arguments as in
part {\bf 1}. In particular, assuming e.g. a non-dissipative periodic
perturbation of the form, for large $|x|$,
$$g(\ep,x,\.x,t)\sim\ \ep x^{-n}\sin\omega t\en(28)$$
it is easy to show that the existence of out/in sets is granted if
$$n>2+\mu\en(29)$$
and to show also that the Melnikov function (17) possesses infinitely
many transversal zeroes. Notice that the Sitnikov $3-$body problem
above mentioned falls into this situation, indeed -- at the first
order in the eccentricity $\ep$ -- the perturbation is given by [4]
$$g=\ep{-3x\over{4\Big(x^2+{1\over 4}\Big)^{5/2}}}\cos t $$
As already remarked in part {\bf 1}, the standard Birkhoff-Smale theorem
cannot be directly used for this \nhy\ case. Here, however, at least for
the case of Sitnikov problem, we can refer to
the arguments used in [10] (see especially the Appendix of [10], and also
[13]), to obtain an equivalence to a ``\nhy\ horseshoe'',
where contracting and expanding actions are not exponential but
``polynomial'' in time. Alternatively, in the general case, one may possibly
resort to the method of ``blowing-up'', devised to
investigate the properties of \nhy\ singularities by means of suitable
changes of coordinates [17,18], but presumably a full and general
treatment of \nhy\ horseshoes is still open.
\bs\bs\pn
{\bf Acknowledgments.}
\pn
We are grateful to prof.s D. Bambusi, U. Bessi, L. Galgani, and
J. Mallet-Paret for useful suggestions and bibliografical indications.
%\vfill\eject
\bs\bs
\Ref
[1] V.K. Melnikov, {\it Trans. Moscow Math. Soc.} {\bf 12} (1963) 1
[2] S. Smale, {\it Bull. Amer. Math. Soc.} {\bf 73} (1967) 747
[3] Z. Nitecki, Differentiable dynamics (MIT Press, Cambridge, Mass. 1971)
[4] J. Moser, Stable and random motions in dynamical systems (Princeton
Univ. Press, Princeton 1973)
[5] S.-N. Chow, J.K. Hale and J. Mallet-Paret, {\it J. Diff. Eq.} {\bf 37}
(1980) 351
[6] J. Guckenheimer and P.J. Holmes, Nonlinear oscillations, dynamical
systems and bifurcations of vector fields (Springer, Berlin 1983)
[7] S. Wiggins, Global bifurcations and chaos (Springer, Berlin 1989)
[8] M. Santoprete, Thesis, Dept. of Physics, Univ. of Pisa
[9] J. Casasayas, E. Fontich and A. Nunes, {\it Nonlinear Diff. Eq.
Appl.} {\bf 4} (1997) 201
[10] H. Dankowicz and P. Holmes, {\it J. Diff. Eq.} {\bf 116} (1995) 468
[11] C. Robinson, {\it Contemp. Math.} {\bf 198} (1996) 45
[12] J. Casasayas, E. Fontich and A. Nunes, {\it Nonlinearity} {\bf 5}
(1992) 1193
[13] Xiao-Biao Lin, {\it Dynamics Reported} {\bf 5} (1996) 99
[14] E.A. Coddington and N. Levinson, Theory of ordinary differential
equations (McGraw-Hill, New York 1955)
[15] V. Rayskin, preprint 1998, Dept. of Math., Texas Univ., Austin.
[16] R. McGehee, {\it J. Diff. Eq.} {\bf 14} (1973) 70
[17] F. Dumortier, {\it J. Diff. Eq.} {\bf 23} (1977) 53
[18] M. Brunella and M. Miari, {\it J. Diff. Eq.} {\bf 85} (1990) 338
\bye