Content-Type: multipart/mixed; boundary="-------------0910271523343" This is a multi-part message in MIME format. ---------------0910271523343 Content-Type: text/plain; name="09-195.keywords" Content-Transfer-Encoding: 7bit Content-Disposition: attachment; filename="09-195.keywords" singularity, non-smooth vector field, structural stability, bifurcation ---------------0910271523343 Content-Type: application/postscript; name="Cod2Singularities_v_6.ps" Content-Transfer-Encoding: 7bit Content-Disposition: inline; filename="Cod2Singularities_v_6.ps" %!PS-Adobe-2.0 %%Creator: dvips(k) 5.96 Copyright 2007 Radical Eye Software %%Title: Cod2Singularities_v_6.dvi %%CreationDate: Tue Oct 20 15:03:32 2009 %%Pages: 54 %%PageOrder: Ascend %%BoundingBox: 0 0 595 842 %%DocumentFonts: CMR17 CMR12 CMSY8 CMSY7 CMR10 CMBX9 CMR9 CMBX12 CMSY6 %%+ CMTT8 MSBM10 CMR7 CMTI10 CMMI10 CMSY10 CMMI7 CMEX10 EUFM10 CMBX10 %%+ CMMI5 CMR5 CMSY5 %%DocumentPaperSizes: a4 %%EndComments %DVIPSWebPage: (www.radicaleye.com) %DVIPSCommandLine: dvips.exe Cod2Singularities_v_6.dvi %DVIPSParameters: dpi=600 %DVIPSSource: TeX output 2009.10.20:1503 %%BeginProcSet: tex.pro 0 0 %! /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 N/hsize 8.5 72 mul N/landplus90{false}def/@rigin{isls{[0 landplus90{1 -1}{-1 1}ifelse 0 0 0]concat}if 72 Resolution div 72 VResolution div neg scale isls{ landplus90{VResolution 72 div vsize mul 0 exch}{Resolution -72 div hsize mul 0}ifelse TR}if Resolution VResolution vsize -72 div 1 add mul TR[ matrix currentmatrix{A A round sub abs 0.00001 lt{round}if}forall round exch round exch]setmatrix}N/@landscape{/isls true N}B/@manualfeed{ statusdict/manualfeed true put}B/@copies{/#copies X}B/FMat[1 0 0 -1 0 0] N/FBB[0 0 0 0]N/nn 0 N/IEn 0 N/ctr 0 N/df-tail{/nn 8 dict N nn begin /FontType 3 N/FontMatrix fntrx N/FontBBox FBB N string/base X array /BitMaps X/BuildChar{CharBuilder}N/Encoding IEn N end A{/foo setfont}2 array copy cvx N load 0 nn put/ctr 0 N[}B/sf 0 N/df{/sf 1 N/fntrx FMat N df-tail}B/dfs{div/sf X/fntrx[sf 0 0 sf neg 0 0]N df-tail}B/E{pop nn A definefont setfont}B/Cw{Cd A length 5 sub get}B/Ch{Cd A length 4 sub get }B/Cx{128 Cd A length 3 sub get sub}B/Cy{Cd A length 2 sub get 127 sub} B/Cdx{Cd A length 1 sub get}B/Ci{Cd A type/stringtype ne{ctr get/ctr ctr 1 add N}if}B/CharBuilder{save 3 1 roll S A/base get 2 index get S /BitMaps get S get/Cd X pop/ctr 0 N Cdx 0 Cx Cy Ch sub Cx Cw add Cy setcachedevice Cw Ch true[1 0 0 -1 -.1 Cx sub Cy .1 sub]{Ci}imagemask restore}B/D{/cc X A type/stringtype ne{]}if nn/base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{A A length 1 sub A 2 index S get sf div put }if put/ctr ctr 1 add N}B/I{cc 1 add D}B/bop{userdict/bop-hook known{ bop-hook}if/SI save N @rigin 0 0 moveto/V matrix currentmatrix A 1 get A mul exch 0 get A mul add .99 lt{/QV}{/RV}ifelse load def pop pop}N/eop{ SI restore userdict/eop-hook known{eop-hook}if showpage}N/@start{ userdict/start-hook known{start-hook}if pop/VResolution X/Resolution X 1000 div/DVImag X/IEn 256 array N 2 string 0 1 255{IEn S A 360 add 36 4 index cvrs cvn put}for pop 65781.76 div/vsize X 65781.76 div/hsize X}N /p{show}N/RMat[1 0 0 -1 0 0]N/BDot 260 string N/Rx 0 N/Ry 0 N/V{}B/RV/v{ /Ry X/Rx X V}B statusdict begin/product where{pop false[(Display)(NeXT) (LaserWriter 16/600)]{A length product length le{A length product exch 0 exch getinterval eq{pop true exit}if}{pop}ifelse}forall}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale Rx Ry false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR Rx Ry scale 1 1 false RMat{BDot} imagemask grestore}}ifelse B/QV{gsave newpath transform round exch round exch itransform moveto Rx 0 rlineto 0 Ry neg rlineto Rx neg 0 rlineto fill grestore}B/a{moveto}B/delta 0 N/tail{A/delta X 0 rmoveto}B/M{S p delta add tail}B/b{S p tail}B/c{-4 M}B/d{-3 M}B/e{-2 M}B/f{-1 M}B/g{0 M} B/h{1 M}B/i{2 M}B/j{3 M}B/k{4 M}B/w{0 rmoveto}B/l{p -4 w}B/m{p -3 w}B/n{ p -2 w}B/o{p -1 w}B/q{p 1 w}B/r{p 2 w}B/s{p 3 w}B/t{p 4 w}B/x{0 S rmoveto}B/y{3 2 roll p a}B/bos{/SS save N}B/eos{SS restore}B end %%EndProcSet %%BeginProcSet: psfrag.pro 0 0 %% %% This is file `psfrag.pro', %% generated with the docstrip utility. %% %% The original source files were: %% %% psfrag.dtx (with options: `filepro') %% %% Copyright (c) 1996 Craig Barratt, Michael C. 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308CF87FE93F02F44BECA5C4FC 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY5 %!PS-AdobeFont-1.1: CMSY5 1.0 %%CreationDate: 1991 Aug 15 07:21:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 48 /prime put readonly def /FontBBox{21 -944 1448 791}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR5 %!PS-AdobeFont-1.1: CMR5 1.00B %%CreationDate: 1992 Feb 19 19:55:02 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 43 /plus put readonly def /FontBBox{-341 -250 1304 965}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI5 %!PS-AdobeFont-1.1: CMMI5 1.100 %%CreationDate: 1996 Aug 02 08:21:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI5) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI5 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /mu put dup 34 /epsilon put dup 115 /s put readonly def /FontBBox{37 -250 1349 750}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMBX10 %!PS-AdobeFont-1.1: CMBX10 1.00B %%CreationDate: 1992 Feb 19 19:54:06 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 66 /B put dup 68 /D put dup 70 /F put dup 71 /G put dup 72 /H put dup 78 /N put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 119 /w put dup 120 /x put dup 121 /y put dup 123 /endash put readonly def /FontBBox{-301 -250 1164 946}readonly def currentdict end currentfile eexec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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: EUFM10 %!PS-AdobeFont-1.1: EUFM10 2.1 %%CreationDate: 1992 Nov 20 17:36:20 % Euler fonts were designed by Hermann Zapf. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (EUFM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /EUFM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 88 /X put readonly def /FontBBox{-26 -224 1055 741}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMEX10 %!PS-AdobeFont-1.1: CMEX10 1.00 %%CreationDate: 1992 Jul 23 21:22:48 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMEX10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMEX10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /parenleftbig put dup 1 /parenrightbig put dup 16 /parenleftBig put dup 17 /parenrightBig put dup 18 /parenleftbigg put dup 19 /parenrightbigg put dup 26 /braceleftbigg put dup 27 /bracerightbigg put dup 32 /parenleftBigg put dup 33 /parenrightBigg put dup 40 /braceleftBigg put dup 41 /bracerightBigg put dup 56 /bracelefttp put dup 58 /braceleftbt put dup 60 /braceleftmid put dup 62 /braceex put dup 112 /radicalbig put dup 114 /radicalbigg put readonly def /FontBBox{-24 -2960 1454 772}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMMI7 %!PS-AdobeFont-1.1: CMMI7 1.100 %%CreationDate: 1996 Jul 23 07:53:53 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /mu put dup 27 /sigma put dup 34 /epsilon put dup 59 /comma put dup 61 /slash put dup 82 /R put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 99 /c put dup 101 /e put dup 105 /i put dup 110 /n put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put readonly def /FontBBox{0 -250 1171 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY10 %!PS-AdobeFont-1.1: CMSY10 1.0 %%CreationDate: 1991 Aug 15 07:20:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 1 /periodcentered put dup 2 /multiply put dup 3 /asteriskmath put dup 6 /plusminus put dup 14 /openbullet put dup 15 /bullet put dup 17 /equivalence put dup 21 /greaterequal put dup 26 /propersubset put dup 33 /arrowright put dup 50 /element put dup 54 /negationslash put dup 67 /C put dup 79 /O put dup 83 /S put dup 85 /U put dup 90 /Z put dup 91 /union put dup 92 /intersection put dup 102 /braceleft put dup 103 /braceright put dup 107 /bardbl put dup 110 /backslash put dup 112 /radical put readonly def /FontBBox{-29 -960 1116 775}readonly def currentdict end currentfile eexec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94C8194FE072E47E32FDEE8B2A4F297B67716558CF0117DFD2422011F49C74C3 EA87C3EC1C49A534975D6E1541A7C24F79299866C23DB48109747AE77A960594 D1CDFC48436ABCD803EE80CF08F7CEB241007DC5FC7844E67F8DAF5333E6F13F 1952058C33564DFD4DF41A1DB570B3F28E4A02AB92E85618836C5F39332EA623 CF1DB961C0837FC7F07835275F13A3F9C987A5DE771C95F7461162B6B4DE670C D2F75DAAB173F4F40227764DAC35BC5411AFDB0B72D830D85FC9A44754860311 79A5BDABD99D5D190AE610A4AEAE582FB576607A2C240A11A780D65D633E1C1F D18E634BC06F9A6496DEE5D8016F20815F4DFD5354F6FA07E4C2AE4B9169DEA9 202E065647BC4FFEB689DC213748AB7B0F23F7368B4FC1A0BF6C241F2651A291 B4384E3683C977FE63BC005D315123138E7802E3597A927A12 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMMI10 %!PS-AdobeFont-1.1: CMMI10 1.100 %%CreationDate: 1996 Jul 23 07:53:57 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.100) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMMI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMMI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /alpha put dup 12 /beta put dup 13 /gamma put dup 17 /eta put dup 21 /lambda put dup 22 /mu put dup 23 /nu put dup 27 /sigma put dup 34 /epsilon put dup 39 /phi1 put dup 58 /period put dup 59 /comma put dup 60 /less put dup 61 /slash put dup 62 /greater put dup 64 /partialdiff put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 70 /F put dup 73 /I put dup 78 /N put dup 80 /P put dup 81 /Q put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 90 /Z put dup 97 /a put dup 98 /b put dup 102 /f put dup 104 /h put dup 107 /k put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put dup 121 /y put readonly def /FontBBox{-32 -250 1048 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 9E394A533A081C36D456A09920001A3D2199583EB9B84B4DEE08E3D12939E321 990CD249827D9648574955F61BAAA11263A91B6C3D47A5190165B0C25ABF6D3E 6EC187E4B05182126BB0D0323D943170B795255260F9FD25F2248D04F45DFBFB DEF7FF8B19BFEF637B210018AE02572B389B3F76282BEB29CC301905D388C721 59616893E774413F48DE0B408BC66DCE3FE17CB9F84D205839D58014D6A88823 D9320AE93AF96D97A02C4D5A2BB2B8C7925C4578003959C46E3CE1A2F0EAC4BF 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cleartomark %%EndFont %%BeginFont: CMTI10 %!PS-AdobeFont-1.1: CMTI10 1.00B %%CreationDate: 1992 Feb 19 19:56:16 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTI10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.04 def /isFixedPitch false def end readonly def /FontName /CMTI10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 11 /ff put dup 12 /fi put dup 14 /ffi put dup 40 /parenleft put dup 41 /parenright put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 91 /bracketleft put dup 93 /bracketright put dup 94 /circumflex put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put readonly def /FontBBox{-163 -250 1146 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA0529731C99A784CCBE85B4993B2EEBDE 3B12D472B7CF54651EF21185116A69AB1096ED4BAD2F646635E019B6417CC77B 532F85D811C70D1429A19A5307EF63EB5C5E02C89FC6C20F6D9D89E7D91FE470 B72BEFDA23F5DF76BE05AF4CE93137A219ED8A04A9D7D6FDF37E6B7FCDE0D90B 986423E5960A5D9FBB4C956556E8DF90CBFAEC476FA36FD9A5C8175C9AF513FE D919C2DDD26BDC0D99398B9F4D03D5993DFC0930297866E1CD0A319B6B1FD958 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0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMBX12 %!PS-AdobeFont-1.1: CMBX12 1.0 %%CreationDate: 1991 Aug 20 16:34:54 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX12) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX12 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 65 /A put dup 66 /B put dup 67 /C put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-53 -251 1139 750}readonly def currentdict end currentfile eexec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A441127A39FE3B2BE468BA77DA0C858545F05C4FBA15A78B1FA47F 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMR7 %!PS-AdobeFont-1.1: CMR7 1.0 %%CreationDate: 1991 Aug 20 16:39:21 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 22 /macron put dup 43 /plus put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 126 /tilde put readonly def /FontBBox{-27 -250 1122 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5CF5B8CABB9FFC6CC3F1E9AE32F234EB60FE7D E34995B1ACFF52428EA20C8ED4FD73E3935CEBD40E0EAD70C0887A451E1B1AC8 47AEDE4191CCDB8B61345FD070FD30C4F375D8418DDD454729A251B3F61DAE7C 8882384282FDD6102AE8EEFEDE6447576AFA181F27A48216A9CAD730561469E4 78B286F22328F2AE84EF183DE4119C402771A249AAC1FA5435690A28D1B47486 1060C8000D3FE1BF45133CF847A24B4F8464A63CEA01EC84AA22FD005E74847E 01426B6890951A7DD1F50A5F3285E1F958F11FC7F00EE26FEE7C63998EA1328B C9841C57C80946D2C2FC81346249A664ECFB08A2CE075036CEA7359FCA1E90C0 F686C3BB27EEFA45D548F7BD074CE60E626A4F83C69FE93A5324133A78362F30 8E8DCC80DD0C49E137CDC9AC08BAE39282E26A7A4D8C159B95F227BDA2A281AF A9DAEBF31F504380B20812A211CF9FEB112EC29A3FB3BD3E81809FC6293487A7 455EB3B879D2B4BD46942BB1243896264722CB59146C3F65BD59B96A74B12BB2 9A1354AF174932210C6E19FE584B1B14C00E746089CBB17E68845D7B3EA05105 EEE461E3697FCF835CBE6D46C75523478E766832751CF6D96EC338BDAD57D53B 52F5340FAC9FE0456AD13101824234B262AC0CABA43B62EBDA39795BAE6CFE97 563A50AAE1F195888739F2676086A9811E5C9A4A7E0BF34F3E25568930ADF80F 0BDDAC3B634AD4BA6A59720EA4749236CF0F79ABA4716C340F98517F6F06D9AB 7ED8F46FC1868B5F3D3678DF71AA772CF1F7DD222C6BF19D8EF0CFB7A76FC6D1 0AD323C176134907AB375F20CFCD667AB094E2C7CB2179C4283329C9E435E7A4 1E042AD0BAA059B3F862236180B34D3FCED833472577BACD472A4EA3A676F820 89827E7A66934F11BA1A6716C3C0CB8567BE6CFF0008423765C291803AA9B249 E0B0C1FF0AD79B718508B0DC65F8262D1A1B32D09FAB9690E005B81A1B27569A 16182B0C1ABA2255395D771F19B5BAA11CA72B608350D6F159B96E09419694FE 4D59C04520BDBD5894189273E762586E498884629FAF85630BC08876F672D608 6F9FA19ECBD77CCAB6C6AC74F3EF1C72479AE81E3C740899B27B9FD7917E36A9 93672C3FA5FD7954E21853C6C878CA1748502BA3C548CF31D50F5F975DA2F0E7 44C74A1D1241925D0C3C8C056361BF207CC4C891406D6882B32357FF5E9AB858 A433E112FEC28749B057FCF4EB10B803F06D12C650F7EAD27C3D2A564E4C2332 111F6E426DF59B2F00AF6BEAD0EF56610ACE01523E717C20E10047B4174DA7A9 4C707E12D94CE098DA2FF04EA37C6626901584ECB42060662D1B3D0840B35DB8 9E33777F5E343218DB61F8C89BEA2B62596708B8644E046204231FE90E3E3CA6 9CFF8ECE855AF9F7D4E21E083158714732E76341164E2B4D2A615D080D32145B C3713232156A0597335BC9F3BE45FA122D45761EB11AA088AA8A16E29434CC13 7B126B5FF40496FCC60F86BBDF82D2A051B3D855FF2483426A5DCB9079835043 50DCDA2F47D22F50ACFAA3A2052AA3323E2CBD393EF8DC27C8285A163EEDE727 DDBB808921279F87F0581A42E801AEE32A7742AA3781AE6F1E8B7D053D84D30F 950612C69DBCDBAA3FA0434371C169798B03B6BE0A34583CE32B57A7AE659B50 9472A3E1CA64BB2E51D4F45B617CCDF688AD5E125315667E6FCDAD65A575A6AB 7DF7363E4B8BD2F9B78271F5B799A437B2CB67E560675E07CD4307DB3C6813B2 1A4C69101E2D45A1B17D9013C2D36960CCE7479A4AA58210B0E8341D14568DF6 ACCD42711590ABE15A87F0E855538B10CE8AE8DC705188E1F5ADC0F1E5B716D1 87E1627F562583936ED7702072770D9E1CFBBBC9F81D2FD667331270E7578516 50026FEA708C74896F25A22CA673324594361A663236914B76BD347F7E653E4D D784090E6983117F5C0DEBC3081C95FB4201D339DCFAFCBEDD56BAD93B9816FF F483C98D4B45F44A274A812EDA88BEFA29655A3D6655B0C883F02E60D3852969 147FC49500B4D55178B6EBB434BDE5B690DFDBAEF7A1315EFE237801730A6BE0 530CA0ECA408E89F97403F23D71C38E151A977DEF5E4483AD2FEABE4F46C83BE EBA5E7DC8199D25F070E86C472086BAD13C2524033F6B0E00140F7ABEEEE8E9F 9EA753C40376E38D432CF4C1A00A53A821B1127B834D464348680B26E98C656E C024BBCBB5229059D1A0BF50FFA5BD3615A9279026B9D2307564AD812B576C42 8D762BF0325BB336A193C5D70CEEB1D093DC9EA62DF15E9F01ED7E87BBA6A30B 75E2418CB9C7BE59768687A2B7C89A3339FED14E64BF73F42373044F0BE88EA8 34DEBA9CFE699099717806E441D16B072FD63AF336612C62FBDC780AE3193409 A5CECAFC63AAFFBEC1DE457BB24CDD8031A6160000B9DC0AC1D63A1023558051 88CF46236A3FAF9C7039833B6A0BDCFCB9D96DC87909F9C94F48474402FE2478 739380733BE03B2272740E31EFF1F5FEAC1381A8FDE5AA5A9F9065E6D12FF200 FB26D45A5083FA4E2AF844E58410DC50DB0ACD1B4460ADCCA16D305B35156550 75F3C27E3DF13E71630B0979F323221A5A134E8F07FB28AEDE25DE8562560B37 5BA383FD9D106DDF38ACF48793A6B9534FD99207BF77897E6376B2CDB53910CD B39C77CD30CCA9D387D3C6C9A6CAAB9FF71F88AB6753AAAA710EBA8FF0B95DFF 60112C66FFB1DB797AE50CA18ED10F802D99519B892AACA7F892540F11EB5B0E 46EC879588DA7F2C4219239CBB419597FEE9E5A0ED110AB03B6DF90C6F4B4100 7089C1A62F1E03F3940234F1D9E8E491F8A9A58CC53D1EE2E85817950C2549CC 5CDEA2C2251D9A5193595FF0A25F54FE70A92B45B7784EE4DDCDE09699E8EBB9 48200DF7B889460F93C91EF58AC9E1D89E77A45CE003B461CCC3398CE666C643 9E124ED92E411D20BC428DBA53F7DC8BD0E19E60BDB7593292640E7D5F6D04AD 8E7D327F42A8EDA5A267672C7F3727189DF04A430002BDE4039E8B4E5DB6974C 6050B3153FE53ED8F208BE0E0CEBD9B5241040683512D1E2419C3FF61086E47A 9F689D49EB661910DAD906C0E2B5305A74B9583751A7E1250AA44E153C950447 BA18DA1E8CFACEE9518081F685FFAC17B032BD7D47C6A92229CAE3C272739BD2 50D5CA2A3B30E6D8E1B4F237FDC0FFA0AAF708C1C2011C49CD96477A046A9A98 D3E93E4996474D96211B96D82F24AD6F8728648D1B6B1EE3AC595C7837127860 E549BA2CCCDBBB8C4AAFF0C8A155480C80F03B34B984421CED17B6AA5307131A E30086F9 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: MSBM10 %!PS-AdobeFont-1.1: MSBM10 2.1 %%CreationDate: 1993 Sep 17 11:10:37 % Math Symbol fonts were designed by the American Mathematical Society. % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (2.1) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (MSBM10) readonly def /FamilyName (Euler) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /MSBM10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 44 /notparallel put dup 82 /R put readonly def /FontBBox{-55 -420 2343 920}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMTT8 %!PS-AdobeFont-1.1: CMTT8 1.0 %%CreationDate: 1991 Aug 20 16:46:05 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMTT8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch true def end readonly def /FontName /CMTT8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 64 /at put dup 67 /C put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 120 /x put readonly def /FontBBox{-5 -232 545 699}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0187316F83DDE3E2D27FCDF6C5CE4F95B6EE 3317BD91B7921F3039DD35FEA387D5CFB6C6E9DC84C178F3432994FC7FAC6E5A ED41A1E2EBA350178FBFEB45944511731BA827167DDAC238FC69A5486B995477 C469E2E27493B0B711DF8E267D3D5613B450011921685147114106C9472580BD F531022F6DF5432B2A4EBC51A8032C7F9689B6FA942D849B29709631613DA68D 4DF7B6F059A19304F40A3C3580CE3B51D79D42984194D4F178801720892FB6E7 61FF43C63F9256B5E9F4227B1378222BAAD4D52C77462DF01892220E11129C16 6C9E45BB9F01ED7C1AD5D8B4D72BE0E12969AFEA90FEF170603CDB91CB243173 B19A56084D10293B80A35275F41BF78A054DDC98F4A1FFF592463D944960FB31 6BE5F03960F9B1F213CBCC7FD448657FE388F10104D42B0715FC9571CC60CF23 C72560CBB8835A0CA208FE06676B3B48B093CB7FB2C0C53AF17EC5B372A9771B BFD52FFB7062B4FE0106A01A2A1A1DD4EF5C8C7623EC9324A2CB3B402FCC1FCE 52BFC8662F8A39D5F1B41C97E7CE34E16AC28A1E94007AEA7D4C519399F1B7A9 48FA7DDB671067244F09C29F95DD60668223F45BBDA8B1C452E930A9F3F341C5 351D59EA87462FFB30277D3B24E2104D4AAB873BB2B16DA5B23BEE25BE2C8128 C4CF2F4F438A4E520CD932BAC455BF8775C27AEA6C73EED3EB2F8DB5E356AE27 41B35C8AEFE73C4CD6A591AAE4F45762EBD6D3636C03F08C552BBFD0A13D11D5 491F8369B4BAB8ED9D6F1DE7DB7AFD383986C4338D3AA71C9AF2B8A0955CFD86 0345F16D9798B25156DDF826A7CB6A0CC4CB43078BEBD3E499DA95562A08EED9 7CA27B7A0CE3FA7EBDAA87A601A29DD05C91B9E3F9F4F6CCDCF61CF87D258F7B 560BAA75420693C50E612A509E9D70CBFE9928D71E117E23ED6D6910D6E2C552 5225B9FE9C28B2E157605FD58FB0631C5D5FD4FE5B72E4178C90278D699BAE97 40C54563B8A4785520D580D79FBE0DE18A516141963F6C4900AA685A899E68B6 C5540C93CFDC6C72A94486914CC4318E6D375B38D88F3702CA27AF15286289AA B1E0E6BC14677B4E91D394FD4D3F5D44B478D036A3CE0C3D1EB945DA01795034 DCF01566F0086440BD5549BADE1A9183142A5A6AF9B803BB9EA1BD216535CB33 11AFE2204B21C034B27C0D6D3823ACA2E9EF89EFBACDC283BED03AFEB63B54C6 31E56510B60D53DC0F8CA7B13C976D63467F5514C3C4A4FE28C8EC54FBB3DE41 5C5B6D3FEF1AD6273483817B7AF554641E96E913816365CDCEE4B7F84E756F4F 420381C30617BD4B153EDB2FB8A0381937777D011C3D77786D6F7D749CCF5239 34768A67F7CFEF9D82C71CA859AB597B8611948E16C7DA3272290FDA4154A83E 240C45B2236872C8104BD638B6FE2C505586CF1D72A3C55BA8AED28F5190865D 66140F4EDD74ED51A50324600DEBC2B6B06B3C88E074BE60F27A0826022E465F 7825EA4C676B3FF65AAD82264748A6643B7B4FAEB728DFFB556AA66C562B6F37 99AFE9BC34F2681973BC7ACA1A3B09E1F3BC4F6E700351E9FDE54A35FDA69255 BF68D8618CF04E503EBB20BC694397B1F087CAEAB55A685C3AA775A3D890D7E8 D9B49B4D617CE0D482ABC304691D950E39AFA0E6E8B5D296C0C9527034F49E7F 4D6737643999FAA23E1FDB5171108A57389ADE1558E0CD6C812E39D6E5476798 06EF42B99A391326354E44F2C320331FAAC0415C4E49D3C56522740DA769CEFB 8570B7C36FBAAE4A6B28FD0B173708CD667EF9C5A2DCED94D36686B60F1AFEBF CBAD7DBF6D95D33F11AB4783191C37A48E6C3A4891EB5C9ADEC85A2EB56E7D42 DEC4A6485629FE20C16382CE96273061407AC05A0F7683BA958C25654B9EB7FB 84D30484D2809E13DB9E2FF6C1DBDEC51FAD85DC398FD2D71B895191CC27C4FC 49501D81A72C8C8ECF5AECC0FC4D0D09F3F4134F705C0F6F62AF6AE0E481036F 5185DEB7C542E531B5FF19380E44C09752AA51AD19DAC95B4A69114B6E75FD80 3DFC9AF0EF4EBA726B05FCDAA1C5853E25F93C49656E528CB0EEFBA4AC878287 ABA0700D178525BF8F03B31487371BA1E8AA34226EE39D248A373C7A4474505D BEE59ED8BD56249FF25A21535849E82C3F82F052EDCFB63D35DE8EDA02EC489D D4A6A499CC22B3B8E6746BF8AB1A41B419DCDD73705B38340D066C8A696BD6A3 E9E2059DBB4B9048E28035B0370A7923ADEEE74FEE713FA99374640CE63D9D2C 8300ED611ABD9FD53AF1E5AE59C75585C293E241FC96E129CC535F1E27AC2FD1 B04C6AC40658FFD40E28CA7C06DE285619329CBF4262559F6CC9664767804EB5 52434775C8B9E87C45673C33B2D7B84786D5EBB6AB791C7FCD86120841389488 4989C6CB0734DE511DB9FD1964777CA0CB3B59DCD689AAF7794C4E7A597B3D9D A210856F867F812E474C679979C007E5AB42E03AFD3975DB38B4E6A72A8FD4AD 654DE1E404746B5141459EACC95B62AAE43C07C1BEB63D19C4188A8000D68E99 58D551EE4945DC31B4D67134640A218D60FDD2886F9823420E5D15E96AFAF443 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cleartomark %%EndFont %%BeginFont: CMSY6 %!PS-AdobeFont-1.1: CMSY6 1.0 %%CreationDate: 1991 Aug 15 07:21:34 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY6) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY6 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-4 -948 1329 786}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR9 %!PS-AdobeFont-1.1: CMR9 1.0 %%CreationDate: 1991 Aug 20 16:39:59 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 12 /fi put dup 40 /parenleft put dup 41 /parenright put dup 45 /hyphen put dup 46 /period put dup 50 /two put dup 70 /F put dup 73 /I put dup 79 /O put dup 84 /T put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put dup 122 /z put readonly def /FontBBox{-39 -250 1036 750}readonly def currentdict end currentfile eexec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All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMBX9) readonly def /FamilyName (Computer Modern) readonly def /Weight (Bold) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMBX9 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 65 /A put dup 97 /a put dup 98 /b put dup 99 /c put dup 114 /r put dup 115 /s put dup 116 /t put readonly def /FontBBox{-58 -250 1195 750}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F0364CD5660F74BEE96790DE35AFA90CCF712 B1805DA88AE375A04D99598EADFC625BDC1F9C315B6CF28C9BD427F32C745C99 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cleartomark %%EndFont %%BeginFont: CMR10 %!PS-AdobeFont-1.1: CMR10 1.00B %%CreationDate: 1992 Feb 19 19:54:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.00B) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR10) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR10 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /Gamma put dup 1 /Delta put dup 5 /Pi put dup 6 /Sigma put dup 11 /ff put dup 12 /fi put dup 13 /fl put dup 18 /grave put dup 19 /acute put dup 22 /macron put dup 34 /quotedblright put dup 40 /parenleft put dup 41 /parenright put dup 43 /plus put dup 44 /comma put dup 45 /hyphen put dup 46 /period put dup 47 /slash put dup 48 /zero put dup 49 /one put dup 50 /two put dup 51 /three put dup 52 /four put dup 53 /five put dup 54 /six put dup 55 /seven put dup 56 /eight put dup 57 /nine put dup 58 /colon put dup 61 /equal put dup 65 /A put dup 66 /B put dup 67 /C put dup 68 /D put dup 69 /E put dup 70 /F put dup 71 /G put dup 72 /H put dup 73 /I put dup 74 /J put dup 75 /K put dup 76 /L put dup 77 /M put dup 78 /N put dup 79 /O put dup 80 /P put dup 82 /R put dup 83 /S put dup 84 /T put dup 85 /U put dup 86 /V put dup 87 /W put dup 88 /X put dup 89 /Y put dup 91 /bracketleft put dup 92 /quotedblleft put dup 93 /bracketright put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 103 /g put dup 104 /h put dup 105 /i put dup 106 /j put dup 107 /k put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 113 /q put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 120 /x put dup 121 /y put dup 122 /z put dup 123 /endash put dup 126 /tilde put readonly def /FontBBox{-251 -250 1009 969}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 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7E66314646A5C03A050EE9C02706E493926EDD2BAC9A78890DF919 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 0000000000000000000000000000000000000000000000000000000000000000 cleartomark %%EndFont %%BeginFont: CMSY7 %!PS-AdobeFont-1.1: CMSY7 1.0 %%CreationDate: 1991 Aug 15 07:21:52 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY7) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY7 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 0 /minus put dup 3 /asteriskmath put dup 6 /plusminus put dup 21 /greaterequal put dup 33 /arrowright put dup 48 /prime put dup 49 /infinity put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-15 -951 1252 782}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMSY8 %!PS-AdobeFont-1.1: CMSY8 1.0 %%CreationDate: 1991 Aug 15 07:22:10 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMSY8) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle -14.035 def /isFixedPitch false def end readonly def /FontName /CMSY8 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 3 /asteriskmath put dup 121 /dagger put dup 122 /daggerdbl put readonly def /FontBBox{-30 -955 1185 779}readonly def currentdict end currentfile eexec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cleartomark %%EndFont %%BeginFont: CMR17 %!PS-AdobeFont-1.1: CMR17 1.0 %%CreationDate: 1991 Aug 20 16:38:24 % Copyright (C) 1997 American Mathematical Society. All Rights Reserved. 11 dict begin /FontInfo 7 dict dup begin /version (1.0) readonly def /Notice (Copyright (C) 1997 American Mathematical Society. All Rights Reserved) readonly def /FullName (CMR17) readonly def /FamilyName (Computer Modern) readonly def /Weight (Medium) readonly def /ItalicAngle 0 def /isFixedPitch false def end readonly def /FontName /CMR17 def /PaintType 0 def /FontType 1 def /FontMatrix [0.001 0 0 0.001 0 0] readonly def /Encoding 256 array 0 1 255 {1 index exch /.notdef put} for dup 70 /F put dup 71 /G put dup 83 /S put dup 97 /a put dup 98 /b put dup 99 /c put dup 100 /d put dup 101 /e put dup 102 /f put dup 105 /i put dup 108 /l put dup 109 /m put dup 110 /n put dup 111 /o put dup 112 /p put dup 114 /r put dup 115 /s put dup 116 /t put dup 117 /u put dup 118 /v put dup 119 /w put dup 121 /y put readonly def /FontBBox{-33 -250 945 749}readonly def currentdict end currentfile eexec D9D66F633B846A97B686A97E45A3D0AA052A014267B7904EB3C0D3BD0B83D891 016CA6CA4B712ADEB258FAAB9A130EE605E61F77FC1B738ABC7C51CD46EF8171 9098D5FEE67660E69A7AB91B58F29A4D79E57022F783EB0FBBB6D4F4EC35014F D2DECBA99459A4C59DF0C6EBA150284454E707DC2100C15B76B4C19B84363758 469A6C558785B226332152109871A9883487DD7710949204DDCF837E6A8708B8 2BDBF16FBC7512FAA308A093FE5F075EA0A10A15B0ED05D5039DA41B32B16E95 A3CE9725A429B35BAD796912FC328E3A28F96FCADA20A598E247755E7E7FF801 BDB00E9B9B086BDBE6EDCF841A3EAFC6F5284FED3C634085BA4EE0FC6A026E96 96D55575481B007BF93CA452EE3F71D83FAAB3D9DEDD2A8F96C5840EAE5BE5DC 9322E81DFF5E250DEB386E12A49FC9FBF9B4C25C3283F3CEA74B8278A1B09DA7 E9AE4FBAAF23EDF5A3E07D39385D521547C3AAAB8EB70549756EBA8EF445AF4A 497CA924ACCC3DD5456F8E2C7E36946A5BF14E2E959895F7C94F49137256BE46 4A238684D52792234869EAE1A6D8ADF4E138B79472D2A90A6CA99E2394CC20CD 3841733046175B20CEBE372327BF13428EED6A3E2FDF84C2DBA4B0AD584EE9DF B51828D3B8F385846158C29C9AC3496CB9692DD10219697B2ED4D425C3957FD8 C4600D76E045C561216EF05D38177243C314877A69A1C22E3BEC611A2EE5A216 9B7C264CF6D1839DBBD78A40610F2C0D7C2FE09FFA9822FF55035AD52546970F 83EED2D30EABB1F303091EBC11A5379B12BB3F405E371519A53EA9D66174ED25 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y(information)h(not)h(only)f(on)h(a)f(concrete)g(system)h(but)g (also)f(on)h(its)g(nearb)n(y)e(systems.)34 b(In)22 b(Section)e(2.3,)i (w)n(e)e(consider)g(b)r(oth)71 2564 y(the)i(classical)f(notion)h(of)f (top)r(ological)g(equiv)-5 b(alence)21 b(and)h(also)f(the)i(notion)e (of)h(\006-equiv)-5 b(alence,)23 b(whic)n(h)f(has)f(b)r(een)h(widely)71 2663 y(used)34 b(in)g(the)g(setting)g(of)g(Filipp)r(o)n(v)g(systems)f (\(see)h([KR)n(G03)o(,)g(BPS01)n(]\).)56 b(This)34 b(last)g (de\014nition)g(is)g(more)f(restrictiv)n(e)71 2763 y(than)26 b(the)h(usual)f(one.)36 b(In)26 b(Section)g(9,)g(a)g(comparativ)n(e)e (analysis)h(b)r(et)n(w)n(een)h(b)r(oth)h(concepts)f(based)f(in)i(some)e (mo)r(dels)i(is)71 2862 y(pro)n(vided.)195 2962 y(On)d(the)g(other)g (hand,)g(ev)n(en)g(if)g(the)g(de\014nition)h(of)e(bifurcation)h(is)g (based)f(on)h(breaking)e(structural)h(stabilit)n(y)-7 b(,)25 b(as)e(far)71 3062 y(as)g(the)g(authors)g(kno)n(w,)g(none)g(of)h 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b(needed)i(to)e(de\014ne)i(the)f(bifurcations)g(and)f(their)h(co)r (dimension.)35 b(The)22 b(results)g(in)g([KR)n(G03)n(])h(w)n(ere)e(ac)n (hiev)n(ed)71 3759 y(mainly)31 b(from)g(certain)f(normal)g(forms.)47 b(Ho)n(w)n(ev)n(er,)31 b(some)f(of)h(those)g(conditions)g(w)n(ere)f (not)h(explicitly)h(stated)f(there.)71 3859 y(Regarding)h(the)j(co)r (dimension-1)e(global)f(bifurcations,)k(w)n(e)d(prop)r(ose)g(a)h (systematic)f(approac)n(h)f(from)i(the)g(p)r(oin)n(t)h(of)71 3958 y(view)27 b(of)h(separatrix)d(connections)i(follo)n(wing)f(the)i (ideas)f(in)h([T)-7 b(ei77)o(].)37 b(The)28 b(authors)e(think)i(that)g (this)g(new)f(approac)n(h,)71 4058 y(b)r(eing)21 b(more)g(systematic,)h (helps)f(more)g(to)g(understand)g(the)h(full)g(classi\014cation)e(of)h (global)f(bifurcations)h(and)g(sharp)r(ens)71 4158 y(some)27 b(results)g(obtained)g(in)h([KR)n(G03)o(].)195 4257 y(Section)35 b(5)g(is)f(dev)n(oted)h(to)f(establish)h(a)f(preliminary)g (classi\014cation)f(of)i(the)g(co)r(dimension-2)f(singularities.)57 b(In)71 4357 y(Sections)26 b(6,)g(7,)f(8,)h(9,)g(10,)g(11,)f(12,)g(13)h (and)g(14)f(the)i(most)f(in)n(teresting)f(cases,)g(as)h(w)n(ell)g(as)f (their)h(bifurcation)g(diagrams,)71 4456 y(are)j(presen)n(ted.)44 b(All)30 b(the)h(lo)r(cal)e(and)h(global)f(co)r(dimension-1)g (bifurcations)h(whic)n(h)g(app)r(ear)f(in)h(their)h(unfoldings)e(are)71 4556 y(also)c(describ)r(ed.)36 b(It)26 b(is)g(w)n(orth)f(to)h(men)n (tion)f(that)i(in)f(this)g(study)g(w)n(e)g(detect)g(sev)n(eral)e(ric)n (h)h(phenomena)h(whic)n(h)g(are)e(not)71 4656 y(presen)n(t)j(in)h(an)n (y)f(co)r(dimension-1)f(singularit)n(y)g(and)i(are)e(gen)n(uinely)h (non-smo)r(oth.)195 4755 y(F)-7 b(or)30 b(instance,)g(in)g(Section)g(9) f(w)n(e)h(sho)n(w)f(a)g(singularit)n(y)f(whose)i(bifurcation)f(diagram) f(di\013ers)i(whether)g(one)f(con-)71 4855 y(siders)36 b(top)r(ological)f(or)h(\006-equiv)-5 b(alence.)65 b(On)36 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Fj(U)32 b Ft(:)23 b Fj(f)9 b Ft(\()p Fj(x;)14 b(y)s Ft(\))24 b Fj(<)e Ft(0)p Fi(g)p Fj(:)195 2683 y Ft(In)28 b(this)g(pap)r(er,)f (the)h(germs)f(of)g(discon)n(tin)n(uous)g(v)n(ector)f(\014elds)i(w)n(e) f(consider)g(are)f(of)i(the)g(form)1328 2911 y Fj(Z)6 b Ft(\()p Fj(x;)14 b(y)s Ft(\))24 b(=)1694 2794 y Fg(\032)1798 2860 y Fj(X)7 b Ft(\()p Fj(x;)14 b(y)s Ft(\))p Fj(;)28 b Ft(\()p Fj(x;)14 b(y)s Ft(\))23 b Fi(2)h Ft(\006)2471 2830 y Fm(+)1798 2960 y Fj(Y)19 b Ft(\()p Fj(x;)14 b(y)s Ft(\))p Fj(;)28 b Ft(\()p Fj(x;)14 b(y)s Ft(\))24 b Fi(2)f Ft(\006)2462 2929 y Fu(\000)2518 2960 y Fj(:)3744 2911 y Ft(\(1\))71 3143 y(F)-7 b(or)27 b(simplicit)n(y)-7 b(,)28 b(w)n(e)f(consider)f(only)i(germs)e(of)i(v)n(ector)e(\014elds)i (in)g(a)f(neigh)n(b)r(orho)r(o)r(d)f(of)i(\(0)p Fj(;)14 b Ft(0\).)195 3243 y(W)-7 b(e)29 b(denote)f Fj(Z)h Ft(=)23 b(\()p Fj(X)r(;)14 b(Y)k Ft(\))29 b(in)f(order)e(to)i(clarify)f(whic)n (h)h(are)f(the)h(comp)r(onen)n(ts)g(of)g(the)g(v)n(ector)e(\014eld.)39 b(F)-7 b(urthermore)71 3342 y(w)n(e)31 b(assume)f(that)h(they)h(are)e Fi(C)1052 3312 y Fh(r)1119 3342 y Ft(for)h Fj(r)g(>)e Ft(1)h(in)p 1585 3272 116 4 v 32 w(\006)1645 3318 y Fm(+)1731 3342 y Ft(and)p 1896 3272 117 4 v 31 w(\006)1956 3318 y Fu(\000)2043 3342 y Ft(resp)r(ectiv)n(ely)-7 b(.)46 b(In)32 b(this)f(assumption)g(w)n(e)f(are)g(using)h(the)71 3442 y(standard)f(con)n(v)n(en)n(tion)g(that)i(for)f(a)f(function)i (de\014ned)g(in)g(a)f(non-op)r(en)g(domain)f Fj(D)r Ft(,)j(b)r(eing)e (class)g Fi(C)3286 3412 y Fh(r)3354 3442 y Ft(means)f(that)i(it)71 3542 y(can)f(b)r(e)g(extended)g(to)g(a)g Fi(C)926 3511 y Fh(r)994 3542 y Ft(function)g(de\014ned)h(on)f(an)g(op)r(en)g(set)g (con)n(taining)f Fj(D)r Ft(,)i(and)f(the)h(same)e(applies)h(to)g(v)n (ector)71 3641 y(\014elds.)195 3741 y(W)-7 b(e)29 b(call)g Fi(Z)560 3711 y Fh(r)625 3741 y Ft(to)g(the)g(space)f(of)g(v)n(ector)g (\014elds)h(of)f(this)h(t)n(yp)r(e.)41 b(It)29 b(can)f(b)r(e)h(tak)n (en)f(as)g Fi(Z)2891 3711 y Fh(r)2953 3741 y Ft(=)c Ff(X)3102 3711 y Fh(r)3158 3741 y Fi(\002)19 b Ff(X)3302 3711 y Fh(r)3367 3741 y Ft(and)28 b(therefore)71 3840 y(w)n(e)f(consider)g(it) h(with)g(the)g(pro)r(duct)f Fi(C)1292 3810 y Fh(r)1356 3840 y Ft(top)r(ology)-7 b(.)195 3940 y(The)28 b(\014rst)g(step)g(is)g (to)g(de\014ne)g(rigorously)e Fj(')1581 3952 y Fh(Z)1635 3940 y Ft(\()p Fj(t;)14 b(p)p Ft(\))28 b(for)f(an)n(y)h Fj(p)23 b Fi(2)h Fj(U)9 b Ft(.)38 b(In)28 b(other)g(w)n(ords,)f(in)h (order)f(to)g(establish)h(the)71 4040 y(dynamics)i(giv)n(en)f(b)n(y)h (the)h(Filipp)r(o)n(v)f(v)n(ector)f(\014eld)i Fj(Z)i Ft(=)27 b(\()p Fj(X)r(;)14 b(Y)k Ft(\))31 b(in)g Fj(U)9 b Ft(,)30 b(\014rst)h(step)f(is)g(to)g(de\014ne)h(the)f(lo)r(cal)g(tra) 5 b(jectory)71 4139 y(through)27 b(a)h(p)r(oin)n(t)h Fj(p)24 b Fi(2)h Fj(U)9 b Ft(.)38 b(T)-7 b(o)28 b(this)h(end,)g(w)n(e)f (need)g(to)g(distinguish)h(whether)f(this)g(p)r(oin)n(t)h(b)r(elongs)f (to)g(\006)3414 4109 y Fm(+)3469 4139 y Ft(,)h(\006)3581 4109 y Fu(\000)3665 4139 y Ft(or)e(\006.)71 4239 y(Since)h(w)n(e)f(are) f(dealing)h(with)i(autonomous)d(systems)h(w)n(e)g(will)h(use)f(tra)5 b(jectory)26 b(and)i(orbit)f(indistinctly)-7 b(.)195 4339 y(F)g(or)33 b(the)h(\014rst)g(t)n(w)n(o)e(regions,)i(the)g(lo)r (cal)f(tra)5 b(jectory)32 b(is)h(de\014ned)h(b)n(y)f(the)h(v)n(ector)f (\014elds)g Fj(X)40 b Ft(and)34 b Fj(Y)52 b Ft(as)33 b(usual.)54 b(In)71 4438 y(order)31 b(to)i(extend)g(its)g(de\014nition) g(to)g(\006)g(w)n(e)f(split)h(it)g(in)g(three)g(parts)f(dep)r(ending)h (whether)g(the)g(v)n(ector)e(\014elds)i(p)r(oin)n(t)71 4538 y(to)n(w)n(ards)26 b(\006)h(or)g(not:)p Black 172 4704 a(1.)p Black 41 w(Crossing)f(Region:)36 b(\006)986 4674 y Fh(c)1043 4704 y Ft(=)23 b Fi(f)p Fj(p)f Fi(2)i Ft(\006)f(:)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))17 b Fi(\001)i Fj(Y)g(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fj(>)f Ft(0)p Fi(g)p Black 172 4870 a Ft(2.)p Black 41 w(Sliding)28 b(Region:)36 b(\006)925 4840 y Fh(s)984 4870 y Ft(=)22 b Fi(f)p Fj(p)h Fi(2)g Ft(\006)g(:)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))23 b Fj(<)f Ft(0)p Fj(;)14 b(Y)k(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fj(>)g Ft(0)p Fi(g)p Black 172 5036 a Ft(3.)p Black 41 w(Escaping)k(Region:)36 b(\006)1001 5006 y Fh(e)1059 5036 y Ft(=)23 b Fi(f)p Fj(p)f Fi(2)i Ft(\006)f(:)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))22 b Fj(>)h Ft(0)p Fj(;)14 b(Y)k(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fj(<)g Ft(0)p Fi(g)71 5202 y Ft(where)k Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))22 b(=)h Fj(X)7 b Ft(\()p Fj(p)p Ft(\))18 b Fi(\001)g Ft(grad)o Fj(f)9 b Ft(\()p Fj(p)p Ft(\))28 b(is)f(the)h(Lie)g(deriv)-5 b(ativ)n(e)26 b(of)i Fj(f)36 b Ft(with)28 b(resp)r(ect)g(to)f(the)h(v)n(ector)e(\014eld)i Fj(X)34 b Ft(at)28 b Fj(p)p Ft(.)195 5302 y(These)g(three)g(regions)f (are)g(relativ)n(ely)f(op)r(en)j(in)f(\006)g(and)g(can)g(ha)n(v)n(e)f (sev)n(eral)f(connected)i(comp)r(onen)n(ts.)38 b(Therefore,)71 5401 y(their)29 b(de\014nitions)h(exclude)f(the)h(so)f(called)g Fk(tangency)i(p)l(oints)p Ft(,)g(that)e(is,)h(p)r(oin)n(ts)g(where)e (one)h(of)h(the)g(t)n(w)n(o)e(v)n(ector)g(\014elds)71 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Fj(p)p Ft(\))18 b Fi(\000)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))2372 1718 y(\()p Fj(Y)19 b(f)9 b Ft(\()p Fj(p)p Ft(\))p Fj(X)e Ft(\()p Fj(p)p Ft(\))18 b Fi(\000)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))p Fj(Y)18 b Ft(\()p Fj(p)p Ft(\)\))398 b(\(2\))71 1945 y(This)23 b(v)n(ector)g(\014eld)g(is)h(called)f Fk(sliding)k(ve)l(ctor)f(\014eld)33 b Ft(indep)r(enden)n(tly)24 b(whether)g(it)g(is)f(de\014ned)h(in)g(the)g(sliding)f(or)g(escaping)71 2044 y(region,)j(and)g(for)g Fj(p)d Fi(2)h Ft(\006)833 2014 y Fh(s)885 2044 y Fi([)17 b Ft(\006)1017 2014 y Fh(e)1079 2044 y Ft(the)27 b(lo)r(cal)g(orbit)f(of)h Fj(p)f Ft(is)h(giv)n(en)f(b)n(y)g(this)h(v)n(ector)f(\014eld)h(\(and)g (therefore)f(is)g(con)n(tained)g(in)71 2144 y(\006)131 2114 y Fh(s)194 2144 y Ft(or)h(\006)356 2114 y Fh(e)391 2144 y Ft(\).)38 b(All)28 b(of)f(this)h(is)g(summarized)e(in)i(next)g (de\014nition.)p Black 71 2310 a Fe(De\014nition)c(2.2.)p Black 35 w Fk(The)h(lo)l(c)l(al)g(tr)l(aje)l(ctory)g(\(or)g(orbital)g 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Fi(\\)c Ft(\006)30 b Fk(c)l(onsists)f(only)h(of)h(cr)l(ossing)f(p)l(oints)g (and)h(r)l(e)l(gular)e(tangency)h(p)l(oints)g(in)g Fj(@)5 b Ft(\006)3517 3701 y Fh(c)3551 3731 y Fk(,)p Black 169 3893 a(3.)p Black 42 w Fj(\015)35 b Fk(is)30 b(maximal)g(with)h(r)l (esp)l(e)l(ct)e(to)h(these)g(c)l(onditions.)p Black 71 4051 a Fe(De\014nition)g(2.8.)p Black 39 w Fk(A)e(sliding)j(orbit)e(of) h Fj(Z)k Fk(is)c(a)f(smo)l(oth)g(curve)g Fj(\015)f Fi(\032)p 2265 3984 96 4 v 22 w Ft(\006)2325 4027 y Fh(s)2377 4051 y Fi([)p 2449 3984 V 17 w Ft(\006)2509 4027 y Fh(e)2574 4051 y Fk(such)h(that)g(it)f(is)i(a)f(maximal)h(tr)l(aje)l(ctory)71 4150 y(of)g Fj(Z)225 4162 y Fh(s)261 4150 y Fk(.)195 4308 y Ft(In)e([BPS01)o(],)f(the)h(sliding)g(orbit)f(is)g(called)h (singular)e(orbit.)p Black 71 4466 a Fe(De\014nition)35 b(2.9.)p Black 42 w Fk(A)e(distinguishe)l(d)h(singularity)g(is)g(a)f(p) l(oint)g Fj(p)g Fk(such)g(that)g Fj(')2592 4478 y Fh(Z)2646 4466 y Ft(\()p Fj(t;)14 b(p)p Ft(\))30 b(=)e Fj(p)p Fk(.)49 b(They)34 b(c)l(an)f(b)l(e)g(classi\014e)l(d)71 4566 y(as:)p Black 195 4723 a Fi(\017)p Black 41 w Fj(p)23 b Fi(2)h Ft(\006)482 4693 y Fu(\006)567 4723 y Fk(such)30 b(that)g Fj(X)7 b Ft(\()p Fj(p)p Ft(\))23 b(=)f(0)29 b Fk(and)i Fj(Y)18 b Ft(\()p Fj(p)p Ft(\))24 b(=)e(0)30 b Fk(r)l(esp)l(e)l(ctively.)p Black 195 4885 a Fi(\017)p Black 41 w Fj(p)23 b Fi(2)h Ft(\006)482 4855 y Fh(s)536 4885 y Fi([)18 b Ft(\006)669 4855 y Fh(e)735 4885 y Fk(such)29 b(that)h Fj(Z)1150 4897 y Fh(s)1185 4885 y Ft(\()p Fj(p)p Ft(\))24 b(=)e(0)p Fk(,)30 b(namely)h(a)f(pseudo)l(e)l(quilibria.)p Black 195 5047 a Fi(\017)p Black 41 w Fj(p)23 b Fi(2)h Fj(@)5 b Ft(\006)531 5017 y Fh(c)583 5047 y Fi([)18 b Fj(@)5 b Ft(\006)765 5017 y Fh(s)819 5047 y Fi([)19 b Fj(@)5 b Ft(\006)1002 5017 y Fh(e)1067 5047 y Fk(such)30 b(that)f(it)h(is)g(a)g(singular)h(tangency)e(p)l(oint.)195 5205 y Ft(As)k(w)n(e)e(ha)n(v)n(e)g(said,)i(these)f(de\014nitions)g (lead)g(to)g(t)n(w)n(o)f(features)g(\(already)g(presen)n(t)g(in)i ([BPS01)n(]\))g(that)f(mak)n(e)f(this)71 5305 y(approac)n(h)j(suitable) i(in)h(the)g(study)f(of)g(the)h(structural)e(stabilit)n(y)h(and)h (generic)e(bifurcations:)53 b(\014rstly)-7 b(,)39 b(uniqueness)71 5404 y(of)c(solutions,)i(that)e(is,)i(an)n(y)e Fj(p)g Fi(2)i Fj(U)44 b Ft(b)r(elongs)34 b(to)h(only)g(one)g(orbit,)i(and)e (secondly)-7 b(,)37 b(an)n(y)d(neigh)n(b)r(orho)r(o)r(d)g Fj(U)44 b Ft(of)35 b Fj(p)h Ft(is)71 5504 y(decomp)r(osed)27 b(in)n(to)g(disjoin)n(t)h(union)f(of)h(orbits.)p Black 1940 5753 a(7)p Black eop end %%Page: 8 8 TeXDict begin 8 7 bop Black Black 71 272 a Fl(2.2)112 b(Separatrices,)39 b(p)s(erio)s(dic)f(orbits)f(and)i(cycles)71 425 y Ft(In)30 b(this)f(section,)h(w)n(e)f(generalize)f(the)i(concepts) f(of)g(separatrix)f(and)h(p)r(erio)r(dic)h(orbit)f(for)g(planar)f (Filipp)r(o)n(v)i(Systems.)71 525 y(F)-7 b(or)27 b(the)h(case)f(of)g (separatrices)f(w)n(e)h(follo)n(w)g(closely)f([BPS01)o(,)h(T)-7 b(ei77].)p Black 71 691 a Fe(De\014nition)31 b(2.10)f Ft(\([BPS01)o(]\))p Fe(.)p Black 41 w Fk(A)n(n)f(unstable)h(sep)l(ar)l (atrix)g(is)g(either)p Black 195 857 a Fi(\017)p Black 41 w Fk(A)f(r)l(e)l(gular)g(orbit)h(such)f(that)f(its)h Fj(\013)p Fk(-limit)h(set)e(is)i(a)f(r)l(e)l(gular)g(sadd)t(le)i(p)l (oint)e Fj(p)23 b Fi(2)p 2763 787 116 4 v 23 w Ft(\006)2823 833 y Fm(+)2907 857 y Fk(of)30 b Fj(X)35 b Fk(or)29 b Fj(p)23 b Fi(2)p 3358 787 117 4 v 24 w Ft(\006)3418 833 y Fu(\000)3502 857 y Fk(of)30 b Fj(Y)19 b Fk(.)38 b(We)278 956 y(denote)31 b(it)e(by)i Fj(W)825 926 y Fh(u)868 956 y Ft(\()p Fj(p)p Ft(\))p Fk(.)p Black 195 1122 a Fi(\017)p Black 41 w Fk(A)26 b(r)l(e)l(gular)f(orbit)h(which)h(dep)l(arts)g(fr)l (om)f(a)f(distinguishe)l(d)i(singularity)g Fj(p)c Fi(2)g Ft(\006)p Fk(.)37 b(We)26 b(denote)g(it)f(by)h Fj(W)3443 1092 y Fh(u)3431 1143 y Fu(\006)3488 1122 y Ft(\()p Fj(p)p Ft(\))p Fk(,)h(wher)l(e)278 1222 y(the)j(subscript)g Fi(\006)g Fk(me)l(ans)f(that)h(it)g(le)l(aves)g Fj(p)g Fk(fr)l(om)g Ft(\006)1932 1192 y Fu(\006)2018 1222 y Fk(r)l(esp)l(e)l(ctively.)195 1388 y Ft(W)-7 b(e)21 b(remark)e(that)i (in)g(the)g(\014rst)g(case,)g(as)f(it)h(is)f(w)n(ell)h(kno)n(wn)e(in)i (smo)r(oth)g(systems,)g(the)g(orbit)f(lying)h(in)f(the)h(separatrix)71 1487 y(only)27 b(reac)n(hes)f Fj(p)h Ft(in)h(in\014nite)h(time)f (whereas)e(in)i(the)g(second)f(case,)g(it)h(ma)n(y)f(reac)n(h)f(the)i (singularit)n(y)e(in)i(\014nite)g(time.)195 1587 y(Stable)c (separatrices)e Fj(W)985 1557 y Fh(s)1021 1587 y Ft(\()p Fj(p)p Ft(\))i(and)g Fj(W)1399 1557 y Fh(s)1387 1607 y Fu(\006)1443 1587 y Ft(\()p Fj(p)p Ft(\))h(are)e(de\014ned)h (analogously)-7 b(.)34 b(If)24 b(a)g(separatrix)e(is)i(sim)n (ultaneously)f(stable)71 1687 y(and)k(unstable)h(it)g(is)f(a)h Fk(sep)l(ar)l(atrix)i(c)l(onne)l(ction)p Ft(.)p Black 71 1852 a Fe(Remark)d(2.11.)p Black 37 w Fk(We)f(p)l(oint)g(out)g(that) g(a)h(pseudono)l(de)h Fj(p)22 b Fi(2)i Ft(\006)2032 1822 y Fh(s)2093 1852 y Fk(do)l(es)j(have)h(sep)l(ar)l(atric)l(es)f(which)h (ar)l(e)e(given)h(by)g(the)f(two)71 1952 y(r)l(e)l(gular)31 b(orbits)h(which)i(arrive)e(at)g(it.)44 b(R)l(e)l(c)l(al)t(l)31 b(that)h(the)f(p)l(oints)h(in)g(these)f(sep)l(ar)l(atric)l(es)i(hit)f (the)f(pseudono)l(de)i(in)f(\014nite)71 2052 y(time,)e(wher)l(e)l(as)h (al)t(l)f(the)g(other)g(p)l(oints)g(in)g Ft(\006)1469 2022 y Fm(+)1554 2052 y Fk(and)g Ft(\006)1775 2022 y Fu(\000)1860 2052 y Fk(close)h(to)e(the)h(pseudono)l(de,)i(\014rst)d (hit)h Ft(\006)3124 2022 y Fh(s)3189 2052 y Fk(in)f(\014nite)h(time,)g (and)71 2151 y(then)f(sp)l(end)i(in\014nite)e(time)h(in)g(the)g (sliding)h(r)l(e)l(gion)f(tending)g(to)f(the)h(pseudono)l(de.)195 2317 y Ft(Regarding)23 b(the)h(generalization)e(of)i(the)g(concept)f (of)h(p)r(erio)r(dic)g(orbit)f(in)h(Filipp)r(o)n(v)g(systems)g(w)n(e)f (ha)n(v)n(e)g(to)g(deal)h(with)71 2417 y(di\013eren)n(t)k(cases.)35 b(The)28 b(\014rst)f(one)h(is)f(the)h Fk(r)l(e)l(gular)i(p)l(erio)l (dic)i(orbit)p Ft(.)p Black 71 2583 a Fe(De\014nition)23 b(2.12.)p Black 34 w Fk(A)g(r)l(e)l(gular)h(p)l(erio)l(dic)i(orbit)e (is)g(a)g(r)l(e)l(gular)g(orbit)g Fj(')p Ft(\()p Fj(t;)14 b(p)p Ft(\))p Fk(,which)26 b(ther)l(efor)l(e)f(b)l(elongs)f(to)g Ft(\006)3454 2553 y Fm(+)3514 2583 y Fi([)5 b Ft(\006)3634 2553 y Fu(\000)3696 2583 y Fi([)p 3756 2516 94 4 v 5 w Ft(\006)3816 2559 y Fh(c)71 2682 y Fk(and)30 b(satis\014es)g Fj(')p Ft(\()p Fj(t)19 b Ft(+)f Fj(T)7 b(;)14 b(p)p Ft(\))22 b(=)h Fj(')p Ft(\()p Fj(t;)14 b(p)p Ft(\))30 b Fk(for)h(some)f Fj(T)k(>)23 b Ft(0)p Fk(.)195 2848 y Ft(The)j(second)f(case)g(is)h(the) g(singular)f(p)r(erio)r(dic)g(orbit.)36 b(This)26 b(case)f(app)r(ears)f (when)i(\006)g(is)g(homeomorphic)e(to)i Fi(S)3653 2818 y Fm(1)3717 2848 y Ft(and)71 2948 y(\006)e(=)f(\006)303 2918 y Fh(s)367 2948 y Ft(or)k(\006)d(=)f(\006)701 2918 y Fh(e)765 2948 y Ft(in)28 b(suc)n(h)g(a)g(w)n(a)n(y)e(that)j(the)f (sliding)g(v)n(ector)f(\014eld)h(do)r(es)g(not)g(ha)n(v)n(e)f(critical) g(p)r(oin)n(ts.)39 b(In)28 b(that)g(case,)71 3047 y(the)e(whole)g(\006) h(is)f(a)g(p)r(erio)r(dic)g(orbit.)36 b(This)26 b(case)f(do)r(es)h(not) g(app)r(ear)g(in)g(this)h(article)e(since)h(in)h(it)f(w)n(e)g(only)g (study)h(planar)71 3147 y(Filipp)r(o)n(v)g(systems)h(lo)r(cally)f(and)g (then)h(\006)g(is)f(alw)n(a)n(ys)f(homeomorphic)g(to)i(an)f(op)r(en)h (segmen)n(t.)195 3247 y(F)-7 b(rom)29 b(De\014nitions)g(2.7)f(and)h (2.8,)g(it)g(is)g(clear)f(that)h(there)g(can)f(not)h(exist)g(p)r(erio)r (dic)g(orbits)f(whic)n(h)h(in)n(v)n(olv)n(e)e(at)i(the)71 3346 y(same)e(time)i(p)r(oin)n(ts)f(in)g(\006)876 3316 y Fm(+)950 3346 y Fi([)19 b Ft(\006)1084 3316 y Fu(\000)1168 3346 y Ft(and)28 b(p)r(oin)n(ts)g(in)g(\006)1737 3316 y Fh(s)1791 3346 y Fi([)19 b Ft(\006)1925 3316 y Fh(e)1989 3346 y Ft(\(that)29 b(is,)f(p)r(erio)r(dic)f(orbits)h(whic)n(h)g(are)f (a)g(com)n(bination)h(of)71 3446 y Fk(r)l(e)l(gular)g(motion)k Ft(and)25 b Fk(sliding)k(motion)6 b Ft(\))27 b(since)e(an)g(orbit)g (can)g(not)g(in)n(tersect)g(b)r(oth)h(sets.)35 b(Th)n(us,)26 b(w)n(e)f(will)g(de\014ne)h(cycles)71 3545 y(to)h(deal)h(with)g Fk(p)l(erio)l(dic)k(motion)i Ft(whic)n(h)28 b(in)n(v)n(olv)n(es)d(at)j (the)g(same)f(time)h(sliding)f(and)h(regular)e(motion.)p Black 71 3711 a Fe(De\014nition)36 b(2.13.)p Black 42 w Fk(A)d(p)l(erio)l(dic)j(cycle)f(is)f(a)g(\014nite)f(set)g(of)i (orbits)f Fj(\015)2283 3723 y Fm(1)2320 3711 y Fj(;)14 b(:)g(:)g(:)g(\015)2511 3723 y Fh(n)2590 3711 y Fk(such)33 b(that)h(the)g(arrival)h(p)l(oint)f(of)h Fj(\015)3730 3723 y Fh(i)3791 3711 y Fk(is)71 3811 y(the)30 b(dep)l(arting)h(p)l (oint)f(of)g Fj(\015)923 3823 y Fh(i)p Fm(+1)1065 3811 y Fk(and)g(the)g(arrival)h(p)l(oint)f(of)h Fj(\015)1984 3823 y Fh(n)2059 3811 y Fk(is)f(the)g(dep)l(arting)h(p)l(oint)f(of)g Fj(\015)3000 3823 y Fm(1)3038 3811 y Fk(.)195 3911 y(We)35 b(de\014ne)f(the)h(p)l(erio)l(d)h(of)f(the)g(cycle)g(as)g(the)g(sum)e (of)j(the)e(times)h(that)f(ar)l(e)h(sp)l(ent)f(to)g(r)l(e)l(ach)h(the)g (arrival)h(p)l(oints)71 4010 y(fr)l(om)30 b(the)g(dep)l(arting)h(p)l (oints)f(of)g(the)g(same)h(orbit.)195 4176 y Ft(In)f([KR)n(G03)o(])f (the)h(regular)d(p)r(erio)r(dic)j(orbits)e(are)h(classi\014ed)f(b)r(et) n(w)n(een)i Fk(standar)l(d)i(p)l(erio)l(dic)h(orbits)k Ft(if)30 b(they)g(sta)n(y)e(in)71 4276 y(\006)131 4246 y Fm(+)206 4276 y Fi([)21 b Ft(\006)342 4246 y Fu(\000)429 4276 y Ft(and)31 b Fk(cr)l(ossing)i(p)l(erio)l(dic)h(orbits)39 b Ft(if)31 b(they)g(in)n(tersect)p 2077 4209 V 30 w(\006)2137 4252 y Fh(c)2171 4276 y Ft(.)46 b(On)30 b(the)h(other)g(hand)f(they)h (call)f Fk(sliding)k(p)l(erio)l(dic)71 4375 y(orbits)h Ft(to)27 b(cycles.)195 4475 y(Besides)32 b(cycles)g(and)g(p)r(erio)r (dic)g(orbits,)h(there)f(exists)g(another)f(distinguished)i(geometric)e (ob)5 b(ject)32 b(whic)n(h)g(is)g(im-)71 4575 y(p)r(ortan)n(t)27 b(when)h(one)f(studies)h(top)r(ological)e(equiv)-5 b(alences)26 b(and)i(bifurcations)f(in)h(Filipp)r(o)n(v)f(systems.)p Black 71 4740 a Fe(De\014nition)34 b(2.14.)p Black 42 w Fk(We)f(de\014ne)39 b Ft(pseudo)r(cycle)32 b 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y Ft(1)2084 2656 y(1)2167 2452 y Fg(!)2276 2594 y Fk(for)g Fj(y)c(>)d Ft(0)1803 2822 y(~)1790 2843 y Fj(Y)42 b Ft(=)1968 2701 y Fg( )2107 2786 y Ft(1)2075 2906 y Fi(\000)p Ft(1)2223 2701 y Fg(!)2332 2843 y Fk(for)30 b Fj(y)c(<)d Ft(0)p Fj(:)278 3059 y Fk(in)30 b Ft(\(0)p Fj(;)14 b Ft(0\))23 b Fi(2)g Fj(V)c Fk(.)p Black 71 3214 a(Pr)l(o)l(of.)p Black 43 w Ft(In)32 b(the)f(\014rst)g(case,)h(the)f(construction)g(of)g (the)h(\006-equiv)-5 b(alence)30 b(is)i(ac)n(hiev)n(ed)e(considering)g Fj(')3282 3226 y Fh(X)3345 3214 y Ft(,)j Fj(')3455 3226 y Fh(Y)3512 3214 y Ft(,)g Fj(')3640 3224 y Fm(~)3622 3239 y Fh(X)3717 3214 y Ft(and)71 3313 y Fj(')135 3323 y Fm(~)125 3338 y Fh(Y)215 3313 y Ft(the)h(\015o)n(ws)d(of)i(the)g(smo) r(oth)g(comp)r(onen)n(ts)f(of)g(b)r(oth)i(v)n(ector)d(\014elds.)52 b(Since)33 b(\(0)p Fj(;)14 b Ft(0\))31 b Fi(2)h Ft(\006)2983 3283 y Fh(c)3050 3313 y Ft(these)g(v)n(ector)g(\014elds)g(are)71 3423 y(transv)n(ersal)22 b(to)j(\006)13 b Fi(\\)g Fj(U)34 b Ft(and)988 3402 y(~)979 3423 y(\006)13 b Fi(\\)g Fj(V)44 b Ft(resp)r(ectiv)n(ely)-7 b(.)36 b(Th)n(us,)25 b(for)f(an)n(y)g(p)r (oin)n(t)h Fj(p)e Fi(2)h Ft(\006)2617 3393 y Fm(+)2685 3423 y Fi(\\)13 b Fj(U)c Ft(,)25 b(using)g(the)g(Implicit)h(F)-7 b(unction)71 3523 y(Theorem)31 b(there)i(exists)f(a)g(time)g Fj(t)p Ft(\()p Fj(p)p Ft(\))g Fi(2)f Fn(R)i Ft(dep)r(ending)f(on)h Fj(p)f Ft(suc)n(h)g(that)g Fj(')2520 3535 y Fh(X)2584 3523 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)14 b(p)p Ft(\)\))32 b Fi(2)f Ft(\006,)j(and)e(analogously)e(for)71 3622 y(\006)131 3592 y Fu(\000)207 3622 y Fi(\\)21 b Fj(U)39 b Ft(and)30 b Fj(')597 3634 y Fh(Y)655 3622 y Ft(.)45 b(Th)n(us,)31 b(imp)r(osing)f(that)g(the)h(\006-equiv)-5 b(alence)30 b(is)g(the)h(iden)n(tit)n(y)f(restricted)g(to)g(\006,)h(it) g(can)f(b)r(e)g(giv)n(en)71 3722 y(b)n(y:)1100 3917 y Fj(h)p Ft(\()p Fj(p)p Ft(\))23 b(=)1364 3746 y Fg(8)1364 3821 y(<)1364 3971 y(:)1480 3816 y Fj(')1552 3826 y Fm(~)1534 3841 y Fh(X)1611 3816 y Ft(\()p Fi(\000)p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)14 b(')1935 3828 y Fh(X)1999 3816 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)g(p)p Ft(\)\))125 b(if)28 b Fj(p)23 b Fi(2)g Ft(\006)2714 3786 y Fm(+)1480 3916 y Fj(p)913 b Ft(if)28 b Fj(p)23 b Fi(2)g Ft(\006)1480 4015 y Fj(')1544 4025 y Fm(~)1534 4040 y Fh(Y)1605 4015 y Ft(\()q Fi(\000)p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)14 b(')1930 4027 y Fh(Y)1987 4015 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)g(p)p Ft(\)\))137 b(if)28 b Fj(p)23 b Fi(2)g Ft(\006)2714 3985 y Fu(\000)3744 3917 y Ft(\(4\))71 4163 y(whic)n(h)k(it)h(can)g(b)r(e)g(seen)f(that)h(is)f Fi(C)1149 4133 y Fm(0)1186 4163 y Ft(,)h(and)f(v)n(eri\014es)g Fj(')1743 4173 y Fm(~)1730 4188 y Fh(Z)1784 4163 y Ft(\()p Fj(t;)14 b(h)p Ft(\()p Fj(p)p Ft(\)\))23 b(=)g Fj(h)p Ft(\()p Fj(')2314 4175 y Fh(Z)2368 4163 y Ft(\()p Fj(t;)14 b(p)p Ft(\)\).)195 4273 y(Finally)-7 b(,)33 b(in)f(this)g(\014rst)g (case,)g(only)f(remains)g(to)h(p)r(oin)n(t)g(out)f(that)h(in)g(fact) 2595 4252 y(~)2577 4273 y Fj(Z)38 b Ft(is)31 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2892 4649 y(\026)2891 4671 y Fj(h)g Ft(whic)n(h)h(lo)r(cally)e (conjugates)71 4771 y(them.)36 b(F)-7 b(or)25 b(p)r(oin)n(ts)f(a)n(w)n (a)n(y)f(from)i(\006,)g(the)h(homeomorphism)d(can)i(b)r(e)g(extended)g (as)f(in)h(the)h(previous)e(cases)f(b)n(y)i(the)g(\015o)n(w)71 4870 y(since)k(for)g(an)n(y)f(p)r(oin)n(t)i Fj(p)25 b Fi(2)i Ft(\006)991 4840 y Fm(+)1065 4870 y Fi(\\)20 b Fj(U)38 b Ft(there)29 b(exists)g(a)g(time)h Fj(t)p Ft(\()p Fj(p)p Ft(\))g(suc)n(h)e(that)i Fj(')2532 4882 y Fh(X)2595 4870 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)14 b(p)p Ft(\)\))27 b Fi(2)f Ft(\006)k(\(and)f(analogously)e(for)71 4970 y Fj(p)c Fi(2)g Ft(\006)274 4940 y Fu(\000)349 4970 y Fi(\\)18 b Fj(U)9 b Ft(\).)37 b(Th)n(us,)28 b(the)g(homeomorphism)e (whic)n(h)i(giv)n(es)e(the)i(\006-equiv)-5 b(alence)27 b(can)g(b)r(e)h(giv)n(en)f(b)n(y)933 5239 y Fj(h)p Ft(\()p Fj(p)p Ft(\))c(=)1198 5069 y Fg(8)1198 5144 y(<)1198 5293 y(:)1313 5139 y Fj(')1386 5149 y Fm(~)1367 5164 y Fh(X)1445 5072 y Fg(\000)1483 5139 y Fi(\000)p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)1721 5117 y Ft(\026)1721 5139 y Fj(h)13 b Ft(\()p Fj(')1868 5151 y Fh(X)1932 5139 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)h(p)p Ft(\)\))2244 5072 y Fg(\001)2406 5139 y Ft(if)28 b Fj(p)23 b Fi(2)h Ft(\006)2686 5109 y Fm(+)2759 5139 y Fi(\\)19 b Fj(U)1314 5217 y Ft(\026)1313 5238 y Fj(h)p Ft(\()p Fj(p)p Ft(\))939 b(if)28 b Fj(p)23 b Fi(2)h Ft(\006)18 b Fi(\\)h Fj(U)1313 5339 y(')1377 5349 y Fm(~)1367 5364 y Fh(Y)1439 5272 y Fg(\000)1477 5339 y Fi(\000)p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)1716 5317 y Ft(\026)1715 5339 y Fj(h)14 b Ft(\()p Fj(')1863 5351 y Fh(Y)1921 5339 y Ft(\()p Fj(t)p Ft(\()p Fj(p)p Ft(\))p Fj(;)g(p)p Ft(\)\))2232 5272 y Fg(\001)2406 5339 y Ft(if)28 b Fj(p)23 b Fi(2)h Ft(\006)2686 5309 y Fu(\000)2760 5339 y Fi(\\)19 b Fj(U)2965 5239 y(:)756 b Ft(\(5\))71 5504 y(The)28 b(third)f(case,)g(can)g(b)r(e)h (studied)g(analogously)e(to)h(the)h(second.)p 3790 5504 4 57 v 3794 5451 50 4 v 3794 5504 V 3843 5504 4 57 v Black 1919 5753 a(11)p Black eop end %%Page: 12 12 TeXDict begin 12 11 bop Black Black 195 272 a Ft(F)-7 b(or)32 b(planar)f(Filipp)r(o)n(v)h(v)n(ector)f(\014elds,)i(there)f (exist)g(the)h(follo)n(wing)e(generic)g(singularities)g(whic)n(h)h(are) f(all)h(distin-)71 372 y(guished)27 b(singularities)g(\(see)g (De\014nition)h(2.5\):)p Black 172 538 a(1.)p Black 41 w(F)-7 b(old)28 b(P)n(oin)n(ts)f(or)g(quadratic)g(tangency)g(p)r(oin)n (ts:)37 b(that)28 b(is)g Fj(p)23 b Fi(2)h Ft(\006)k(suc)n(h)g(that)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))23 b(=)g(0)k(and)h Fj(X)3343 508 y Fm(2)3380 538 y Fj(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fi(6)p Ft(=)g(0)28 b(and)278 637 y Fj(Y)19 b(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fi(6)p Ft(=)g(0)k(or)g(p)r(oin)n(ts)g (suc)n(h)g(that)h Fj(Y)19 b(f)9 b Ft(\()p Fj(p)p Ft(\))23 b(=)g(0)k(and)g Fj(Y)2030 607 y Fm(2)2067 637 y Fj(f)9 b Ft(\()p Fj(p)p Ft(\))23 b Fi(6)p Ft(=)g(0)k(and)g Fj(X)7 b(f)i Ft(\()p Fj(p)p Ft(\))23 b Fi(6)p Ft(=)f(0.)p Black 172 803 a(2.)p Black 41 w(Hyp)r(erb)r(olic)k(\014xed)f(p)r(oin)n(ts)g (of)g(the)h(sliding)f(v)n(ector)f(\014eld:)36 b(that)25 b(is)g(p)r(oin)n(ts)g Fj(p)e Fi(2)h Ft(\006)2807 773 y Fh(s)2856 803 y Fi([)14 b Ft(\006)2985 773 y Fh(e)3046 803 y Ft(suc)n(h)25 b(that)g Fj(X)7 b Ft(\()p Fj(p)p Ft(\))23 b Fi(k)f Fj(Y)d Ft(\()p Fj(p)p Ft(\))278 903 y(and)28 b(hence)f Fj(Z)727 915 y Fh(s)763 903 y Ft(\()p Fj(p)p Ft(\))c(=)g(0.)36 b(Moreo)n(v)n(er,)25 b(w)n(e)i(imp)r(ose)h (the)g(generic)e(condition)1878 1086 y Fj(Z)1941 1051 y Fu(0)1935 1106 y Fh(s)1970 1086 y Ft(\()p Fj(p)p Ft(\))d Fi(6)p Ft(=)g(0)p Fj(:)1492 b Ft(\(6\))195 1302 y(Next)28 b(prop)r(osition)f(deals)g(with)h(the)g(normal)f(forms)g(of)g(these)h (generic)e(singularities.)p Black 71 1468 a Fe(Prop)s(osition)k(3.2.)p Black 40 w Fk(The)h(fol)t(lowing)h Ft(\006)p Fk(-e)l(quivalenc)l(es)e (hold:)p Black 169 1634 a(1.)p Black 42 w(If)j Ft(\(0)p Fj(;)14 b Ft(0\))27 b Fi(2)g Ft(\006)32 b Fk(is)g(a)h(fold)g(p)l(oint)g (of)g(the)f(ve)l(ctor)g(\014eld)h Fj(Z)f Ft(=)27 b(\()p Fj(X)r(;)14 b(Y)19 b Ft(\))27 b Fi(2)h(Z)2561 1604 y Fh(r)2629 1634 y Fk(de\014ne)l(d)k(in)h(a)f(neighb)l(orho)l(o)l(d)j Fj(U)40 b Fk(of)33 b Fj(p)p Fk(,)278 1733 y(then)d(it)g(is)g Ft(\006)p Fk(-e)l(quivalent)g(in)g(a)g(neighb)l(orho)l(o)l(d)i Fj(V)49 b Fk(of)30 b Ft(\(0)p Fj(;)14 b Ft(0\))29 b Fk(to)1403 2115 y Fj(Z)1460 2127 y Fh(a;b)1572 2115 y Ft(=)1660 1845 y Fg(8)1660 1920 y(>)1660 1945 y(>)1660 1970 y(>)1660 1995 y(>)1660 2020 y(<)1660 2169 y(>)1660 2194 y(>)1660 2219 y(>)1660 2244 y(>)1660 2269 y(:)1733 1991 y Fj(X)1802 2003 y Fh(a)1865 1991 y Ft(=)1953 1849 y Fg( )2085 1934 y Ft(1)2060 2054 y Fj(ax)2193 1849 y Fg(!)2302 1991 y Fk(for)i Fj(y)26 b(>)c Ft(0)1733 2240 y Fj(Y)1781 2252 y Fh(b)1838 2240 y Ft(=)1926 2098 y Fg( )2033 2183 y Ft(0)2036 2303 y Fj(b)2116 2098 y Fg(!)2225 2240 y Fk(for)31 b Fj(y)25 b(<)e Ft(0)3744 2115 y(\(7\))278 2505 y Fk(wher)l(e)31 b Fj(a)23 b Ft(=)f(sgn)802 2438 y Fg(\000)840 2505 y Fj(X)916 2475 y Fm(2)952 2505 y Fj(f)9 b Ft(\()p 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