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cleartomark

%%EndFont 
%%BeginFont: CMMI12
%!PS-AdobeFont-1.1: CMMI12 1.100
%%CreationDate: 1996 Jul 27 08:57:55

% 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 (CMMI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI12 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 14 /delta put
dup 18 /theta put
dup 24 /xi put
dup 25 /pi put
dup 28 /tau put
dup 32 /psi put
dup 33 /omega put
dup 34 /epsilon 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 66 /B put
dup 67 /C put
dup 68 /D put
dup 70 /F put
dup 71 /G put
dup 72 /H put
dup 73 /I put
dup 74 /J put
dup 78 /N put
dup 79 /O put
dup 83 /S put
dup 84 /T put
dup 98 /b put
dup 99 /c put
dup 100 /d put
dup 104 /h put
dup 105 /i put
dup 107 /k put
dup 110 /n put
dup 116 /t put
dup 117 /u put
dup 120 /x put
dup 121 /y put
readonly def
/FontBBox{-30 -250 1026 750}readonly def
/UniqueXX 5087386 def
currentdict end
currentfile eexec
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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 11 /ff put
dup 13 /fl 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 65 /A put
dup 67 /C put
dup 73 /I put
dup 74 /J put
dup 77 /M put
dup 82 /R put
dup 84 /T put
dup 87 /W 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 120 /x put
dup 121 /y put
readonly def
/FontBBox{-53 -251 1139 750}readonly def
/UniqueXX 5000769 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI6
%!PS-AdobeFont-1.1: CMMI6 1.100
%%CreationDate: 1996 Jul 23 07:53:52

% 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 (CMMI6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI6 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 105 /i put
readonly def
/FontBBox{11 -250 1241 750}readonly def
/UniqueXX 5087381 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR6
%!PS-AdobeFont-1.1: CMR6 1.0
%%CreationDate: 1991 Aug 20 16:39:02

% 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 (CMR6) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR6 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 49 /one put
dup 50 /two put
readonly def
/FontBBox{-20 -250 1193 750}readonly def
/UniqueXX 5000789 def
currentdict end
currentfile eexec
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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
readonly def
/FontBBox{-4 -948 1329 786}readonly def
/UniqueXX 5000816 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMMI8
%!PS-AdobeFont-1.1: CMMI8 1.100
%%CreationDate: 1996 Jul 23 07:53:54

% 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 (CMMI8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMMI8 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 25 /pi put
dup 28 /tau put
dup 59 /comma put
dup 78 /N put
dup 84 /T put
dup 97 /a put
dup 99 /c put
dup 102 /f put
dup 105 /i put
dup 109 /m put
dup 110 /n put
dup 116 /t put
dup 120 /x put
readonly def
/FontBBox{-24 -250 1110 750}readonly def
/UniqueXX 5087383 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 12 /vextendsingle put
dup 16 /parenleftBig put
dup 17 /parenrightBig put
dup 18 /parenleftbigg put
dup 19 /parenrightbigg put
dup 88 /summationdisplay put
dup 90 /integraldisplay put
dup 104 /bracketleftBig put
dup 105 /bracketrightBig put
readonly def
/FontBBox{-24 -2960 1454 772}readonly def
/UniqueXX 5000774 def
currentdict end
currentfile eexec
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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 20 /lessequal put
dup 21 /greaterequal put
dup 24 /similar put
dup 25 /approxequal put
dup 28 /lessmuch put
dup 29 /greatermuch put
dup 33 /arrowright put
dup 50 /element put
dup 55 /mapsto put
dup 74 /J put
dup 106 /bar put
dup 112 /radical put
readonly def
/FontBBox{-29 -960 1116 775}readonly def
/UniqueXX 5000820 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMR8
%!PS-AdobeFont-1.1: CMR8 1.0
%%CreationDate: 1991 Aug 20 16:39:40

% 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 (CMR8) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle 0 def
/isFixedPitch false def
end readonly def
/FontName /CMR8 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 1 /Delta 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 56 /eight put
dup 61 /equal put
readonly def
/FontBBox{-36 -250 1070 750}readonly def
/UniqueXX 5000791 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 0 /minus put
dup 3 /asteriskmath put
dup 20 /lessequal put
dup 48 /prime put
dup 50 /element put
readonly def
/FontBBox{-30 -955 1185 779}readonly def
/UniqueXX 5000818 def
currentdict end
currentfile eexec
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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 34 /epsilon put
dup 61 /slash put
readonly def
/FontBBox{-32 -250 1048 750}readonly def
/UniqueXX 5087385 def
currentdict end
currentfile eexec
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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 40 /parenleft put
dup 41 /parenright put
dup 44 /comma put
dup 45 /hyphen put
dup 46 /period put
dup 49 /one put
dup 50 /two put
dup 73 /I put
dup 84 /T put
dup 87 /W 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 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
readonly def
/FontBBox{-251 -250 1009 969}readonly def
/UniqueXX 5000793 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 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{-301 -250 1164 946}readonly def
/UniqueXX 5000768 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
%%BeginFont: CMTI12
%!PS-AdobeFont-1.1: CMTI12 1.0
%%CreationDate: 1991 Aug 18 21:06:53

% 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 (CMTI12) readonly def
/FamilyName (Computer Modern) readonly def
/Weight (Medium) readonly def
/ItalicAngle -14.04 def
/isFixedPitch false def
end readonly def
/FontName /CMTI12 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 44 /comma 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 55 /seven put
dup 56 /eight put
dup 65 /A put
dup 68 /D put
dup 69 /E put
dup 73 /I put
dup 77 /M put
dup 80 /P put
dup 82 /R put
dup 83 /S put
dup 84 /T put
dup 85 /U put
dup 97 /a 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 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 120 /x put
dup 121 /y put
dup 122 /z put
readonly def
/FontBBox{-36 -251 1103 750}readonly def
/UniqueXX 5000829 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 44 /comma put
dup 46 /period put
dup 65 /A put
dup 73 /I put
dup 76 /L put
dup 78 /N put
dup 80 /P put
dup 82 /R put
dup 86 /V put
dup 97 /a put
dup 100 /d put
dup 101 /e put
dup 104 /h put
dup 105 /i put
dup 108 /l put
dup 110 /n put
dup 115 /s put
dup 116 /t put
dup 118 /v put
readonly def
/FontBBox{-33 -250 945 749}readonly def
/UniqueXX 5000795 def
currentdict end
currentfile eexec
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cleartomark

%%EndFont 
TeXDict begin 40258431 52099146 1000 600 600 (paper.dvi)
@start /Fa 140[46 3[59 65 4[33 5[52 37[91 11[59 5[91
1[46 46 40[{}10 119.552 /CMR12 rf /Fb 184[92 4[89 37[51
28[{}3 119.552 /CMMI12 rf /Fc 202[67 2[67 67 2[37 46[{}4
119.552 /CMBX12 rf /Fd 150[22 105[{}1 49.8132 /CMMI6
rf /Fe 205[30 30 49[{}2 49.8132 /CMR6 rf /Ff 252[32 3[{}1
49.8132 /CMSY6 rf /Fg 135[40 3[25 5[43 62 3[24 2[34 2[31
1[37 12[41 5[56 18[20 30[31 2[41 25[{}13 66.4176 /CMMI8
rf /Fh 150[39 39 13[46 1[120 68[61 61 50 50 3[28 12[{}9
83.022 /CMEX10 rf /Fi 143[83 5[28 31[68 18[0 4[66 16[100
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%%EndDocument
@endspecial
0 2873 a Fu(Figure)29 b(1:)42 b(Phase)31 b(p)s(ortraits)e(of)g(the)h
(system)i(with)e(Hamiltonian)g(\(4\))f(frozen)h(at)f(di\013eren)m(t)i
(v)-5 b(alues)30 b(of)0 2993 y Fj(\034)11 b Fu(:)44 b(a\))32
b Fj(G)p Fu(\()p Fj(\034)11 b Fu(\))28 b(=)g(0;)k(b\))h
Fj(G)p Fu(\()p Fj(\034)11 b Fu(\))28 b Fj(>)f Fu(0;)33
b(c\))f Fj(G)p Fu(\()p Fj(\034)11 b Fu(\))28 b Fj(<)g
Fu(0.)0 3284 y(closed)41 b(and)e(are)g(con\014ned)i(within)f(square)h
(separatrix)f(cells.)65 b(If)40 b Fj(G)p Fu(\()5 b(~)-54
b Fj(\034)11 b Fu(\))39 b Fj(>)g Fu(0,)i(the)f(phase)g(p)s(ortrait)0
3404 y(is)35 b(qualitativ)m(ely)h(di\013eren)m(t)g(\(Fig.)48
b(1b\).)h(Some)35 b(tra)5 b(jectories)35 b(are)g(closed)g(and)g
(corresp)s(ond)g(to)f(\014nite)0 3525 y(motion)40 b(within)g(the)g
(cells)h(con\014ned)g(b)m(y)g(separatrices.)66 b(Other)40
b(tra)5 b(jectories)41 b(are)f(not)f(closed)i(and)0 3645
y(corresp)s(ond)34 b(to)f(un)m(b)s(ounded)i(motion)e(in)g(the)h(p)s
(ositiv)m(e)g(direction)h(along)d(the)i Fj(x)p Fu(-axis.)46
b(If)33 b(w)m(e)i(c)m(ho)s(ose)5 3765 y(~)-54 b Fj(\034)55
b Fu(suc)m(h)44 b(that)f Fj(G)p Fu(\()5 b(~)-54 b Fj(\034)11
b Fu(\))46 b Fj(<)g Fu(0,)f(the)f(picture)g(is)g(similar)f(to)g(the)h
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y(un)m(b)s(ounded)34 b(motion)f(is)g(op)s(en)f(in)h(the)g(negativ)m(e)h
(direction)f(along)g(the)g Fj(x)p Fu(-axis)g(\(see)h(Fig.)43
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(the)f(area)g Fj(S)1759 3970 y Fm(\003)1799 4006 y Fu(\()p
Fj(\034)11 b Fu(\))27 b(b)s(ounded)i(b)m(y)f(a)f(separatrix)h(lo)s(op)f
(in)h(the)g(system)0 4126 y(frozen)33 b(at)f(a)g(certain)i(v)-5
b(alue)33 b(of)f Fj(\034)11 b Fu(.)44 b(One)33 b(obtains:)1153
4367 y Fj(S)1219 4326 y Fm(\003)1258 4367 y Fu(\()p Fj(\034)11
b Fu(\))28 b(=)g Fj(\031)1578 4326 y Fl(2)1639 4367 y
Fi(\000)23 b Fu(2)p Fj(\031)t Fi(j)p Fj(G)p Fu(\()p Fj(\034)11
b Fu(\))p Fj(")p Fu(ln)o Fj(")p Fi(j)22 b Fu(+)g Fj(O)s
Fu(\()p Fj(")p Fu(\))p Fj(:)1027 b Fu(\(5\))146 4542
y(Consider)41 b(no)m(w)f(a)f(phase)h(tra)5 b(jectory)40
b(on)g(a)f(phase)h(p)s(ortrait)f(frozen)g(at)g(a)g(certain)h(v)-5
b(alue)40 b(of)f Fj(\034)11 b Fu(.)0 4662 y(If)39 b(the)g(tra)5
b(jectory)40 b(is)f(closed,)j(the)e(area)e Fj(S)45 b
Fu(inside)40 b(of)f(it)g(is)g(connected)i(with)e(the)h(action)f
Fj(I)47 b Fu(of)38 b(the)0 4782 y(system)f(b)m(y)f(a)g(simple)g
(relation)g Fj(S)i Fu(=)32 b(2)p Fj(\031)t(I)43 b Fu(\(Fig.)51
b(1a\).)g(If)35 b(the)h(tra)5 b(jectory)36 b(is)g(not)f(closed)i
(\(Fig.)51 b(1b\))0 4903 y(and)28 b(crosses)i(the)f(line)g
Fj(x)f Fu(=)f(0)h(at)g Fj(y)j Fi(\024)d Fj(\031)t(=)p
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b Fu(=)27 b Fj(S)6 b Fu(,)29 b(where)g Fj(S)34 b Fu(is)29
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b(jectory)33 b(and)f(the)g(segmen)m(ts)i(of)d(the)i(lines)g
Fj(y)d Fu(=)e(0)p Fj(;)17 b(x)28 b Fu(=)f(0)p Fj(;)17
b(x)28 b Fu(=)f(2)p Fj(\031)t Fu(.)43 b(If)32 b(it)g(crosses)h(the)g
(line)1865 5281 y(4)p eop
%%Page: 5 5
5 4 bop 0 407 a Fj(x)28 b Fu(=)g(0)k(at)g Fj(y)f(>)c(\031)t(=)p
Fu(2,)32 b(w)m(e)i(put)f(2)p Fj(\031)t(I)i Fu(=)27 b(2)p
Fj(\031)1505 371 y Fl(2)1567 407 y Fi(\000)22 b Fj(S)6
b Fu(.)43 b(De\014ned)34 b(in)e(this)h(w)m(a)m(y)-8 b(,)34
b Fj(I)41 b Fu(is)33 b(a)f(con)m(tin)m(uous)i(function)0
527 y(of)e(the)h(co)s(ordinates.)146 648 y(In)38 b(system)i(\(4\))d
(with)h Fj(\034)47 b Fu(=)36 b Fj("t)p Fu(,)j(the)f(action)f
Fj(I)46 b Fu(is)38 b(an)f(adiabatic)h(in)m(v)-5 b(arian)m(t)38
b(of)f(motion)g(\(see,)j(for)0 768 y(example,)34 b([10]\).)43
b(F)-8 b(ar)32 b(from)g(the)h(separatrices,)i(it)d(undergo)s(es)i(v)-5
b(ariations)32 b(of)g(order)h Fj(")3217 732 y Fl(2)3256
768 y Fu(.)146 888 y(In)48 b(the)g(follo)m(wing,)k(w)m(e)c(will)g(use)h
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b(arian)m(t)48 b Fj(J)63 b Fu(=)53 b Fj(I)40 b Fu(+)0
1009 y Fj("u)p Fu(\()p Fj(x;)17 b(y)t(;)g(\034)11 b Fu(\))37
b(rather)i(than)g Fj(I)8 b Fu(.)62 b(The)39 b(function)g
Fj(u)g Fu(is)g(de\014ned)h(in)f(\(17\))f(in)h(the)g(App)s(endix.)64
b(F)-8 b(ar)38 b(from)0 1129 y(the)47 b(separatrices)h(v)-5
b(ariations)47 b(of)f Fj(J)56 b Fu(are)47 b(of)f(order)h
Fj(")2042 1093 y Fl(3)2081 1129 y Fu(.)86 b(As)47 b(the)g(slo)m(w)h
(time)f Fj(\034)59 b Fu(gro)m(ws,)51 b(the)c(area)0 1249
y(b)s(ounded)25 b(b)m(y)h(a)e(separatrix)i Fj(S)1108
1213 y Fm(\003)1147 1249 y Fu(\()p Fj(\034)11 b Fu(\))25
b(slo)m(wly)h(c)m(hanges.)43 b(On)24 b(the)h(other)g(hand,)i(a)d(v)-5
b(alue)25 b(of)f(the)h(adiabatic)0 1370 y(in)m(v)-5 b(arian)m(t)43
b(asso)s(ciated)g(with)g(a)f(certain)h(phase)g(tra)5
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%%EndDocument
@endspecial
0 1928 a Fu(Figure)27 b(3:)40 b(Plots)28 b(of)f(the)g(functions)h
Fj(x)p Fu(\()p Fj(t)p Fu(\))f(and)h Fj(J)9 b Fu(\()p
Fj(t)p Fu(\))27 b(obtained)g(b)m(y)h(n)m(umerical)h(in)m(tegration)e
(of)g(system)0 2048 y(\(4\))32 b(with)h(the)g(p)s(erturbation)g(of)f
(example)i(5.1.)43 b Fj(")28 b Fu(=)f(0)p Fj(:)p Fu(005)p
Fj(;)17 b(B)32 b Fu(=)c(0)p Fj(:)p Fu(6.)0 2339 y(in)m(v)-5
b(arian)m(t)35 b Fj(J)44 b Fu(on)35 b(a)f(time)i(in)m(terv)-5
b(al)35 b(not)g(long)f(enough)h(for)g(a)f(signi\014can)m(t)i(c)m(hange)
g(of)e Fj(J)44 b Fu(\(see)36 b(Section)0 2459 y(3\).)68
b(If)41 b(the)h(equation)f Fj(S)937 2423 y Fm(\003)976
2459 y Fu(\()p Fj(\034)11 b Fu(\))43 b(=)e(2)p Fj(\031)t(J)50
b Fu(has)42 b(only)f(t)m(w)m(o)h(ro)s(ots)e Fj(\034)2372
2474 y Fl(1)2412 2459 y Fj(;)17 b(\034)2498 2474 y Fl(2)2578
2459 y Fu(on)41 b(the)g(in)m(terv)-5 b(al)42 b(\()p Fi(\000)p
Fj(\031)t(;)17 b(\031)t Fu(\),)43 b(the)0 2580 y(phase)36
b(p)s(oin)m(t)f(drifts)g(only)g(in)g(the)h(p)s(ositiv)m(e)g(direction)g
(of)e(the)h Fj(x)p Fu(-axis)h(\(see)g(Section)f(4\).)50
b(Numerical)0 2700 y(sim)m(ulation)41 b(of)f(suc)m(h)i(b)s(eha)m(vior)f
(is)g(sho)m(wn)g(in)f(Fig.)66 b(6a.)g(When)41 b Fj(\034)53
b Fi(2)41 b Fu(\()p Fj(\034)2763 2715 y Fl(1)2830 2700
y Fu(+)27 b(2)p Fj(\031)t(k)s(;)17 b(\034)3181 2715 y
Fl(2)3248 2700 y Fu(+)27 b(2)p Fj(\031)t(k)s Fu(\),)42
b Fj(k)h Fu(is)0 2820 y(an)38 b(in)m(teger,)h(the)g(phase)f(p)s(oin)m
(t)g(p)s(erforms)g(regular)g(drift)f(to)h(the)g(righ)m(t,)h(otherwise)g
(it)f(is)g(k)m(ept)h(in)f(a)0 2941 y(con)m(v)m(ectiv)m(e)30
b(cell.)43 b(If)27 b(the)h(equation)g Fj(S)1387 2905
y Fm(\003)1426 2941 y Fu(\()p Fj(\034)11 b Fu(\))28 b(=)f(2)p
Fj(\031)t(J)36 b Fu(has)28 b(six)g(ro)s(ots)f Fj(\034)2481
2956 y Fl(3)2520 2941 y Fj(;)17 b(:::;)g(\034)2731 2956
y Fl(8)2798 2941 y Fu(on)27 b(the)h(in)m(terv)-5 b(al)28
b(\()p Fi(\000)p Fj(\031)t(;)17 b(\031)t Fu(\),)0 3061
y(the)34 b(phase)h(p)s(oin)m(t)f(can)g(drift)g(in)g(b)s(oth)f
(directions.)49 b(The)35 b(results)g(of)e(computer)i(sim)m(ulations)h
(in)e(this)0 3182 y(case)k(are)f(sho)m(wn)h(in)f(Fig.)55
b(6b.)i(One)37 b(can)g(also)g(see)h(in)f(Fig.)55 b(6)37
b(that)f(the)i(adiabatic)f(in)m(v)-5 b(arian)m(t)37 b
Fj(J)46 b Fu(is)0 3302 y(nearly)33 b(constan)m(t)h(in)f(the)g(b)s(oth)f
(cases.)146 3422 y(The)f(time)g(ev)m(olution)g(of)f(a)f(drop)h(of)g(dy)
m(e)h(in)f(this)h(case)g(w)m(as)g(describ)s(ed)h(in)e(the)h(end)f(of)g
(Section)h(4.)709 3608 y
@beginspecial @setspecial
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%%EndDocument
@endspecial
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@beginspecial @setspecial
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%%EndDocument
@endspecial
760 4858 a(Figure)33 b(5:)43 b(Plot)33 b(of)f(the)h(function)g
Fj(S)2127 4822 y Fm(\003)2166 4858 y Fu(\()p Fj(\034)11
b Fu(\))33 b(in)g(example)h(5.2.)1841 5281 y(12)p eop
%%Page: 13 13
13 12 bop 945 349 a
@beginspecial @setspecial
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%%EndDocument
@endspecial
1426 x
@beginspecial @setspecial
%%BeginDocument: em:graph fig6b.pcx
126.00 79.44
scale /picstr  132 string def 1056 662 1
[ 1056 0 0 -662 0 0 ]
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%%EndDocument
@endspecial
0 3159 a Fu(Figure)36 b(6:)51 b(Plots)37 b(of)e(the)i(functions)g
Fj(x)p Fu(\()p Fj(t)p Fu(\))g(and)f Fj(J)9 b Fu(\()p
Fj(t)p Fu(\))37 b(obtained)f(b)m(y)i(n)m(umerical)f(in)m(tegration)g
(of)f(sys-)0 3279 y(tem)41 b(\(4\))f(with)h(the)g(p)s(erturbation)g(of)
f(example)i(5.2.)67 b Fj(")41 b Fu(=)g(0)p Fj(:)p Fu(005)p
Fj(;)17 b(B)45 b Fu(=)d(0)p Fj(:)p Fu(6.)67 b(Figures)41
b(a\))f(and)g(b\))0 3400 y(corresp)s(ond)33 b(to)g(di\013eren)m(t)g
(initial)g(conditions.)945 3586 y
@beginspecial @setspecial
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%%EndDocument
@endspecial
428 4970 a(Figure)f(7:)44 b(Time)33 b(ev)m(olution)h(of)e(\001)p
Fj(x)h Fu(in)g(example)h(5.2.)43 b Fj(")28 b Fu(=)f(0)p
Fj(:)p Fu(005)p Fj(;)17 b(B)32 b Fu(=)c(0)p Fj(:)p Fu(6.)1841
5281 y(13)p eop
%%Page: 14 14
14 13 bop 472 307 a
@beginspecial @setspecial
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%%EndDocument
@endspecial
0 1455 a Fu(Figure)30 b(8:)42 b(A)31 b(sc)m(hematical)h(picture)g(of)d
(separatrix)j(crossing.)44 b(Note,)31 b(that)f(actually)h(the)g
(separatrix)0 1576 y(enclosed)j(areas)f(c)m(hange)h(and)e(the)h(turns)h
(of)e(the)h(tra)5 b(jectory)33 b(sta)m(y)g(roughly)g(constan)m(t)h(in)f
(size.)1059 2006 y Fj(u)28 b Fu(=)f Fj(u)p Fu(\()p Fj(x;)17
b(y)t(;)g(\034)11 b Fu(\))27 b(=)1796 1938 y(1)p 1766
1983 108 4 v 1766 2074 a(2)p Fj(\031)1901 1889 y Fh(Z)1984
1915 y Fg(T)1947 2077 y Fl(0)2055 1885 y Fh(\022)2126
1938 y Fj(T)p 2126 1983 71 4 v 2137 2074 a Fu(2)2229
2006 y Fi(\000)c Fj(t)2364 1885 y Fh(\023)2452 1938 y
Fj(@)5 b(H)p 2452 1983 146 4 v 2469 2074 a(@)g(\034)2607
2006 y(dt:)886 b Fu(\(17\))0 2217 y(The)39 b(in)m(tegral)f(in)g(\(17\))
f(is)i(computed)g(along)e(the)h(unp)s(erturb)s(ed)h(tra)5
b(jectory)39 b(passing)f(through)g(the)0 2338 y(p)s(oin)m(t)g(\()p
Fj(x;)17 b(y)t Fu(\))37 b(at)g(the)i(time)f(momen)m(t)h
Fj(t)e Fu(=)f(0;)41 b Fj(T)51 b Fu(is)38 b(the)h(p)s(erio)s(d)e(of)h
(motion)g(on)f(this)i(tra)5 b(jectory)-8 b(.)60 b(If)0
2458 y(the)36 b(tra)5 b(jectory)35 b(is)h(not)f(closed,)i
Fj(T)49 b Fu(is)35 b(the)h(time)g(necessary)h(for)e(a)g(phase)h(p)s
(oin)m(t)f(on)g(this)h(tra)5 b(jectory)0 2579 y(to)32
b(co)m(v)m(er)i(the)f(distance)h(of)e(2)p Fj(\031)k Fu(along)d(the)g
Fj(x)p Fu(-axis.)146 2699 y(Consider)46 b(the)e(motion)g(in)g(the)g
(system)i(with)f(Hamiltonian)f(\(4\))f(frozen)h(at)g(a)g(certain)g(v)-5
b(alue)0 2819 y(of)46 b(the)g(slo)m(w)i(time)f Fj(\034)62
b Fu(=)56 b(\026)-54 b Fj(\034)11 b Fu(.)85 b(As)47 b(the)f(unp)s
(erturb)s(ed)i(tra)5 b(jectory)46 b(tends)i(to)e(the)g(separatrix,)51
b(the)0 2940 y(p)s(erio)s(d)38 b Fj(T)52 b Fu(gro)m(ws)39
b(logarithmically)-8 b(.)62 b(On)38 b(a)g(phase)i(tra)5
b(jectory)39 b Fj(H)45 b Fu(=)37 b Fj(h)h Fu(one)h(obtains)g(in)f(the)h
(main)0 3060 y(appro)m(ximation:)1291 3301 y Fj(T)i Fu(=)28
b Fi(\000)p Fu(2ln)p Fi(j)p Fj(h)p Fi(j)22 b(\000)g Fu(2ln)p
Fi(j)p Fj("\031)t(G)p Fi(j)f Fu(+)h Fj(b:)1118 b Fu(\(18\))0
3475 y(Here)38 b Fj(b)g Fu(is)f(a)g(constan)m(t)i(whic)m(h)f(will)g
(not)f(b)s(e)h(imp)s(ortan)m(t)f(for)g(the)h(rest)g(of)e(the)i(argumen)
m(t.)58 b(In)38 b(\(18\),)0 3596 y(it)33 b(is)h(assumed)h(that)e
Fi(j)p Fj(h)p Fi(j)28 b Fj(<)g("\031)t Fi(j)p Fj(G)p
Fi(j)p Fj(c)1325 3559 y Fm(\000)p Fl(1)1418 3596 y Fu(,)34
b(where)g Fj(c)f Fu(is)h(a)f(large)g(p)s(ositiv)m(e)h(constan)m(t.)46
b(\(F)-8 b(or)33 b(de\014niteness,)0 3716 y(w)m(e)j(consider)h
(crossing)f(the)g(lo)m(w)m(er)h(separatrix)f Fj(H)k Fu(=)32
b(0;)k(for)f(the)h(upp)s(er)g(separatrix)g Fj(H)k Fu(=)32
b Fj("\031)t(G)j Fu(the)0 3836 y(\014nal)f(result)g(ob)m(viously)h(is)f
(the)g(same\).)47 b(In)34 b(agreemen)m(t)h(with)f(the)g(form)m(ula)g
Fj(T)43 b Fu(=)29 b(2)p Fj(\031)t(@)5 b(I)j(=@)d(h)34
b Fu(w)m(e)h(\014nd)0 3957 y(for)d(the)h(adiabatic)g(in)m(v)-5
b(arian)m(t)33 b Fj(I)40 b Fu(in)33 b(the)g(main)g(appro)m(ximation)965
4198 y(2)p Fj(\031)t(I)i Fu(=)28 b Fj(S)1321 4156 y Fm(\003)1382
4198 y Fi(\000)23 b Fu(2)p Fj(h)p Fu(ln)p Fi(j)p Fj(h)p
Fi(j)f(\000)h Fu(2)p Fj(h)p Fu(ln)p Fi(j)p Fj("\031)t(G)p
Fi(j)e Fu(+)h(\(2)g(+)g Fj(b)p Fu(\))p Fj(h:)792 b Fu(\(19\))0
4372 y(W)-8 b(e)44 b(no)m(w)h(compute)g(the)f(function)h
Fj(u)e Fu(at)h(a)g(p)s(oin)m(t)g(of)f(the)i(v)m(ertex)g(bisecting)h
(the)e(angle)g(b)s(et)m(w)m(een)0 4492 y(incoming)39
b(and)g(outgoing)g(separatrices)h(of)e(a)h(saddle)g(p)s(oin)m(t)g
(\(see)h(Fig.)e(8\).)62 b(F)-8 b(rom)38 b(\(17\),)i(\(18\))e(one)0
4613 y(obtains:)1298 4878 y(2)p Fj(\031)t(u)27 b Fu(=)1603
4811 y Fj(\022)p 1603 4855 49 4 v 1603 4946 a Fu(2)1661
4878 y(ln)q Fi(j)p Fj(h)p Fi(j)p Fj(;)211 b(\022)31 b
Fu(=)2282 4811 y Fj(@)5 b(S)2404 4774 y Fm(\003)p 2282
4855 163 4 v 2308 4946 a Fj(@)g(\034)2454 4878 y(:)1125
b Fu(\(20\))1841 5281 y(14)p eop
%%Page: 15 15
15 14 bop 146 407 a Fu(Consider)32 b(no)m(w)f(a)f(separatrix)h
(crossing)h(in)e(the)h(exact)g(system.)45 b(Assume)32
b(for)e(de\014niteness,)k(that)0 527 y(on)27 b(the)g(relev)-5
b(an)m(t)27 b(in)m(terv)-5 b(al)28 b(of)e(the)h(slo)m(w)h(time)f
Fj(G)p Fu(\()p Fj(\034)11 b Fu(\))28 b Fj(>)g Fu(0)p
Fj(;)33 b(\022)s Fu(\()p Fj(\034)11 b Fu(\))28 b Fj(<)g
Fu(0.)41 b(Let)27 b(a)f(phase)i(p)s(oin)m(t)f(b)s(e)g(initially)0
648 y(captured)38 b(in)f(a)f(separatrix)i(cell)g(b)s(ounded)f(b)m(y)h
(the)f(separatrix)h Fj(H)43 b Fu(=)35 b(0)h(\(see)i(Fig.)e(8\).)56
b(The)38 b(initial)0 768 y(v)-5 b(alues)42 b(of)f(the)h(Hamiltonian)f
(and)h(the)g(impro)m(v)m(ed)h(adiabatic)e(in)m(v)-5 b(arian)m(t)42
b(at)f Fj(\034)54 b Fu(=)43 b Fj(\034)3209 783 y Fm(\000)3309
768 y Fu(are)f Fj(h)3537 783 y Fm(\000)3596 768 y Fj(;)17
b(J)3694 783 y Fm(\000)3752 768 y Fu(.)0 888 y(The)37
b(phase)g(p)s(oin)m(t)g(rotates)f(close)h(to)f(the)h(separatrix.)55
b(As)37 b(the)g(area)f(b)s(ounded)h(b)m(y)g(the)g(separatrix)0
1009 y(decreases,)i(the)d(p)s(oin)m(t)h(comes)g(closer)g(to)e(the)i
(separatrix)f(with)h(eac)m(h)g(turn.)54 b(Denote)36 b(b)m(y)h
Fj(h)3453 1024 y Fg(n)3500 1009 y Fj(;)17 b(I)3587 1024
y Fg(n)3633 1009 y Fj(;)g(u)3733 1024 y Fg(n)0 1129 y
Fu(v)-5 b(alues)30 b(of)e(the)h(corresp)s(onding)h(functions)f(at)g
(time)g(momen)m(ts)h Fj(\034)2384 1144 y Fg(n)2460 1129
y Fu(when)g(the)f(tra)5 b(jectory)30 b(crosses)g(the)0
1249 y(v)m(ertex)f(bisecting)g(the)f(angle)f(b)s(et)m(w)m(een)j
(incoming)e(and)f(outgoing)g(separatrices)i(of)e(the)h(saddle)g(p)s
(oin)m(t)0 1370 y Fj(C)34 b Fu(\(see)28 b(Fig.)e(8\).)41
b(W)-8 b(e)28 b(en)m(umerate)g Fj(\034)1315 1385 y Fg(n)1389
1370 y Fu(as)f(follo)m(ws:)42 b Fj(\034)1901 1385 y Fl(0)1967
1370 y Fu(is)28 b(the)f(time)g(of)g(the)g(last)g(crossing)h(of)e(the)i
(v)m(ertex)0 1490 y(b)s(efore)h(crossing)g(the)g(separatrix,)i(other)d
Fj(\034)1593 1505 y Fg(n)1669 1490 y Fu(ha)m(v)m(e)i(negativ)m(e)g(n)m
(um)m(b)s(ers)g Fj(\034)2702 1505 y Fl(0)2770 1490 y
Fj(>)d(\034)2915 1505 y Fm(\000)p Fl(1)3038 1490 y Fj(>)g(:::)h(>)g
(\034)3396 1505 y Fm(\000)p Fg(N)3546 1490 y Fi(\025)g
Fj(\034)3693 1505 y Fm(\000)3752 1490 y Fu(.)0 1611 y(Here)36
b Fj(N)43 b Fi(\035)32 b Fu(1)j(is)h(a)f(large)h(in)m(teger.)52
b(Its)36 b(exact)g(v)-5 b(alue)36 b(do)s(es)g(not)f(in\015uence)j(the)d
(result)i(in)e(the)h(main)0 1731 y(appro)m(ximation.)79
b(After)44 b(crossing)h(the)f(separatrix,)k(the)d(phase)g(p)s(oin)m(t)f
(shifts)h(along)e(the)i Fj(x)p Fu(-axis.)0 1851 y(A)m(t)39
b(time)g(momen)m(ts)h Fj(\034)838 1866 y Fg(n)886 1851
y Fj(;)33 b Fu(1)38 b Fi(\024)g Fj(n)g Fi(\024)h Fj(N)10
b Fu(,)40 b(the)f(tra)5 b(jectory)40 b(subsequen)m(tly)i(crosses)e(the)
f(v)m(ertices)i(near)0 1972 y(the)d(saddles)h Fj(C)585
1987 y Fg(n)669 1972 y Fu(suc)m(h)g(that)e(the)g(distance)i(b)s(et)m(w)
m(een)h Fj(C)k Fu(and)37 b Fj(C)2427 1987 y Fg(n)2511
1972 y Fu(equals)i(2)p Fj(\031)t(n)e Fu(\(see)i(Fig.)57
b(8\).)h(The)0 2092 y(corresp)s(onding)33 b(v)-5 b(alues)34
b(of)e Fj(h;)17 b(I)8 b(;)17 b(u)32 b Fu(are)g Fj(h)1529
2107 y Fg(n)1576 2092 y Fj(;)17 b(I)1663 2107 y Fg(n)1710
2092 y Fj(;)g(u)1810 2107 y Fg(n)1856 2092 y Fu(.)146
2213 y(It)35 b(follo)m(ws)h(from)e(\(18\))h(-)f(\(20\),)h(that)f(in)h
(the)h(main)f(appro)m(ximation)g(the)h(follo)m(wing)f(expressions)0
2333 y(are)e(v)-5 b(alid:)478 2673 y Fj(h)534 2688 y
Fg(n)p Fl(+1)755 2673 y Fu(=)82 b Fj(h)969 2688 y Fg(n)1039
2673 y Fi(\000)22 b Fj("\022)s(;)492 2839 y(\034)534
2854 y Fg(n)p Fl(+1)755 2839 y Fu(=)82 b Fj(\034)955
2854 y Fg(n)1025 2839 y Fi(\000)1136 2771 y Fj(")p 1134
2815 49 4 v 1134 2907 a Fu(2)1210 2839 y([ln)p Fi(j)p
Fj(h)1402 2854 y Fg(n)1449 2839 y Fi(j)22 b Fu(+)g(3ln)p
Fi(j)p Fj(h)1811 2854 y Fg(n)1880 2839 y Fi(\000)g Fj("\022)s
Fi(j)g Fu(+)g(4ln)p Fj("\031)t(G)g Fi(\000)g Fu(2)p Fj(b)p
Fu(])17 b Fj(;)791 b Fu(\(21\))474 3015 y(2)p Fj(\031)t(I)625
3030 y Fg(n)755 3015 y Fu(=)82 b Fj(S)979 2974 y Fm(\003)1019
3015 y Fu(\()p Fj(\034)1099 3030 y Fl(0)1138 3015 y Fu(\))22
b Fi(\000)h Fj(\022)s Fu(\()p Fj(\034)1426 3030 y Fl(0)1488
3015 y Fi(\000)g Fj(\034)1630 3030 y Fg(n)1677 3015 y
Fu(\))f Fi(\000)h Fu(2)p Fj(h)1942 3030 y Fg(n)1989 3015
y Fu(ln)p Fi(j)p Fj(h)2154 3030 y Fg(n)2201 3015 y Fi(j)f(\000)g
Fu(2)p Fj(h)2455 3030 y Fg(n)2502 3015 y Fu(ln)p Fj("\031)t(G)g
Fu(+)g(\(2)g(+)g Fj(b)p Fu(\))p Fj(h)3227 3030 y Fg(n)3274
3015 y Fj(;)461 3210 y Fu(2)p Fj(\031)t(u)625 3225 y
Fg(n)755 3210 y Fu(=)924 3142 y Fj(\022)p 923 3186 V
923 3278 a Fu(2)982 3210 y(ln)p Fi(j)p Fj(h)1147 3225
y Fg(n)1194 3210 y Fi(j)p Fj(:)0 3455 y Fu(Here)27 b(and)f(b)s(elo)m(w)
g(the)h(v)-5 b(alue)26 b(of)g Fj(\022)j Fu(is)d(calculated)h(at)f
Fj(\034)39 b Fu(=)28 b Fj(\034)2147 3470 y Fm(\003)2186
3455 y Fu(,)g(the)e(time)h(of)e(separatrix)i(crossing)g(in)f(the)0
3575 y(adiabatic)37 b(appro)m(ximation.)56 b(Summing)37
b(the)g(ab)s(o)m(v)m(e)g(expressions)j(\(22\))35 b(from)i
Fj(n)d Fu(=)g Fi(\000)p Fj(N)48 b Fu(to)36 b Fj(n)e Fu(=)h(0,)0
3696 y(w)m(e)i(\014nd)f(the)h(c)m(hange)g(of)e(the)i(impro)m(v)m(ed)g
(adiabatic)f(in)m(v)-5 b(arian)m(t)37 b(b)s(efore)f(the)g(separatrix)h
(crossing)g(in)0 3816 y(the)c(main)g(appro)m(ximation:)699
4057 y(2)p Fj(\031)t Fu(\()p Fj(J)899 4072 y Fl(0)960
4057 y Fi(\000)23 b Fj(J)1114 4072 y Fm(\000)p Fg(N)1236
4057 y Fu(\))k(=)h Fj("\022)1515 3961 y Fh(h)1555 4057
y Fu(ln)p Fj(\030)e Fi(\000)d Fu(2)p Fj(\030)5 b Fu(ln)o
Fj(\030)27 b Fu(+)22 b(2)p Fj(\030)k Fi(\000)d Fu(2ln\()2536
3970 y Fi(p)p 2619 3970 108 4 v 87 x Fu(2)p Fj(\031)t(=)p
Fu(\000\()p Fj(\030)5 b Fu(\)\))2999 3961 y Fh(i)3053
4057 y Fj(;)526 b Fu(\(22\))0 4244 y(where)34 b Fj(\030)e
Fu(=)27 b Fi(j)p Fj(h)544 4259 y Fl(0)584 4244 y Fj(=)p
Fu(\()p Fj("\022)s Fu(\))p Fi(j)p Fu(,)k(\000\()p Fi(\001)p
Fu(\))h(is)i(the)f(gamma)f(function.)44 b(A)m(t)33 b(the)g(separatrix)g
(crossing)h(w)m(e)g(obtain:)446 4485 y(2)p Fj(\031)t
Fu(\()p Fj(J)646 4500 y Fl(1)707 4485 y Fi(\000)22 b
Fj(J)860 4500 y Fl(0)900 4485 y Fu(\))27 b(=)h Fj("\022)19
b Fu([)p Fi(\000)p Fu(ln)q Fj(\030)27 b Fi(\000)22 b
Fu(ln)q(\(1)g Fi(\000)g Fj(\030)5 b Fu(\))22 b(+)g(2\(1)f
Fi(\000)i Fj(\030)5 b Fu(\)ln\(1)21 b Fi(\000)i Fj(\030)5
b Fu(\))22 b(+)g(2)p Fj(\030)5 b Fu(ln)o Fj(\030)27 b
Fi(\000)22 b Fu(2])17 b Fj(:)272 b Fu(\(23\))0 4659 y(F)-8
b(or)32 b(the)h(c)m(hange)g(of)f Fj(J)42 b Fu(after)32
b(the)h(separatrix)h(crossing)f(w)m(e)h(\014nd)100 4999
y(2)p Fj(\031)t Fu(\()p Fj(J)300 5014 y Fg(N)388 4999
y Fi(\000)23 b Fj(J)542 5014 y Fl(1)581 4999 y Fu(\))28
b(=)g Fj("\022)861 4903 y Fh(h)900 4999 y Fu(ln\(1)22
b Fi(\000)h Fj(\030)5 b Fu(\))21 b Fi(\000)i Fu(2\(1)f
Fi(\000)g Fj(\030)5 b Fu(\)ln\(1)22 b Fi(\000)g Fj(\030)5
b Fu(\))22 b Fi(\000)h Fu(2)p Fj(\030)j Fu(+)c(2)g Fi(\000)g
Fu(2ln)q(\()2792 4912 y Fi(p)p 2874 4912 V 2874 4999
a Fu(2)p Fj(\031)t(=)p Fu(\000\(1)g Fi(\000)g Fj(\030)5
b Fu(\)\))3424 4903 y Fh(i)3479 4999 y Fj(:)100 b Fu(\(24\))1841
5281 y(15)p eop
%%Page: 16 16
16 15 bop 146 407 a Fu(F)-8 b(ar)30 b(from)h(the)h(separatrix,)g(the)f
(v)-5 b(ariation)31 b(of)f Fj(J)41 b Fu(is)31 b(of)g(order)g
Fj(")2482 371 y Fl(3)2552 407 y Fu(on)g(time)g(p)s(erio)s(ds)h(of)e
(order)h(1)p Fj(=")p Fu(.)0 527 y(Hence,)g(to)e(obtain)g(in)g(the)h
(main)f(appro)m(ximation)h(the)f(jump)h(of)e Fj(J)38
b Fu(at)29 b(the)g(separatrix)h(crossing,)h(one)0 648
y(has)i(to)f(sum)i(up)f(expression)h(\(22\))e(-)g(\(24\).)43
b(Th)m(us,)35 b(w)m(e)e(\014nd)g(form)m(ula)g(\(6\).)1841
5281 y(16)p eop
%%Page: 17 17
17 16 bop 0 407 a Fw(References)49 626 y Fu([1])49 b(T.H.Solomon)24
b(and)f(J.P)-8 b(.Gollub,)26 b(Chaotic)e(particle)g(transp)s(ort)g(in)g
(time-dep)s(enden)m(t)h(Ra)m(yleigh-)201 746 y(Benard)33
b(con)m(v)m(ection,)h(Ph)m(ys.)h(Rev.)e(A)g Fk(38)f Fu(\(1988\))g
(6280-6286.)49 950 y([2])49 b(T.H.Solomon,)32 b(S.T)-8
b(omas,)32 b(and)f(J.L.W)-8 b(arner,)32 b(Chaotic)g(mixing)g(of)e
(immiscible)k(impurities)e(in)201 1070 y(a)g(t)m(w)m(o-dimensional)i
(\015o)m(w,)f(Ph)m(ys.Fluids)j Fk(10)c Fu(\(1998\))g(342-350.)49
1273 y([3])49 b(A.A.Chernik)m(o)m(v,)37 b(A.I.Neish)m(tadt,)g
(A.V.Rogal'sky)-8 b(,)36 b(and)e(V.Z.Y)-8 b(akhnin,)36
b(Lagrangian)e(turbu-)201 1394 y(lence)f(in)g(nonstationary)g(2-D)f
(\015o)m(ws,)h(Chaos)h Fk(1)e Fu(\(1991\))g(206-211.)49
1597 y([4])49 b(P)-8 b(.Castiglione,)72 b(A.Mazzino,)h(P)-8
b(.Muratore-Ginannesc)m(hi,)73 b(and)64 b(A.V)-8 b(ulpiani,)72
b(On)63 b(strong)201 1718 y(anomalous)32 b(di\013usion,)i(Ph)m(ysica)h
(D,)d Fk(34)g Fu(\(1999\))g(75-93.)49 1921 y([5])49 b(P)-8
b(.Castiglione,)46 b(R.F)-8 b(esta,)45 b(and)e(A.Mazzino,)j(Di\013eren)
m(t)d(transp)s(ort)f(regimes)i(in)f(a)f(spatially-)201
2041 y(extended)50 b(recirculating)f(bac)m(kground,)54
b(Electronic)c(preprin)m(t)f(c)m(hao-dyn/9910029)e(20)h(Oct)201
2162 y(1999.)49 2365 y([6])h(H.Aref,)33 b(Stirring)f(b)m(y)i(c)m
(haotic)f(adv)m(ection,)h(J.Fluid.Mec)m(h.)h Fk(143)e
Fu(\(1984\))e(1-21.)49 2569 y([7])49 b(A.V.Timofeev,)43
b(On)c(the)h(constancy)h(of)e(an)g(adiabatic)g(in)m(v)-5
b(arian)m(t)40 b(when)h(the)e(nature)h(of)f(the)201 2689
y(motion)32 b(c)m(hanges,)i(So)m(v.)f(Ph)m(ys.)i(JETP)f
Fk(48)e Fu(\(1978\))g(656-659.)49 2892 y([8])49 b(J.R.Cary)-8
b(,)36 b(D.F.Escande,)h(and)e(J.T)-8 b(enn)m(yson,)38
b(Adiabatic)e(in)m(v)-5 b(arian)m(t)36 b(c)m(hange)g(due)g(to)e
(separa-)201 3013 y(trix)e(crossing,)i(Ph)m(ys.)h(Rev.)e
Fk(A34)f Fu(\(1986\))g(4256-4275.)49 3216 y([9])49 b(A.I.Neish)m(tadt,)
33 b(Change)g(of)e(an)h(adiabatic)g(in)m(v)-5 b(arian)m(t)33
b(at)e(a)h(separatrix,)h(So)m(v.J.Plasma)h(Ph)m(ys.)201
3337 y Fk(12)e Fu(\(1986\))g(568-573.)0 3540 y([10])49
b(V.I.Arnold,)g(V.V.Kozlo)m(v,)g(and)c(A.I.Neish)m(tadt,)50
b Fr(Mathematic)-5 b(al)46 b(asp)-5 b(e)g(cts)46 b(of)g(classic)-5
b(al)45 b(and)201 3660 y(c)-5 b(elestial)48 b(me)-5 b(chanics)47
b(\(Encyclop)-5 b(ae)g(dia)49 b(of)g(mathematic)-5 b(al)49
b(scienc)-5 b(es)48 b(3\))g Fu(\(Springer,)54 b(Berlin,)201
3781 y(1988\).)0 3984 y([11])49 b(V.I.Arnol'd,)31 b(Small)f
(denominators)h(and)f(problems)h(of)e(stabilit)m(y)i(of)e(motion)h(in)g
(classical)h(and)201 4104 y(celestial)j(mec)m(hanics)g
Fr(R)n(uss.)h(Math.)g(Surveys)e Fk(18)f Fu(\(1963\))g(85-192.)0
4308 y([12])49 b(A.I.Neish)m(tadt,)39 b(On)e(the)g(c)m(hange)h(in)f
(the)g(adiabatic)g(in)m(v)-5 b(arian)m(t)38 b(on)e(crossing)i(a)f
(separatrix)g(in)201 4428 y(systems)e(with)e(t)m(w)m(o)g(degrees)h(of)e
(freedom,)h(PMM)h(U.S.S.R.)f Fk(51)g Fu(\(1987\))e(586-592.)0
4632 y([13])49 b(A.P)-8 b(.Itin,)47 b(A.I.Neish)m(tadt,)i(and)44
b(A.A.V)-8 b(asiliev,)48 b(Captures)d(in)m(to)g(resonance)g(and)f
(scattering)201 4752 y(on)i(resonance)i(in)f(dynamics)h(of)e(a)h(c)m
(harged)g(relativistic)h(particle)g(in)f(magnetic)g(\014eld)g(and)201
4872 y(electrostatic)33 b(w)m(a)m(v)m(e,)i(Ph)m(ysica)g(D)d
Fk(141)g Fu(\(2000\))g(281-296.)1841 5281 y(17)p eop
%%Trailer
end
userdict /end-hook known{end-hook}if
%%EOF