%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: pap1.dvi %%Pages: 18 -1 %%PageOrder: Descend %%BoundingBox: 0 0 612 792 %%EndComments %DVIPSCommandLine: dvips -pp1-18 pap1.dvi -o pap1.ps %DVIPSParameters: dpi=300, comments removed %DVIPSSource: TeX output 1999.03.12:1031 %%BeginProcSet: tex.pro /TeXDict 250 dict def TeXDict begin /N{def}def /B{bind def}N /S{exch}N /X{S N}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{dup dup 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 /IE 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 IE N end dup{/foo setfont}2 array copy cvx N load 0 nn put /ctr 0 N[}B /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 dup definefont setfont}B /ch-width{ch-data dup length 5 sub get}B /ch-height{ch-data dup length 4 sub get}B /ch-xoff{ 128 ch-data dup length 3 sub get sub}B /ch-yoff{ch-data dup length 2 sub get 127 sub}B /ch-dx{ch-data dup length 1 sub get}B /ch-image{ch-data dup type /stringtype ne{ctr get /ctr ctr 1 add N}if}B /id 0 N /rw 0 N /rc 0 N /gp 0 N /cp 0 N /G 0 N /sf 0 N /CharBuilder{save 3 1 roll S dup /base get 2 index get S /BitMaps get S get /ch-data X pop /ctr 0 N ch-dx 0 ch-xoff ch-yoff ch-height sub ch-xoff ch-width add ch-yoff setcachedevice ch-width ch-height true[1 0 0 -1 -.1 ch-xoff sub ch-yoff .1 sub]{ch-image}imagemask restore}B /D{/cc X dup type /stringtype ne{]} if nn /base get cc ctr put nn /BitMaps get S ctr S sf 1 ne{dup dup length 1 sub dup 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 dup 1 get dup mul exch 0 get dup 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 /IE 256 array N 0 1 255{IE S 1 string dup 0 3 index put cvn put}for 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 /rulex 0 N /ruley 0 N /v{/ruley X /rulex X V}B /V {}B /RV statusdict begin /product where{pop product dup length 7 ge{0 7 getinterval dup(Display)eq exch 0 4 getinterval(NeXT)eq or}{pop false} ifelse}{false}ifelse end{{gsave TR -.1 .1 TR 1 1 scale rulex ruley false RMat{BDot}imagemask grestore}}{{gsave TR -.1 .1 TR rulex ruley scale 1 1 false RMat{BDot}imagemask grestore}}ifelse B /QV{gsave newpath transform round exch round exch itransform moveto rulex 0 rlineto 0 ruley neg rlineto rulex neg 0 rlineto fill grestore}B /a{moveto}B /delta 0 N /tail {dup /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: special.pro TeXDict begin /SDict 200 dict N SDict begin /@SpecialDefaults{/hs 612 N /vs 792 N /ho 0 N /vo 0 N /hsc 1 N /vsc 1 N /ang 0 N /CLIP 0 N /rwiSeen false N /rhiSeen false N /letter{}N /note{}N /a4{}N /legal{}N}B /@scaleunit 100 N /@hscale{@scaleunit div /hsc X}B /@vscale{@scaleunit div /vsc X}B /@hsize{/hs X /CLIP 1 N}B /@vsize{/vs X /CLIP 1 N}B /@clip{ /CLIP 2 N}B /@hoffset{/ho X}B /@voffset{/vo X}B /@angle{/ang X}B /@rwi{ 10 div /rwi X /rwiSeen true N}B /@rhi{10 div /rhi X /rhiSeen true N}B /@llx{/llx X}B /@lly{/lly X}B /@urx{/urx X}B /@ury{/ury X}B /magscale true def end /@MacSetUp{userdict /md known{userdict /md get type /dicttype eq{userdict begin md length 10 add md maxlength ge{/md md dup length 20 add dict copy def}if end md begin /letter{}N /note{}N /legal{} N /od{txpose 1 0 mtx defaultmatrix dtransform S atan/pa X newpath clippath mark{transform{itransform moveto}}{transform{itransform lineto} }{6 -2 roll transform 6 -2 roll transform 6 -2 roll transform{ itransform 6 2 roll itransform 6 2 roll itransform 6 2 roll curveto}}{{ closepath}}pathforall newpath counttomark array astore /gc xdf pop ct 39 0 put 10 fz 0 fs 2 F/|______Courier fnt invertflag{PaintBlack}if}N /txpose{pxs pys scale ppr aload pop por{noflips{pop S neg S TR pop 1 -1 scale}if xflip yflip and{pop S neg S TR 180 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{pop S neg S TR pop 180 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{ppr 1 get neg ppr 0 get neg TR}if}{noflips{TR pop pop 270 rotate 1 -1 scale}if xflip yflip and{TR pop pop 90 rotate 1 -1 scale ppr 3 get ppr 1 get neg sub neg ppr 2 get ppr 0 get neg sub neg TR}if xflip yflip not and{TR pop pop 90 rotate ppr 3 get ppr 1 get neg sub neg 0 TR}if yflip xflip not and{TR pop pop 270 rotate ppr 2 get ppr 0 get neg sub neg 0 S TR}if}ifelse scaleby96{ppr aload pop 4 -1 roll add 2 div 3 1 roll add 2 div 2 copy TR .96 dup scale neg S neg S TR}if}N /cp {pop pop showpage pm restore}N end}if}if}N /normalscale{Resolution 72 div VResolution 72 div neg scale magscale{DVImag dup scale}if 0 setgray} N /psfts{S 65781.76 div N}N /startTexFig{/psf$SavedState save N userdict maxlength dict begin /magscale true def normalscale currentpoint TR /psf$ury psfts /psf$urx psfts /psf$lly psfts /psf$llx psfts /psf$y psfts /psf$x psfts currentpoint /psf$cy X /psf$cx X /psf$sx psf$x psf$urx psf$llx sub div N /psf$sy psf$y psf$ury psf$lly sub div N psf$sx psf$sy scale psf$cx psf$sx div psf$llx sub psf$cy psf$sy div psf$ury sub TR /showpage{}N /erasepage{}N /copypage{}N /p 3 def @MacSetUp}N /doclip{ psf$llx psf$lly psf$urx psf$ury currentpoint 6 2 roll newpath 4 copy 4 2 roll moveto 6 -1 roll S lineto S lineto S lineto closepath clip newpath moveto}N /endTexFig{end psf$SavedState restore}N /@beginspecial{SDict begin /SpecialSave save N gsave normalscale currentpoint TR @SpecialDefaults count /ocount X /dcount countdictstack N}N /@setspecial {CLIP 1 eq{newpath 0 0 moveto hs 0 rlineto 0 vs rlineto hs neg 0 rlineto closepath clip}if ho vo TR hsc vsc scale ang rotate rwiSeen{rwi urx llx sub div rhiSeen{rhi ury lly sub div}{dup}ifelse scale llx neg lly neg TR }{rhiSeen{rhi ury lly sub div dup scale llx neg lly neg TR}if}ifelse CLIP 2 eq{newpath llx lly moveto urx lly lineto urx ury lineto llx ury lineto closepath clip}if /showpage{}N /erasepage{}N /copypage{}N newpath }N /@endspecial{count ocount sub{pop}repeat countdictstack dcount sub{ end}repeat grestore SpecialSave restore end}N /@defspecial{SDict begin} N /@fedspecial{end}B /li{lineto}B /rl{rlineto}B /rc{rcurveto}B /np{ /SaveX currentpoint /SaveY X N 1 setlinecap newpath}N /st{stroke SaveX SaveY moveto}N /fil{fill SaveX SaveY moveto}N /ellipse{/endangle X /startangle X /yrad X /xrad X /savematrix matrix currentmatrix N TR xrad yrad scale 0 0 1 startangle endangle arc savematrix setmatrix}N end %%EndProcSet TeXDict begin 40258431 52099146 1000 300 300 (pap1.dvi) @start /Fa 14 120 df<3078FCFC7830060676851A>46 D<0000C00001E00001E00003 E00003C00007C0000780000F80000F00000F00001F00001E00003E00003C00007C000078 0000780000F80000F00001F00001E00003E00003C00003C00007C0000780000F80000F00 001F00001E00001E00003E00003C00007C0000780000F80000F00000F000006000001327 7DA21A>I<3078FCFC78300000000000000000003078FCFC7830061576941A>58 D<1FF0003FFC007FFE00780F00300700000380000380007F8007FF801FFF803F83807803 80700380E00380E00380E00380700780780F803FFFFC1FFDFC07F0FC16157D941A>97 D<00FF8003FFC00FFFE01F01E03C00C0780000700000700000E00000E00000E00000E000 00E000007000007000007800703C00701F01F00FFFE003FFC000FE0014157D941A>99 D104 D107 D<7CE0E000FFFBF8007FFFF8001F1F1C001E1E1C001E1E1C001C1C1C001C1C1C001C1C1C 001C1C1C001C1C1C001C1C1C001C1C1C001C1C1C001C1C1C001C1C1C001C1C1C001C1C1C 007F1F1F00FF9F9F807F1F1F00191580941A>109 D<01F00007FC001FFF003E0F803C07 807803C07001C0E000E0E000E0E000E0E000E0E000E0E000E0F001E07001C07803C03C07 803E0F801FFF0007FC0001F00013157D941A>111 DI<7F83F0FF8FF87FBFFC03FC3C03F01803E00003 C00003C0000380000380000380000380000380000380000380000380000380000380007F FF00FFFF007FFF0016157E941A>114 D<07FB801FFF807FFF80780780E00380E00380E0 03807800007FC0003FFC0007FE00003F800007806001C0E001C0E001C0F003C0FC0780FF FF00EFFE00E3F80012157C941A>I<00C00001C00001C00001C00001C00001C00001C000 7FFFE0FFFFE0FFFFE001C00001C00001C00001C00001C00001C00001C00001C00001C000 01C00001C07001C07001C07001C07000E0E000FFE0007FC0001F00141C7F9B1A>I119 D E /Fb 1 108 df<3C000C000C00180018001800187031903230340038007F00618061 906190C1A0C0C00C117E9010>107 D E /Fc 1 49 df<1F00318060C04040C060C060C0 60C060C060C060C060C060404060C031801F000B107F8F0F>48 D E /Fd 27 120 df<387CFEFEFE7C3807077C8610>46 D<00FE0007FFC00F83E01F01F03E 00F83E00F87C007C7C007C7C007CFC007CFC007EFC007EFC007EFC007EFC007EFC007EFC 007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C7C007C3E00F83E00F81F 01F00F83E007FFC000FE0017207E9F1C>48 D<00180000780001F800FFF800FFF80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 0001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F80001F8 007FFFE07FFFE013207C9F1C>I<03FC000FFF003C1FC07007E07C07F0FE03F0FE03F8FE 03F8FE01F87C01F83803F80003F80003F00003F00007E00007C0000F80001F00003E0000 380000700000E01801C0180380180700180E00380FFFF01FFFF03FFFF07FFFF0FFFFF0FF FFF015207D9F1C>I<00FE0007FFC00F07E01E03F03F03F03F81F83F81F83F81F81F03F8 1F03F00003F00003E00007C0001F8001FE0001FF000007C00001F00001F80000FC0000FC 3C00FE7E00FEFF00FEFF00FEFF00FEFF00FC7E01FC7801F81E07F00FFFC001FE0017207E 9F1C>I<0000E00001E00003E00003E00007E0000FE0001FE0001FE00037E00077E000E7 E001C7E00187E00307E00707E00E07E00C07E01807E03807E07007E0E007E0FFFFFEFFFF FE0007E00007E00007E00007E00007E00007E00007E000FFFE00FFFE17207E9F1C>I<10 00201E01E01FFFC01FFF801FFF001FFE001FF8001BC00018000018000018000018000019 FC001FFF001E0FC01807E01803E00003F00003F00003F80003F83803F87C03F8FE03F8FE 03F8FC03F0FC03F07007E03007C01C1F800FFF0003F80015207D9F1C>I<001F8000FFE0 03F07007C0F00F01F81F01F83E01F83E01F87E00F07C00007C0000FC0800FC7FC0FCFFE0 FD80F0FF00F8FE007CFE007CFC007EFC007EFC007EFC007E7C007E7C007E7C007E3C007C 3E007C1E00F80F00F00783E003FFC000FF0017207E9F1C>I<6000007800007FFFFE7FFF FE7FFFFC7FFFF87FFFF87FFFF0E00060E000C0C00180C00300C00300000600000C00001C 0000180000380000780000780000F00000F00000F00001F00001F00001F00003F00003F0 0003F00003F00003F00003F00003F00001E00017227DA11C>I<00FE0003FFC00703E00E 00F01C00F01C00783C00783E00783F00783F80783FE0F01FF9E01FFFC00FFF8007FFC003 FFE007FFF01E7FF83C1FFC7807FC7801FEF000FEF0003EF0001EF0001EF0001CF8001C78 00383C00381F01F00FFFC001FF0017207E9F1C>I<01FE0007FF800F83E01E01F03E00F0 7C00F87C0078FC007CFC007CFC007CFC007EFC007EFC007EFC007E7C00FE7C00FE3E01FE 1E037E0FFE7E07FC7E00207E00007C00007C1E007C3F00F83F00F83F00F03F01E01E03C0 1C0F800FFE0003F80017207E9F1C>I<000070000000007000000000F800000000F80000 0000F800000001FC00000001FC00000003FE00000003FE00000003FE00000006FF000000 067F0000000E7F8000000C3F8000000C3F800000183FC00000181FC00000381FE0000030 0FE00000300FE00000600FF000006007F00000E007F80000FFFFF80000FFFFF800018001 FC00018001FC00038001FE00030000FE00030000FE000600007F000600007F00FFE00FFF F8FFE00FFFF825227EA12A>65 D76 D<3FFFFFE03FFFFFE03F801FC03E003FC03C003F803800 7F007000FF007000FE007001FE006003FC006003F8006007F8000007F000000FE000001F E000001FC000003FC000007F8000007F000000FF000000FE006001FC006003FC006003F8 006007F800E00FF000E00FE000E01FE001C01FC001C03F8003C07F8007C07F003FC0FFFF FFC0FFFFFFC01B227DA122>90 D<07FC001FFF803F07C03F03E03F01E03F01F01E01F000 01F00001F0003FF003FDF01FC1F03F01F07E01F0FC01F0FC01F0FC01F0FC01F07E02F07E 0CF81FF87F07E03F18167E951B>97 D<00FF8007FFE00F83F01F03F03E03F07E03F07C01 E07C0000FC0000FC0000FC0000FC0000FC0000FC00007C00007E00007E00003E00301F00 600FC0E007FF8000FE0014167E9519>99 D<0001FE000001FE0000003E0000003E000000 3E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0001FC 3E0007FFBE000F81FE001F007E003E003E007E003E007C003E00FC003E00FC003E00FC00 3E00FC003E00FC003E00FC003E00FC003E00FC003E007C003E007C003E003E007E001E00 FE000F83BE0007FF3FC001FC3FC01A237EA21F>I<00FE0007FF800F87C01E01E03E01F0 7C00F07C00F8FC00F8FC00F8FFFFF8FFFFF8FC0000FC0000FC00007C00007C00007E0000 3E00181F00300FC07003FFC000FF0015167E951A>I<03FC1E0FFF7F1F0F8F3E07CF3C03 C07C03E07C03E07C03E07C03E07C03E03C03C03E07C01F0F801FFF0013FC003000003000 003800003FFF801FFFF00FFFF81FFFFC3800FC70003EF0001EF0001EF0001EF0001E7800 3C7C007C3F01F80FFFE001FF0018217E951C>103 D107 DIII<00FE0007FFC00F83E01E00F03E00F87C007C7C007C7C 007CFC007EFC007EFC007EFC007EFC007EFC007EFC007E7C007C7C007C3E00F81F01F00F 83E007FFC000FE0017167E951C>I<0FF3003FFF00781F00600700E00300E00300F00300 FC00007FE0007FF8003FFE000FFF0001FF00000F80C00780C00380E00380E00380F00700 FC0E00EFFC00C7F00011167E9516>115 D<018000018000018000018000038000038000 0780000780000F80003F8000FFFF00FFFF000F80000F80000F80000F80000F80000F8000 0F80000F80000F80000F80000F80000F81800F81800F81800F81800F81800F830007C300 03FE0000F80011207F9F16>I119 D E /Fe 5 107 df0 D<0000300000F00001C0000700001E00007800 01E0000380000E00003C0000F00000F000003800000E000007800001E000007800001C00 0007000003C00000F00000300000000000000000000000000000000000007FFFE0FFFFF0 141E7D951B>20 D<0000010000000080000000800000004000000020FFFFFFFCFFFFFFFC 00000020000000400000008000000080000001001E0C7E8D23>33 D<0F8007C019E01C202070301040184008C00C8004800780048007000480038004800780 048004C00C400860082030381010E01E600F8007C01E0E7E8D23>49 D106 D E /Ff 28 122 df<0001FF0000001FFFC000007F81E00000FC01E00001 F807F00003F807F00007F007F00007F007F00007F007F00007F007F00007F001C00007F0 00000007F000000007F000000007F03FF800FFFFFFF800FFFFFFF800FFFFFFF80007F003 F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003F8 0007F003F80007F003F80007F003F80007F003F80007F003F80007F003F80007F003F800 07F003F80007F003F80007F003F80007F003F80007F003F80007F003F8007FFF3FFF807F FF3FFF807FFF3FFF80212A7FA925>12 D<1C007F007F00FF80FF80FF807F007F001C0009 097B8813>46 D<000E00001E00007E0007FE00FFFE00FFFE00F8FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE0000FE 0000FE0000FE007FFFFE7FFFFE7FFFFE17277BA622>49 D<00FF800007FFF0000FFFFC00 1E03FE003800FF807C003F80FE003FC0FF001FC0FF001FE0FF000FE0FF000FE07E000FE0 3C001FE000001FE000001FC000001FC000003F8000003F0000007E000000FC000000F800 0001F0000003E00000078000000F0000001E0000003C00E0007000E000E000E001C001C0 038001C0060001C00FFFFFC01FFFFFC03FFFFFC07FFFFFC0FFFFFF80FFFFFF80FFFFFF80 1B277DA622>I<007F800003FFF00007FFFC000F80FE001F007F003F807F003F803F803F 803F803F803F801F803F801F003F8000007F0000007F0000007E000000FC000001F80000 07F00000FFC00000FFC0000001F80000007E0000003F0000003F8000001FC000001FC000 001FE000001FE03C001FE07E001FE0FF001FE0FF001FE0FF001FC0FF003FC0FE003F807C 007F003F00FE001FFFFC0007FFF00000FF80001B277DA622>I<0007F800003FFE0000FF FF0001FC078003F00FC007C01FC00F801FC01F801FC01F001FC03F000F803F0000007E00 00007E0000007E000000FE020000FE1FF000FE3FFC00FE603E00FE801F00FF801F80FF00 0FC0FF000FC0FE000FE0FE000FE0FE000FE0FE000FE07E000FE07E000FE07E000FE07E00 0FE03E000FE03F000FC01F000FC01F001F800F801F0007E07E0003FFFC0001FFF800003F C0001B277DA622>54 D<00007FE0030007FFFC07001FFFFF0F007FF00F9F00FF0001FF01 FC0000FF03F800007F07F000003F0FE000001F1FC000001F1FC000000F3F8000000F3F80 0000077F800000077F800000077F00000000FF00000000FF00000000FF00000000FF0000 0000FF00000000FF00000000FF00000000FF00000000FF000000007F000000007F800000 007F800000073F800000073F800000071FC00000071FC000000E0FE000000E07F000001C 03F800003C01FC00007800FF0001F0007FF007C0001FFFFF800007FFFE0000007FF00028 297CA831>67 D76 D<00FF00C003FFE1C00FFFF9C01F80FFC03F003FC03E000FC07C0007C07C0007C0 FC0003C0FC0003C0FC0001C0FE0001C0FE0001C0FF000000FFC000007FFC00007FFFE000 3FFFF8001FFFFE001FFFFF0007FFFF8003FFFFC000FFFFC0000FFFE000007FE000001FF0 00000FF0000007F0E00003F0E00003F0E00003F0E00003F0F00003E0F00003E0F80007E0 FC0007C0FF000F80FFE01F80E3FFFF00E1FFFC00C01FF0001C297CA825>83 D<03FF80000FFFF0001F01FC003F80FE003F807F003F803F003F803F801F003F8000003F 8000003F8000003F8000003F80003FFF8001FC3F800FE03F801F803F803F003F807E003F 80FC003F80FC003F80FC003F80FC003F80FC005F807E00DF803F839FFC1FFE0FFC03F803 FC1E1B7E9A21>97 DI<003FF00001FFFC0003F03E000FC07F001F807F003F007F003F007F007F003E00 7E0000007E000000FE000000FE000000FE000000FE000000FE000000FE000000FE000000 7E0000007E0000007F0000003F0003803F8003801F8007000FE00E0003F83C0001FFF800 003FC000191B7E9A1E>I<003FC00001FFF00003E07C000F803E001F801F001F001F003F 000F807E000F807E000FC07E000FC0FE0007C0FE0007C0FFFFFFC0FFFFFFC0FE000000FE 000000FE0000007E0000007E0000007F0000003F0001C01F0001C00F80038007C0070003 F01E0000FFFC00003FE0001A1B7E9A1F>101 D<0007F8003FFC007E3E01FC7F03F87F03 F07F07F07F07F03E07F00007F00007F00007F00007F00007F00007F000FFFFC0FFFFC0FF FFC007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F0007FFF807FFF807F FF80182A7EA915>I104 D<07000F801FC03FE03FE03FE01FC00F800700000000000000000000000000 0000FFE0FFE0FFE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE00FE0 0FE00FE00FE00FE00FE00FE00FE0FFFEFFFEFFFE0F2B7EAA12>I108 DII<00 3FE00001FFFC0003F07E000FC01F801F800FC03F0007E03F0007E07E0003F07E0003F07E 0003F0FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE0003F8FE0003F87E 0003F07E0003F03F0007E03F0007E01F800FC00FC01F8007F07F0001FFFC00003FE0001D 1B7E9A22>II114 D<03FE300FFFF03E03F07800F07000F0F00070F00070F80070FE0000FFE0 007FFF007FFFC03FFFE01FFFF007FFF800FFF80007FC0000FCE0007CE0003CF0003CF000 38F80038FC0070FF01E0E7FFC0C1FF00161B7E9A1B>I<00700000700000700000700000 F00000F00000F00001F00003F00003F00007F0001FFFE0FFFFE0FFFFE007F00007F00007 F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F00007F07007 F07007F07007F07007F07007F07007F07003F0E001F8C000FFC0003F0014267FA51A>I< FFE07FF0FFE07FF0FFE07FF00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0 0FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F00FE007F0 0FE007F00FE007F00FE007F00FE00FF00FE00FF007E017F003F067FF01FFC7FF007F87FF 201B7D9A25>II120 DI E /Fg 6 89 df12 D<00000180000003000000060000000C00000018 0000003000000060000000E0000001C000000180000003800000070000000E0000000E00 00001C0000001C000000380000007800000070000000F0000000E0000001E0000001C000 0001C0000003C00000038000000780000007800000070000000F0000000F0000000E0000 001E0000001E0000001E0000001C0000003C0000003C0000003C0000003C000000380000 007800000078000000780000007800000078000000780000007800000070000000F00000 00F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00000 00F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00000 00F0000000F0000000700000007800000078000000780000007800000078000000780000 0078000000380000003C0000003C0000003C0000003C0000001C0000001E0000001E0000 001E0000000E0000000F0000000F0000000700000007800000078000000380000003C000 0001C0000001C0000001E0000000E0000000F00000007000000078000000380000001C00 00001C0000000E0000000E00000007000000038000000180000001C0000000E000000060 00000030000000180000000C000000060000000300000001801977768125>18 DI<000000C000000180000003000000070000000E00 00000C00000018000000380000007000000060000000E0000001C0000001C00000038000 0007800000070000000F0000000E0000001E0000001C0000003C00000038000000780000 0078000000F0000000F0000000F0000001E0000001E0000001E0000003C0000003C00000 03C000000780000007800000078000000F8000000F0000000F0000000F0000001F000000 1F0000001E0000001E0000001E0000003E0000003E0000003E0000003E0000003C000000 3C0000007C0000007C0000007C0000007C0000007C0000007C0000007C0000007C000000 78000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000 F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000 F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000 F8000000F8000000780000007C0000007C0000007C0000007C0000007C0000007C000000 7C0000007C0000003C0000003C0000003E0000003E0000003E0000003E0000001E000000 1E0000001E0000001F0000001F0000000F0000000F0000000F0000000F80000007800000 078000000780000003C0000003C0000003C0000001E0000001E0000001E0000000F00000 00F0000000F00000007800000078000000380000003C0000001C0000001E0000000E0000 000F00000007000000078000000380000001C0000001C0000000E0000000600000007000 000038000000180000000C0000000E00000007000000030000000180000000C01A947581 27>32 DI88 D E /Fh 2 83 df<0003FC02001FFF86007F03FE 01EC00FE03980066063000360C30001E1860000E1860000E30C0000660C0000660C00002 61800000C1800000C1800000C1800000C1800000C1800000C1800000C1800000C1800000 C1800000C18000006180000060C0000060C0000030C0000018C00001186000030C600006 0630000C03B0001801FC0070007F01E0001FFF800003FE0020247EA218>67 D82 D E /Fi 9 62 df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j 15 115 df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k 18 121 df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l 41 121 df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m 50 122 df<00008000010000020000040000080000100000300000600000C00000 C0000180000300000300000600000600000E00000C00001C000018000018000038000030 0000300000700000700000600000600000E00000E00000E00000C00000C00000C00000C0 0000C00000C00000C00000C00000C00000C00000C00000C00000C0000040000060000060 0000200000300000100000080000113278A414>40 D<0008000004000006000002000003 000003000001000001800001800001800001800001800001800001800001800001800001 800001800001800001800003800003800003800003000003000007000007000006000006 00000E00000C00000C00001C0000180000380000300000300000600000600000C0000180 00018000030000060000040000080000100000200000400000800000113280A414>I<0E 1E1E1E1E02020404080810204080070F7D840F>44 DI<70 F8F8F0E005057A840F>I<0001000300030006001E002E03CE001C001C001C001C003800 3800380038007000700070007000E000E000E000E001C001C001C001C003800380038003 800780FFFC10217AA019>49 D<000FC000106000603800801800801C01001C02201E0210 1E04101E04101E04101E08203C08203C0840380840780880F00700E00001C00003000006 0000180000200000C0000100000200000400100800301000202000605F80C063FFC040FF 80807F00801E0017227CA019>I<000FC000307000C01801001C02001C04000C04401C08 201C08201C08201C08403808C0380700700000600001C000070000FC0000070000038000 03800001C00001C00001C00003C06003C0F003C0F00380E00780800700800E00801C0040 380020F0001F800016227BA019>I<07000F800F800F000E000000000000000000000000 00000000000000000000007000F800F800F000E00009157A940F>58 D<0000030000000300000007000000070000000F0000000F0000001F0000002F0000002F 0000004F0000004F80000087800000878000010780000207800002078000040780000407 800008078000080780001007800030078000200780007FFF80004007C0008007C0008003 C0010003C0030003C0020003C0040003C0040003C00C0003C03C0007C0FF003FFC1E237D A224>65 D<00FFFFE0000F0038000F001C000F001E001E000E001E000F001E000F001E00 0F003C000E003C001E003C001E003C003C00780078007800F0007801E00078078000FFFF 8000F001E000F000F000F0007801E0007801E0003801E0003C01E0003C03C0007803C000 7803C0007803C000F0078000F0078001E0078003C0078007000F801E00FFFFF00020227D A122>I<00007F00800003808100000E00630000380027000070001F0000E0000E0001C0 000E000380000E000700000E000F000004000E000004001E000004003C000004003C0000 0800780000000078000000007800000000F000000000F000000000F000000000F0000000 00F000000000E000000000E000002000E000002000E000004000E000004000F000008000 70000080007000010000380002000018000400001C0008000006003000000381C0000000 FE000000212479A223>I<00FFFFF000000F003C00000F000E00000F000700001E000380 001E000380001E0001C0001E0001C0003C0001C0003C0001E0003C0001E0003C0001E000 780001E000780001E000780001E000780001E000F00003C000F00003C000F00003C000F0 0003C001E000078001E000078001E000070001E0000F0003C0000E0003C0001C0003C000 3C0003C00038000780007000078000E000078001C00007800700000F801C0000FFFFF000 0023227DA125>I<00FFFFFF80000F000780000F000180000F000180001E000180001E00 0180001E000100001E000100003C000100003C000100003C010100003C01000000780200 000078020000007806000000780E000000FFFC000000F00C000000F00C000000F00C0000 01E008000001E008000001E008040001E000080003C000080003C000080003C000100003 C000100007800020000780006000078000C000078001C0000F8007C000FFFFFF80002122 7DA121>I<00FFFFFF000F000F000F0003000F0003001E0003001E0003001E0002001E00 02003C0002003C0002003C0102003C010000780200007802000078060000780E0000FFFC 0000F00C0000F00C0000F00C0001E0080001E0080001E0080001E0000003C0000003C000 0003C0000003C00000078000000780000007800000078000000F800000FFFC000020227D A120>I<00007F00800003808100000E00630000380027000070001F0000E0000E0001C0 000E000380000E000700000E000F000004000E000004001E000004003C000004003C0000 0800780000000078000000007800000000F000000000F000000000F000000000F0000000 00F0003FFC00E00001E000E00001E000E00001E000E00003C000E00003C000F00003C000 700003C0007000078000380007800018000F80001C0013800006002300000381C1000000 FE000000212479A226>I<00FFF8000F00000F00000F00001E00001E00001E00001E0000 3C00003C00003C00003C0000780000780000780000780000F00000F00000F00000F00001 E00001E00001E00001E00003C00003C00003C00003C0000780000780000780000780000F 8000FFF80015227DA113>73 D<0007FFC000003C0000003C0000003C0000007800000078 0000007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E0 000001E0000003C0000003C0000003C0000003C000000780000007800000078000000780 00000F0000000F0000380F0000780F0000F81E0000F81E0000F03C000040380000407000 0021E000001F8000001A237CA11A>I<00FFFC00000F8000000F0000000F0000001E0000 001E0000001E0000001E0000003C0000003C0000003C0000003C00000078000000780000 007800000078000000F0000000F0000000F0000000F0000001E0000001E0000001E00020 01E0002003C0004003C0004003C0008003C0008007800180078001000780030007800F00 0F803E00FFFFFE001B227DA11F>76 D<00FF800007FC000F80000F80000F80001780000F 80001780001780002F000013C0002F000013C0004F000013C0008F000023C0009E000023 C0011E000023C0011E000023C0021E000043C0043C000043C0043C000043C0083C000041 E0083C000081E01078000081E02078000081E02078000081E04078000101E040F0000101 E080F0000101E100F0000101E100F0000200F201E0000200F201E0000200F401E0000200 F801E0000400F803C0000400F003C0000400F003C0000C00E003C0001E00C007C000FFC0 C07FFC002E227DA12C>I<00FF000FFC000F8001E0000F800180000FC000800013C00100 0013C001000011E001000011E001000021E002000020F002000020F002000020F0020000 407804000040780400004078040000403C040000803C080000803E080000801E08000080 1E080001001F100001000F100001000F10000100079000020007A000020007A000020003 E000020003E000040003C000040001C000040001C0000C0001C0001E00008000FFC00080 0026227DA124>I<0000FE0000078380000C00E0003800700070003800E0003801C0001C 0380001C0700001C0F00001E1E00001E1C00001E3C00001E3C00001E7800001E7800001E 7800001EF000003CF000003CF000003CF0000078F0000078E0000078E00000F0E00000F0 E00001E0E00001C0F00003C0F00007807000070078000E0038001C001C0038000E00E000 0703800001FC00001F2479A225>I<00FFFFE0000F0038000F001E000F000E001E000700 1E0007001E0007001E0007003C000F003C000F003C000F003C001E0078001E0078003C00 780078007800E000F003C000FFFE0000F0000000F0000001E0000001E0000001E0000001 E0000003C0000003C0000003C0000003C00000078000000780000007800000078000000F 800000FFF8000020227DA121>I<00FFFFC0000F0070000F003C000F001C001E000E001E 000E001E000F001E000F003C001E003C001E003C001E003C003C00780038007800700078 01E00078078000FFFC0000F00E0000F0070000F0038001E003C001E003C001E003C001E0 03C003C0078003C0078003C0078003C0078007800F0007800F0107800F01078007020F80 0702FFF8038C000000F020237DA124>82 D<0001F020000E0C40001802C0003001C00060 01C000E0018000C0018001C0018001C0018003C0010003C0010003C0000003C0000003E0 000001F8000001FF000000FFE000007FF000001FF8000003FC0000007C0000003C000000 1E0000001E0000001E0020001C0020001C0020001C002000180060003800600030007000 60007000C000C8018000C607000081FC00001B247DA21B>I<1FFFFFF81E03C0381803C0 183003C01820078018200780184007801040078010400F0010800F0010800F0010000F00 00001E0000001E0000001E0000001E0000003C0000003C0000003C0000003C0000007800 0000780000007800000078000000F0000000F0000000F0000000F0000001E0000001E000 0001E0000001E0000003E00000FFFF00001D2277A123>I86 D<00F8C00185C00705C00E03800E03801C03803C0380380700780700780700780700F00E 00F00E00F00E00F00E10F01C20701C20703C20305C40308C400F078014157B9419>97 D<03C03F8003800380038007000700070007000E000E000E000E001C001CF81D0C1E0E3C 0638073807380F700F700F700F700FE01EE01EE01EE03CE038E038607060E031C01F0010 237BA216>I<007E0001C1000301800703800E07801C07803C0000380000780000780000 780000F00000F00000F00000F00000F00100700100700200300C001830000FC00011157B 9416>I<00003C0003F80000380000380000380000700000700000700000700000E00000 E00000E00000E00001C000F9C00185C00705C00E03800E03801C03803C03803807007807 00780700780700F00E00F00E00F00E00F00E10F01C20701C20703C20305C40308C400F07 8016237BA219>I<00F803840E021C023C0238027804F018FFE0F000F000E000E000E000 E000E002E0026004701830600F800F157A9416>I<00003E0000470000CF00018F000186 000380000380000380000700000700000700000700000700000E0000FFF0000E00000E00 000E00001C00001C00001C00001C00001C00003800003800003800003800003800007000 00700000700000700000700000E00000E00000E00000E00000C00001C00001C000718000 F18000F300006200003C0000182D82A20F>I<001F180030B800E0B801C07001C0700380 700780700700E00F00E00F00E00F00E01E01C01E01C01E01C01E01C01E03800E03800E07 80060B8006170001E700000700000700000E00000E00000E00701C00F01800F0300060E0 003F8000151F7E9416>I<00F0000FE00000E00000E00000E00001C00001C00001C00001 C000038000038000038000038000070000071F0007218007C0C00F00E00F00E00E00E00E 00E01C01C01C01C01C01C01C01C0380380380380380380380704700708700E08700E1070 0610E006206003C016237DA219>I<00C001E001C001C000000000000000000000000000 0000001C002300430043008700870087000E000E001C001C001C00380038003840708070 807080710032001C000B217BA00F>I<00F0000FE00000E00000E00000E00001C00001C0 0001C00001C0000380000380000380000380000700000701E0070210070C700E10F00E10 F00E20600E40001D80001E00001FC0001C7000383800383800381C00381C207038407038 40703840701880E01880600F0014237DA216>107 D<01E01FC001C001C001C003800380 0380038007000700070007000E000E000E000E001C001C001C001C003800380038003800 7000700070007100E200E200E200E200640038000B237CA20C>I<1C0F80F8002610C10C 00476066060087807807008780780700870070070087007007000E00E00E000E00E00E00 0E00E00E000E00E00E001C01C01C001C01C01C001C01C01C001C01C03820380380384038 038070403803807080380380308070070031003003001E0023157B9428>I<1C0F002631 C04740C08780E08780E08700E08700E00E01C00E01C00E01C00E01C01C03801C03801C03 801C0704380708380E08380E103806107006203003C016157B941B>I<007E0001C30003 81800701C00E01C01C01E03C01E03801E07801E07801E07801E0F003C0F003C0F00380F0 0780700700700E00700C0030180018700007C00013157B9419>I<01C1F002621804741C 08780C08700E08700E08701E00E01E00E01E00E01E00E01E01C03C01C03C01C03C01C078 03807003807003C0E003C1C0072380071E000700000700000E00000E00000E00000E0000 1C00001C00001C0000FFC000171F7F9419>I<00F8400184C00705C00E03800E03801C03 803C0380380700780700780700780700F00E00F00E00F00E00F00E00F01C00701C00703C 00305C0030B8000F380000380000380000700000700000700000700000E00000E00000E0 000FFE00121F7B9416>I<1C1F002620804741C08783C08703C08701808700000E00000E 00000E00000E00001C00001C00001C00001C000038000038000038000038000070000030 000012157B9415>I<00FC000183000200800401800C03800C03000C00000F00000FF000 07FC0003FE00003E00000F00000700700700F00600F00600E004004008002030001FC000 11157D9414>I<00C001C001C001C001C003800380038003800700FFF8070007000E000E 000E000E001C001C001C001C003800380038003810702070207040708031001E000D1F7C 9E10>I<1E00602300E04380E04381C08381C08701C08701C00703800E03800E03800E03 801C07001C07001C07001C07081C0E10180E101C0E101C1E200C262007C3C015157B941A >I<1E03802307C04387C04383C08381C08700C08700C00700800E00800E00800E00801C 01001C01001C01001C02001C02001C04001C08001C08000C300003C00012157B9416>I< 1E0060E02300E1F04380E1F04381C0F08381C0708701C0308701C030070380200E038020 0E0380200E0380201C0700401C0700401C0700401C0700801C0700801C0701001C0F0100 0C0F020006138C0003E0F0001C157B9420>I<1E00302300704380704380E08380E08700 E08700E00701C00E01C00E01C00E01C01C03801C03801C03801C03801C07001C07001C07 001C0F000C3E0003CE00000E00000E00001C00601C00F03800F03000E0600080C0004380 003E0000141F7B9418>121 D E /Fn 38 122 df<0001C0000003C000000FC000007FC0 001FFFC000FFFFC000FFBFC000E03FC000003FC000003FC000003FC000003FC000003FC0 00003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC0 00003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC0 00003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC0 00003FC000003FC000003FC0007FFFFFF07FFFFFF07FFFFFF01C2E7AAD28>49 D<003FE00001FFFE0007FFFF800F80FFC01E003FE038001FF07C000FF87E0007FCFF0007 FCFF8007FEFF8007FEFF8003FEFF8003FE7F0003FE3E0007FE000007FE000007FC000007 FC00000FF800000FF800000FF000001FE000001FC000003F8000007F0000007E000000F8 000001F0000003E0000007C000000F0000001E000E003C000E0038000E0070001E00E000 1C01C0001C0300003C07FFFFFC0FFFFFFC1FFFFFFC3FFFFFFC7FFFFFF8FFFFFFF8FFFFFF F8FFFFFFF81F2E7CAD28>I<001FF8000000FFFF000003FFFFC00007E01FF0000F0007F8 001F8007FC003FC007FC003FC003FE003FC003FE003FC003FE003FC003FE001F8003FE00 0F0007FE00000007FC00000007FC00000007F80000000FF00000001FE00000003F800000 00FF0000003FF80000003FFF800000001FE00000000FF000000007F800000003FC000000 03FE00000001FF00000001FF00000001FF80000001FF80000001FF801C0001FF803E0001 FF807F0001FF80FF8001FF80FF8001FF00FF8001FF00FF8003FE007F0003FE007E0007FC 003C0007F8001FC01FF0000FFFFFC00003FFFF0000003FF80000212E7DAD28>I<000000 7000000000F000000001F000000003F000000007F00000000FF00000000FF00000001FF0 0000003FF000000077F0000000F7F0000000E7F0000001C7F000000387F000000707F000 000F07F000000E07F000001C07F000003807F000007007F00000F007F00000E007F00001 C007F000038007F000070007F0000F0007F0000E0007F0001C0007F000380007F0007000 07F000E00007F000FFFFFFFFE0FFFFFFFFE0FFFFFFFFE000000FF00000000FF00000000F F00000000FF00000000FF00000000FF00000000FF00000000FF00000000FF000000FFFFF E0000FFFFFE0000FFFFFE0232E7EAD28>I<0C0000300FC007F00FFFFFE00FFFFFC00FFF FF800FFFFF000FFFFE000FFFF8000FFFF0000FFF80000E0000000E0000000E0000000E00 00000E0000000E0000000E0000000E0000000E1FF0000EFFFE000FE03F800F000FC00E00 07E00C0007F0000007F8000003F8000003FC000003FC000003FE000003FE180003FE3E00 03FE7F0003FEFF0003FEFF0003FEFF0003FCFF0003FCFE0003FC780007F8780007F03C00 0FE01E001FC00FC07F8007FFFF0001FFFC00003FE0001F2E7CAD28>I<0000FF00000007 FFE000001FFFF000007F80F80000FE003C0001F8007C0003F000FE0007F001FE000FE001 FE000FE001FE001FC001FE003FC000FC003FC00078003FC00000007F800000007F800000 007F80000000FF83FC0000FF8FFF8000FF9C0FC000FFB003F000FFB001F800FFE001FC00 FFC001FE00FFC000FE00FFC000FF00FFC000FF00FF8000FF80FF8000FF80FF8000FF80FF 8000FF807F8000FF807F8000FF807F8000FF807F8000FF803F8000FF003FC000FF001FC0 00FF001FC000FE000FC001FC0007E001FC0003F003F80001FC0FE00000FFFFC000003FFF 0000000FFC0000212E7DAD28>I<38000000003E000000003FFFFFFFC03FFFFFFFC03FFF FFFFC03FFFFFFF807FFFFFFF007FFFFFFE007FFFFFFC007FFFFFF8007800003800700000 7000700000E000F00001C000E000038000E000070000E00007000000000E000000001C00 000000380000000078000000007000000000F000000001E000000001E000000003E00000 0003C000000007C000000007C00000000FC00000000FC00000001FC00000001F80000000 1F800000003F800000003F800000003F800000003F800000007F800000007F800000007F 800000007F800000007F800000007F800000007F800000007F800000003F000000001E00 000022307CAF28>I<000003FF80018000003FFFF003800001FFFFFC07800007FF003F0F 80001FF800079F80003FC00001FF8000FF800000FF8001FE0000007F8003FC0000003F80 07FC0000001F8007F80000000F800FF00000000F801FF000000007801FF000000007803F E000000007803FE000000003807FE000000003807FE000000003807FC000000000007FC0 0000000000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC000 00000000FFC00000000000FFC00000000000FFC00000000000FFC000000000007FC00000 0000007FC000000000007FE000000000007FE000000003803FE000000003803FE0000000 03801FF000000003801FF000000007800FF0000000070007F8000000070007FC0000000E 0003FC0000001E0001FE0000001C0000FF8000007800003FC00000F000001FF80003E000 0007FF003F80000001FFFFFE000000003FFFF80000000003FF80000031317CB03A>67 D69 DI<000003FF00030000 007FFFF007000001FFFFFC0F000007FF007E1F00001FF0000FBF00007FC00003FF0000FF 800001FF0001FE0000007F0003FC0000007F0007FC0000003F000FF80000001F000FF000 00001F001FF00000000F001FF00000000F003FE000000007003FE000000007007FE00000 0007007FE000000007007FC00000000000FFC00000000000FFC00000000000FFC0000000 0000FFC00000000000FFC00000000000FFC00000000000FFC00000000000FFC000000000 00FFC00000000000FFC00000000000FFC00007FFFFF87FC00007FFFFF87FE00007FFFFF8 7FE0000001FF003FE0000001FF003FE0000001FF001FF0000001FF001FF0000001FF000F F0000001FF000FF8000001FF0007FC000001FF0003FC000001FF0001FE000001FF0000FF 800001FF00007FC00003FF00001FF800077F000007FF003E3F000001FFFFFC1F0000007F FFF00F00000003FF80030035317CB03F>I73 D76 DI< FFFF800001FFFFC0FFFFC00001FFFFC0FFFFE00001FFFFC000FFF0000003E00000FFF800 0001C00000EFFC000001C00000E7FC000001C00000E7FE000001C00000E3FF000001C000 00E1FF800001C00000E0FFC00001C00000E07FE00001C00000E03FE00001C00000E03FF0 0001C00000E01FF80001C00000E00FFC0001C00000E007FE0001C00000E003FE0001C000 00E001FF0001C00000E001FF8001C00000E000FFC001C00000E0007FE001C00000E0003F F001C00000E0001FF001C00000E0001FF801C00000E0000FFC01C00000E00007FE01C000 00E00003FF01C00000E00001FF81C00000E00000FF81C00000E00000FFC1C00000E00000 7FE1C00000E000003FF1C00000E000001FF9C00000E000000FFDC00000E0000007FDC000 00E0000007FFC00000E0000003FFC00000E0000001FFC00000E0000000FFC00000E00000 007FC00000E00000003FC00000E00000003FC00000E00000001FC00000E00000000FC000 01F000000007C000FFFFE0000003C000FFFFE0000001C000FFFFE0000001C0003A317EB0 3F>I82 D<001FF8018000FFFF038003FFFFC78007F007EF800F8000FF801F00007F 803E00001F803E00000F807C00000F807C00000780FC00000780FC00000780FC00000380 FE00000380FE00000380FF00000000FFC00000007FF00000007FFF8000003FFFF800003F FFFF80001FFFFFF0000FFFFFF80007FFFFFC0003FFFFFF0000FFFFFF00003FFFFF800001 FFFFC000001FFFE0000001FFE00000003FE00000001FF00000000FF000000007F0600000 07F0E0000003F0E0000003F0E0000003F0E0000003E0F0000003E0F0000003E0F8000007 C0FC000007C0FF00000F80FFC0001F00FBFC00FE00F1FFFFF800E03FFFF000C003FF8000 24317CB02D>I<00FFF0000003FFFF00000F803F80000FC00FE0001FE007F0001FE007F0 001FE003F8000FC003FC00078003FC00000003FC00000003FC00000003FC00000003FC00 0000FFFC00001FFFFC0000FFE3FC0003FC03FC000FF003FC001FC003FC003FC003FC007F 8003FC007F8003FC00FF0003FC00FF0003FC00FF0003FC00FF0007FC00FF0007FC007F80 0DFC003FC01DFE001FE078FFF007FFE07FF000FF803FF024207E9F27>97 D<000FFF00007FFFC001FC01F003F003F007E007F80FE007F81FC007F83FC003F03FC001 E07F8000007F8000007F800000FF800000FF800000FF800000FF800000FF800000FF8000 00FF800000FF8000007F8000007F8000007F8000003FC0001C3FC0001C1FC000380FE000 3807E0007003F001E001FC07C0007FFF00000FF8001E207D9F24>99 D<0000000FC0000007FFC0000007FFC0000007FFC00000007FC00000003FC00000003FC0 0000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC000 00003FC00000003FC00000003FC00000003FC00007F83FC0003FFF3FC000FE07BFC003F8 01FFC007E0007FC00FE0007FC01FC0003FC03FC0003FC03FC0003FC07F80003FC07F8000 3FC07F80003FC0FF80003FC0FF80003FC0FF80003FC0FF80003FC0FF80003FC0FF80003F C0FF80003FC0FF80003FC07F80003FC07F80003FC07F80003FC03FC0003FC03FC0003FC0 1FC0003FC00FE0007FC007E000FFC003F003FFE001FC0F3FFE007FFE3FFE000FF03FFE27 327DB12D>I<000FFC00007FFF8001FC0FC003F003E007E001F00FE001F81FC000FC3FC0 00FE3FC000FE7F80007E7F80007F7F80007FFF80007FFF80007FFFFFFFFFFFFFFFFFFF80 0000FF800000FF800000FF8000007F8000007F8000007F8000003FC000071FC000071FC0 000E0FE0000E07F0001C03F8007800FE03E0003FFFC00007FE0020207E9F25>I<0001FE 00000FFF80001FC3C0007F07E000FE0FF001FE0FF001FC0FF003FC0FF003FC07E003FC01 8003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC0000FFFFFC 00FFFFFC00FFFFFC0003FC000003FC000003FC000003FC000003FC000003FC000003FC00 0003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC00 0003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC00 0003FC00007FFFF0007FFFF0007FFFF0001C327EB119>I<001FF007C000FFFE3FE001F8 3F79F007E00FC3F00FE00FE1F00FC007E0E01FC007F0001FC007F0003FC007F8003FC007 F8003FC007F8003FC007F8003FC007F8001FC007F0001FC007F0000FC007E0000FE00FE0 0007E00FC00003F83F000006FFFE00000E1FF000000E000000001E000000001E00000000 1F000000001F800000001FFFFF80000FFFFFF0000FFFFFFC0007FFFFFE0003FFFFFF0003 FFFFFF800FFFFFFFC03F00007FC07E00001FE07C00000FE0FC000007E0FC000007E0FC00 0007E0FC000007E07E00000FC03E00000F803F00001F800FC0007E0007F803FC0001FFFF F000001FFF0000242F7E9F28>I<01F8000000FFF8000000FFF8000000FFF80000000FF8 00000007F800000007F800000007F800000007F800000007F800000007F800000007F800 000007F800000007F800000007F800000007F800000007F800000007F800000007F807F8 0007F83FFE0007F8783F0007F8C03F8007F9801FC007FB001FC007FE001FE007FC001FE0 07FC001FE007FC001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007 F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8 001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE0FFFFC3 FFFFFFFFC3FFFFFFFFC3FFFF28327DB12D>I<03C00007E0000FF0001FF8001FF8001FF8 001FF8000FF00007E00003C0000000000000000000000000000000000000000000000000 0000000001F8007FF8007FF8007FF80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F8 0007F80007F80007F80007F80007F80007F800FFFF80FFFF80FFFF8011337DB217>I<01 F800FFF800FFF800FFF8000FF80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007 F80007F80007F80007F80007F80007F80007F80007F80007F80007F80007F800FFFFC0FF FFC0FFFFC012327DB117>108 D<03F007F8001FE000FFF03FFE00FFF800FFF0783F01E0 FC00FFF0C03F8300FE000FF1801FC6007F0007F3001FCC007F0007F6001FF8007F8007FC 001FF0007F8007FC001FF0007F8007FC001FF0007F8007F8001FE0007F8007F8001FE000 7F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8 001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE000 7F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8 001FE0007F8007F8001FE0007F8007F8001FE0007F8007F8001FE0007F80FFFFC3FFFF0F FFFCFFFFC3FFFF0FFFFCFFFFC3FFFF0FFFFC3E207D9F43>I<03F007F800FFF03FFE00FF F0783F00FFF0C03F800FF1801FC007F3001FC007F6001FE007FC001FE007FC001FE007FC 001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F800 1FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001F E007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE0FFFFC3FFFFFFFFC3FFFF FFFFC3FFFF28207D9F2D>I<0007FC0000007FFFC00001FC07F00003F001F80007E000FC 000FC0007E001FC0007F003FC0007F803F80003F807F80003FC07F80003FC07F80003FC0 FF80003FE0FF80003FE0FF80003FE0FF80003FE0FF80003FE0FF80003FE0FF80003FE0FF 80003FE07F80003FC07F80003FC07F80003FC03FC0007F803FC0007F801FC0007F000FE0 00FE0007E000FC0003F803F80001FE0FF000007FFFC0000007FC000023207E9F28>I<01 F83FE000FFF8FFFC00FFFBE07F00FFFF003F8007FE001FC007FC000FE007F8000FF007F8 0007F807F80007F807F80007FC07F80003FC07F80003FC07F80003FE07F80003FE07F800 03FE07F80003FE07F80003FE07F80003FE07F80003FE07F80003FE07F80003FC07F80007 FC07F80007FC07F80007F807F80007F807F8000FF007FC000FE007FE001FC007FF003F80 07FBC0FE0007F8FFF80007F83FC00007F800000007F800000007F800000007F800000007 F800000007F800000007F800000007F800000007F800000007F800000007F8000000FFFF C00000FFFFC00000FFFFC00000272E7E9F2D>I<0007F801C0003FFE03C000FE0707C003 F8018FC007F000CFC00FE000FFC01FE0007FC03FC0007FC03FC0003FC07FC0003FC07FC0 003FC07F80003FC0FF80003FC0FF80003FC0FF80003FC0FF80003FC0FF80003FC0FF8000 3FC0FF80003FC0FF80003FC07F80003FC07F80003FC07FC0003FC03FC0003FC03FC0003F C01FE0007FC00FE0007FC007F000FFC003F803BFC001FC0F3FC0007FFE3FC0000FF03FC0 0000003FC00000003FC00000003FC00000003FC00000003FC00000003FC00000003FC000 00003FC00000003FC00000003FC00000003FC0000007FFFE000007FFFE000007FFFE272E 7D9F2B>I<03F03F00FFF07FC0FFF1C3E0FFF187E00FF30FF007F60FF007F60FF007FC07 E007FC03C007FC000007FC000007F8000007F8000007F8000007F8000007F8000007F800 0007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F8000007F800 0007F8000007F8000007F80000FFFFE000FFFFE000FFFFE0001C207E9F21>I<01FF8600 07FFFE001F00FE003C003E0078001E0078000E00F8000E00F8000E00F8000E00FC000000 FF800000FFFC00007FFFC0003FFFF0003FFFF8001FFFFC0007FFFE0001FFFF00003FFF00 0000FF8000003F8060001F80E0000F80E0000F80F0000F80F0000F00F8000F00FC001E00 FE001C00FF807800F3FFF000C07F800019207D9F20>I<001C0000001C0000001C000000 1C0000001C0000003C0000003C0000003C0000007C0000007C000000FC000001FC000003 FC000007FC00001FFFFE00FFFFFE00FFFFFE0003FC000003FC000003FC000003FC000003 FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003FC000003 FC000003FC000003FC000003FC038003FC038003FC038003FC038003FC038003FC038003 FC038001FC038001FC070000FE0700007F0E00003FFC000007F000192E7FAD1F>I<01F8 0007E0FFF803FFE0FFF803FFE0FFF803FFE00FF8003FE007F8001FE007F8001FE007F800 1FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001F E007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE007F8001FE0 07F8001FE007F8001FE007F8003FE007F8003FE003F8007FE003F8007FE001FC00DFF000 FE039FFF007FFF1FFF000FFC1FFF28207D9F2D>II<7FFF807FFC7FFF807FFC7FFF807FFC03FC000F0001FE001E0000FF00 3C0000FF803800007FC07800003FC0F000001FE1E000000FF3C000000FFF80000007FF00 000003FE00000001FE00000000FF00000000FF80000000FFC0000001FFC0000003DFE000 00078FF00000078FF800000F07FC00001E03FE00003C01FE00007800FF0000F000FF8001 E0007FC003E0003FE0FFFC01FFFFFFFC01FFFFFFFC01FFFF28207F9F2B>120 DI E /Fo 90 125 df 0 D<0002000000070000000700000007000000070000000F8000000F8000000F80000013 C0000013C0000013C0000021E0000021E0000021E0000040F0000040F0000040F0000080 F800008078000080780001007C0001003C0001003C0002003E0002001E0002001E000400 1F0004000F0004000F0008000F800800078008000780180007C03C000FC0FF807FF81D23 7EA222>3 D6 D<03FFFC00001F8000000F0000000F0000000F0000000F0000003FC00001EF 7800078F1E000E0F07001E0F07803C0F03C07C0F03E0780F01E0F80F01F0F80F01F0F80F 01F0F80F01F0F80F01F0F80F01F0780F01E07C0F03E03C0F03C01E0F07800E0F0700078F 1E0001EF7800003FC000000F0000000F0000000F0000000F0000001F800003FFFC001C22 7DA123>8 D<001F83E000706E3000C07C780180F8780380F07807007000070070000700 7000070070000700700007007000070070000700700007007000FFFFFFC0070070000700 700007007000070070000700700007007000070070000700700007007000070070000700 700007007000070070000700700007007000070070000700700007007000070078007FE3 FF801D2380A21C>11 D<001FC0000070200000C010000180380003807800070078000700 300007000000070000000700000007000000070000000700000007000000FFFFF8000700 780007003800070038000700380007003800070038000700380007003800070038000700 380007003800070038000700380007003800070038000700380007003800070038000700 38007FE1FF80192380A21B>I<001FD8000070380000C078000180780003807800070038 000700380007003800070038000700380007003800070038000700380007003800FFFFF8 000700380007003800070038000700380007003800070038000700380007003800070038 000700380007003800070038000700380007003800070038000700380007003800070038 00070038007FF3FF80192380A21B>I<000FC07F00007031C08000E00B004001801E00E0 03803E01E007003C01E007001C00C007001C000007001C000007001C000007001C000007 001C000007001C000007001C0000FFFFFFFFE007001C01E007001C00E007001C00E00700 1C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C 00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00E007001C00 E007001C00E07FF1FFCFFE272380A229>I<7038F87CFC7EFC7E743A0402040204020402 0804080410081008201040200F0F7EA218>34 D<003C000000006200000000C200000001 810000000181000000038100000003810000000381000000038100000003820000000382 0000000384000000038800000001C800000001D000000001E003FF8001C0007C0000E000 380001E000300001F0002000027000400004700040000838008000183C008000301C0100 00701E020000700E020000F007040000F007880000F003880000F001D00100F000E00100 78007003003800B802003C031C04000E0C0E0C0003F003F00021257EA326>38 D<70F8FCFC7404040404080810102040060F7CA20E>I<00200040008001000300060004 000C000C00180018003000300030007000600060006000E000E000E000E000E000E000E0 00E000E000E000E000E000E000E0006000600060007000300030003000180018000C000C 0004000600030001000080004000200B327CA413>I<800040002000100018000C000400 060006000300030001800180018001C000C000C000C000E000E000E000E000E000E000E0 00E000E000E000E000E000E000E000C000C000C001C00180018001800300030006000600 04000C00180010002000400080000B327DA413>I<000180000001800000018000000180 000001800000018000000180000001800000018000000180000001800000018000000180 00000180000001800000018000FFFFFFFEFFFFFFFE000180000001800000018000000180 000001800000018000000180000001800000018000000180000001800000018000000180 000001800000018000000180001F227D9C26>43 D<70F8FCFC7404040404080810102040 060F7C840E>II<70F8F8F87005057C840E>I<00008000018000 0180000300000300000300000600000600000600000C00000C00000C0000180000180000 180000300000300000300000600000600000600000C00000C00000C00001800001800001 80000180000300000300000300000600000600000600000C00000C00000C000018000018 0000180000300000300000300000600000600000600000C00000C00000C0000011317DA4 18>I<01F000071C000C06001803003803803803807001C07001C07001C07001C0F001E0 F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0F001E0 F001E07001C07001C07001C07803C03803803803801C07000C0600071C0001F00013227E A018>I<008003800F80F380038003800380038003800380038003800380038003800380 03800380038003800380038003800380038003800380038003800380038007C0FFFE0F21 7CA018>I<03F0000C1C001007002007804003C04003C08003E0F003E0F801E0F801E0F8 01E02003E00003E00003C00003C0000780000700000E00001C0000180000300000600000 C0000180000100000200200400200800201800603000403FFFC07FFFC0FFFFC013217EA0 18>I<03F8000C1E001007002007804007C07807C07803C07807C03807C0000780000780 000700000F00000E0000380003F000001C00000F000007800007800003C00003C00003E0 2003E07003E0F803E0F803E0F003C04003C0400780200780100F000C1C0003F00013227E A018>I<000200000600000E00000E00001E00001E00002E00004E00004E00008E00008E 00010E00020E00020E00040E00040E00080E00100E00100E00200E00200E00400E00800E 00FFFFF8000E00000E00000E00000E00000E00000E00000E00001F0001FFF015217FA018 >I<1000801E07001FFF001FFE001FF80013E00010000010000010000010000010000010 000010F800130E001407001803801003800001C00001C00001E00001E00001E00001E070 01E0F001E0F001E0E001C08001C04003C04003802007001006000C1C0003F00013227EA0 18>I<007E0001C1000300800601C00E03C01C03C0180180380000380000780000700000 700000F0F800F30C00F40600F40300F80380F801C0F001C0F001E0F001E0F001E0F001E0 F001E07001E07001E07001E03801C03801C01803801C03000C0600070C0001F00013227E A018>I<4000006000007FFFE07FFFC07FFFC0400080C001008001008002008002000004 0000080000080000100000300000200000600000600000600000E00000C00000C00001C0 0001C00001C00001C00003C00003C00003C00003C00003C00003C00003C00003C0000180 0013237DA118>I<01F800060E000803001001802001802000C06000C06000C06000C070 00C07801803E01003F02001FC4000FF80003F80003FC00067F00083F80100F803007C060 01C06000E0C000E0C00060C00060C00060C000606000406000C03000801803000E0E0003 F00013227EA018>I<01F000060C000C0600180700380380700380700380F001C0F001C0 F001C0F001E0F001E0F001E0F001E0F001E07001E07003E03803E01805E00C05E00619E0 03E1E00001C00001C00001C0000380000380300300780700780600700C00201800103000 0FC00013227EA018>I<70F8F8F870000000000000000000000070F8F8F87005157C940E> I<70F8F8F870000000000000000000000070F8F8F87808080808101010204040051F7C94 0E>I61 D<07E01838201C400E800FF00FF0 0FF00F000F000E001C00380030006000C000C00080008001800100010001000100010001 0000000000000000000000038007C007C007C0038010237DA217>63 D<0001800000018000000180000003C0000003C0000003C0000005E0000005E000000DF0 000008F0000008F0000010F800001078000010780000203C0000203C0000203C0000401E 0000401E0000401E0000800F0000800F0000FFFF000100078001000780030007C0020003 C0020003C0040003E0040001E0040001E00C0000F00C0000F03E0001F8FF800FFF20237E A225>65 DI<0007E0100038183000E0063001C00170038000F0070000F00E0000701E000070 1C0000303C0000303C0000307C0000107800001078000010F8000000F8000000F8000000 F8000000F8000000F8000000F8000000F800000078000000780000107C0000103C000010 3C0000101C0000201E0000200E000040070000400380008001C0010000E0020000381C00 0007E0001C247DA223>IIII<0007F008003C0C1800E0021801C001B8038000F8070000780F000038 1E0000381E0000183C0000183C0000187C0000087800000878000008F8000000F8000000 F8000000F8000000F8000000F8000000F8000000F8001FFF780000F8780000787C000078 3C0000783C0000781E0000781E0000780F00007807000078038000B801C000B800E00318 003C0C080007F00020247DA226>III<03FFF0001F00000F00000F00000F00000F0000 0F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F00000F0000 0F00000F00000F00000F00000F00000F00000F00000F00700F00F80F00F80F00F80E00F0 1E00401C0020380018700007C00014237EA119>IIIII<000FE00000783C0000E00E0003C00780078003C00F0001 E00E0000E01E0000F03C0000783C0000787C00007C7C00007C7800003C7800003CF80000 3EF800003EF800003EF800003EF800003EF800003EF800003EF800003EF800003E780000 3C7C00007C7C00007C3C0000783E0000F81E0000F00F0001E00F0001E0078003C003C007 8000E00E0000783C00000FE0001F247DA226>II<000FE00000783C0000E00E0003C0078007 8003C00F0001E00E0000E01E0000F03E0000F83C0000787C00007C7C00007C7800003C78 00003CF800003EF800003EF800003EF800003EF800003EF800003EF800003EF800003EF8 00003E7800003C7C00007C7C00007C3C0000783C0000781E0380F00E0420E00F0831E007 8833C003C83F8000E83E00007C3C02000FFC0200001C0200001C0200001E0600000F0E00 000FFC000007FC000007F8000003F8000001E01F2D7DA226>II<03F0200C0C601802 603001E07000E0600060E00060E00060E00020E00020E00020F00000F000007800007F00 003FF0001FFE000FFF0003FF80003FC00007E00001E00000F00000F00000708000708000 70800070800070C00060C00060E000C0F000C0C80180C6070081FC0014247DA21B>I<7F FFFFF87807807860078018400780084007800840078008C007800C800780048007800480 078004800780040007800000078000000780000007800000078000000780000007800000 078000000780000007800000078000000780000007800000078000000780000007800000 07800000078000000780000007800000078000000FC00003FFFF001E227EA123>I III<7FF807FF0007E001F80003C000E00003E000C00001E000800000F001000000F8 0300000078020000007C040000003E0C0000001E080000001F100000000FB000000007A0 00000007C000000003E000000001E000000001F000000003F80000000278000000047C00 00000C3E000000081E000000101F000000300F80000020078000004007C00000C003E000 008001E000010001F000030000F000070000F8001F8001FC00FFE007FFC022227FA125> II<7FFF FE7E003E78003C7000786000784000F0C000F0C001E08003C08003C0800780000780000F 00001F00001E00003C00003C0000780000780000F00001F00001E00103C00103C0010780 010780030F00031E00021E00023C00063C000E78001EF8007EFFFFFE18227DA11E>II<0804100820102010402040208040804080 408040B85CFC7EFC7E7C3E381C0F0F7AA218>II<70F8F8F87005057CA10E>95 D<0FE0001838003C0C003C0E0018070000070000 070000070000FF0007C7001E07003C0700780700700700F00708F00708F00708F00F0878 17083C23900FC1E015157E9418>97 D<0E0000FE00001E00000E00000E00000E00000E00 000E00000E00000E00000E00000E00000E00000E00000E1F000E61C00E80600F00300E00 380E003C0E001C0E001E0E001E0E001E0E001E0E001E0E001E0E001E0E001C0E003C0E00 380F00700C80600C41C0083F0017237FA21B>I<01FE000703000C07801C078038030078 0000700000F00000F00000F00000F00000F00000F00000F000007000007800403800401C 00800C010007060001F80012157E9416>I<0000E0000FE00001E00000E00000E00000E0 0000E00000E00000E00000E00000E00000E00000E00000E001F8E00704E00C02E01C01E0 3800E07800E07000E0F000E0F000E0F000E0F000E0F000E0F000E0F000E07000E07800E0 3800E01801E00C02E0070CF001F0FE17237EA21B>I<01FC000707000C03801C01C03801 C07801E07000E0F000E0FFFFE0F00000F00000F00000F00000F000007000007800203800 201C00400E008007030000FC0013157F9416>I<003C00C6018F038F030F070007000700 070007000700070007000700FFF807000700070007000700070007000700070007000700 070007000700070007000700070007807FF8102380A20F>I<00007001F198071E180E0E 181C07001C07003C07803C07803C07803C07801C07001C07000E0E000F1C0019F0001000 001000001800001800001FFE000FFFC00FFFE03800F0600030400018C00018C00018C000 186000306000303800E00E038003FE0015217F9518>I<0E0000FE00001E00000E00000E 00000E00000E00000E00000E00000E00000E00000E00000E00000E00000E1F800E60C00E 80E00F00700F00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E 00700E00700E00700E00700E00700E0070FFE7FF18237FA21B>I<1C003E003E003E001C 00000000000000000000000000000000000E00FE001E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E00FFC00A227FA10E>I<01C003E003E0 03E001C00000000000000000000000000000000001E00FE001E000E000E000E000E000E0 00E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E0 60E0F0C0F18061803E000B2C82A10F>I<0E0000FE00001E00000E00000E00000E00000E 00000E00000E00000E00000E00000E00000E00000E00000E03FC0E01F00E01C00E01800E 02000E04000E08000E10000E38000EF8000F1C000E1E000E0E000E07000E07800E03C00E 01C00E01E00E00F00E00F8FFE3FE17237FA21A>I<0E00FE001E000E000E000E000E000E 000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E 000E000E000E000E000E000E000E000E00FFE00B237FA20E>I<0E1FC07F00FE60E18380 1E807201C00F003C00E00F003C00E00E003800E00E003800E00E003800E00E003800E00E 003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E003800E00E00 3800E00E003800E00E003800E00E003800E0FFE3FF8FFE27157F942A>I<0E1F80FE60C0 1E80E00F00700F00700E00700E00700E00700E00700E00700E00700E00700E00700E0070 0E00700E00700E00700E00700E00700E0070FFE7FF18157F941B>I<01FC000707000C01 801800C03800E0700070700070F00078F00078F00078F00078F00078F00078F000787000 707800F03800E01C01C00E038007070001FC0015157F9418>I<0E1F00FE61C00E80600F 00700E00380E003C0E001C0E001E0E001E0E001E0E001E0E001E0E001E0E001E0E003C0E 003C0E00380F00700E80E00E41C00E3F000E00000E00000E00000E00000E00000E00000E 00000E00000E0000FFE000171F7F941B>I<01F8200704600E02601C01603801E07800E0 7800E0F000E0F000E0F000E0F000E0F000E0F000E0F000E07000E07800E03801E01C01E0 0C02E0070CE001F0E00000E00000E00000E00000E00000E00000E00000E00000E00000E0 000FFE171F7E941A>I<0E3CFE461E8F0F0F0F060F000E000E000E000E000E000E000E00 0E000E000E000E000E000E000F00FFF010157F9413>I<0F8830786018C018C008C008E0 08F0007F803FE00FF001F8003C801C800C800CC00CC008E018D0308FC00E157E9413>I< 02000200020002000600060006000E001E003E00FFF80E000E000E000E000E000E000E00 0E000E000E000E000E040E040E040E040E040E040708030801F00E1F7F9E13>I<0E0070 FE07F01E00F00E00700E00700E00700E00700E00700E00700E00700E00700E00700E0070 0E00700E00700E00700E00F00E00F006017003827800FC7F18157F941B>IIIII<3FFFC0380380300780200700600E00401C00403C0040380000700000E00001E00001 C0000380400700400F00400E00C01C0080380080780180700780FFFF8012157F9416>I< FFFFFE1701808C18>II E /Fp 41 122 df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q 2 122 df<040004000400C460E4E03F800E003F80 E4E0C4600400040004000B0D7E8D11>3 D<0C000C000C000C000C000C00FFC0FFC00C00 0C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000C000A1A 7E9310>121 D E /Fr 41 124 df<001F83E000F06E3001C078780380F8780300F03007 007000070070000700700007007000070070000700700007007000FFFFFF800700700007 007000070070000700700007007000070070000700700007007000070070000700700007 007000070070000700700007007000070070000700700007007000070070007FE3FF001D 20809F1B>11 D<003F0000E0C001C0C00381E00701E00701E00700000700000700000700 00070000070000FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700 E00700E00700E00700E00700E00700E00700E00700E00700E00700E07FC3FE1720809F19 >I<003FE000E0E001C1E00381E00700E00700E00700E00700E00700E00700E00700E007 00E0FFFFE00700E00700E00700E00700E00700E00700E00700E00700E00700E00700E007 00E00700E00700E00700E00700E00700E00700E00700E07FE7FE1720809F19>I<002000 4000800100020006000C000C00180018003000300030007000600060006000E000E000E0 00E000E000E000E000E000E000E000E000E0006000600060007000300030003000180018 000C000C000600020001000080004000200B2E7DA112>40 D<800040002000100008000C 00060006000300030001800180018001C000C000C000C000E000E000E000E000E000E000 E000E000E000E000E000E000C000C000C001C001800180018003000300060006000C0008 0010002000400080000B2E7DA112>I<70F8FCFC74040404080810102040060E7C840D> 44 D<70F8F8F87005057C840D>46 D<018003800F80F380038003800380038003800380 038003800380038003800380038003800380038003800380038003800380038003800380 07C0FFFE0F1E7C9D17>49 D<000FC040007030C001C009C0038005C0070003C00E0001C0 1E0000C01C0000C03C0000C07C0000407C00004078000040F8000000F8000000F8000000 F8000000F8000000F8000000F8000000F8000000F8000000780000007C0000407C000040 3C0000401C0000401E0000800E000080070001000380020001C0040000703800000FC000 1A217D9F21>67 D<000FE0200078186000E004E0038002E0070001E00F0000E01E000060 1E0000603C0000603C0000207C00002078000020F8000000F8000000F8000000F8000000 F8000000F8000000F8000000F8007FFCF80003E0780001E07C0001E03C0001E03C0001E0 1E0001E01E0001E00F0001E0070001E0038002E000E0046000781820000FE0001E217D9F 24>71 D76 D<001F800000F0F00001C0380007801E000F000F000E0007001E0007803C0003C03C0003 C07C0003E0780001E0780001E0F80001F0F80001F0F80001F0F80001F0F80001F0F80001 F0F80001F0F80001F0F80001F0780001E07C0003E07C0003E03C0003C03C0003C01E0007 800E0007000F000F0007801E0001C0380000F0F000001F80001C217D9F23>79 D<07E0800C1980100780300380600180600180E00180E00080E00080E00080F00000F000 007800007F00003FF0001FFC000FFE0003FF00001F800007800003C00003C00001C08001 C08001C08001C08001C0C00180C00380E00300F00600CE0C0081F80012217D9F19>83 D<7FFFFFE0780F01E0600F0060400F0020400F0020C00F0030800F0010800F0010800F00 10800F0010000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F0000000F00 00000F0000000F0000001F800007FFFE001C1F7E9E21>I87 D<1FE000303000781800781C00300E00000E00000E00000E0000FE0007 8E001E0E00380E00780E00F00E10F00E10F00E10F01E10781E103867200F83C014147E93 17>97 D<0E0000FE00000E00000E00000E00000E00000E00000E00000E00000E00000E00 000E00000E3E000EC3800F01C00F00E00E00E00E00700E00700E00780E00780E00780E00 780E00780E00780E00700E00700E00E00F00E00D01C00CC300083E0015207F9F19>I<03 F80E0C1C1E381E380C70007000F000F000F000F000F000F00070007000380138011C020E 0C03F010147E9314>I<000380003F800003800003800003800003800003800003800003 8000038000038000038003E380061B801C0780380380380380700380700380F00380F003 80F00380F00380F00380F003807003807003803803803807801C07800E1B8003E3F81520 7E9F19>I<03F0000E1C001C0E00380700380700700700700380F00380F00380FFFF80F0 0000F00000F000007000007000003800801800800C010007060001F80011147F9314>I< 007C00C6018F038F07060700070007000700070007000700FFF007000700070007000700 07000700070007000700070007000700070007000700070007007FF01020809F0E>I<00 00E003E3300E3C301C1C30380E00780F00780F00780F00780F00780F00380E001C1C001E 380033E0002000002000003000003000003FFE001FFF800FFFC03001E0600070C00030C0 0030C00030C000306000603000C01C038003FC00141F7F9417>I<0E0000FE00000E0000 0E00000E00000E00000E00000E00000E00000E00000E00000E00000E3E000E43000E8180 0F01C00F01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0 0E01C00E01C00E01C00E01C0FFE7FC16207F9F19>I<1C003E003E003E001C0000000000 00000000000000000E007E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E000E000E00FFC00A1F809E0C>I<0E0000FE00000E00000E00000E00000E00 000E00000E00000E00000E00000E00000E00000E0FF00E03C00E03000E02000E04000E08 000E10000E30000E70000EF8000F38000E1C000E1E000E0E000E07000E07800E03800E03 C00E03E0FFCFF815207F9F18>107 D<0E00FE000E000E000E000E000E000E000E000E00 0E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E000E00 0E000E000E00FFE00B20809F0C>I<0E1F01F000FE618618000E81C81C000F00F00E000F 00F00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00 E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E00E000E00E0 0E00FFE7FE7FE023147F9326>I<0E3E00FE43000E81800F01C00F01C00E01C00E01C00E 01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C0FF E7FC16147F9319>I<01F800070E001C03803801C03801C07000E07000E0F000F0F000F0 F000F0F000F0F000F0F000F07000E07000E03801C03801C01C0380070E0001F80014147F 9317>I<0E3E00FEC3800F01C00F00E00E00E00E00F00E00700E00780E00780E00780E00 780E00780E00780E00700E00F00E00E00F01E00F01C00EC3000E3E000E00000E00000E00 000E00000E00000E00000E00000E0000FFE000151D7F9319>I<03E0800619801C05803C 0780380380780380700380F00380F00380F00380F00380F00380F0038070038078038038 03803807801C0B800E138003E38000038000038000038000038000038000038000038000 0380003FF8151D7E9318>I<0E78FE8C0F1E0F1E0F0C0E000E000E000E000E000E000E00 0E000E000E000E000E000E000E00FFE00F147F9312>I<1F9030704030C010C010C010E0 0078007F803FE00FF00070803880188018C018C018E030D0608F800D147E9312>I<0200 02000200060006000E000E003E00FFF80E000E000E000E000E000E000E000E000E000E00 0E000E080E080E080E080E080610031001E00D1C7F9B12>I<0E01C0FE1FC00E01C00E01 C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01C00E01 C00E03C00603C0030DC001F1FC16147F9319>III<7FC3FC0F01E0 0701C007018003810001C20000E40000EC00007800003800003C00007C00004E00008700 0107000303800201C00601E01E01E0FF07FE1714809318>II<3FFF380E200E201C40384078407000E001E001C00380078007 010E011E011C0338027006700EFFFE10147F9314>II E /Fs 7 117 df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t 2 122 df<01800180018001804182F18F399C0F F003C003C00FF0399CF18F4182018001800180018010127E9215>3 D<02000700070007000700070007000200020002007270FFF87270020002000700070007 000700070007000700070007000700070007000700070007000700070002000200020002 00020002000D267E9C12>121 D E /Fu 39 122 df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v 32 124 df<00001FF3800000F01F800003C00F800007 800F80000F001F80001E001F80003C001F80007C000F80007C000F8000F8000F8000F800 0F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F 8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F80FFFFFFFF80FFFFFFFF80 00F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000 F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8 000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F800 0F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F8000F 8000F8000F8000F8000F8000F8000F8001FC001FC03FFFE3FFFE3FFFE3FFFE273C7FBB2B >13 D<387CFEFEFE7C380707798615>46 D<387CFEFEFE7C380000000000000000000000 000000000000000000000000387CFEFEFE7C38072579A415>58 D<000001FF0000800000 1FFFE0018000007F007801800001F8000E03800003E000070780000FC000018780001F00 0000CF80003E0000006F80007C0000003F8000F80000003F8001F00000001F8003F00000 000F8007E00000000F8007C000000007800FC000000007800FC000000007801F80000000 03801F8000000003803F8000000003803F0000000001803F0000000001807F0000000001 807F0000000001807E0000000000007E000000000000FE000000000000FE000000000000 FE000000000000FE000000000000FE000000000000FE000000000000FE000000000000FE 000000000000FE000000000000FE000000000000FE0000000000007E0000000000007E00 00000000007F0000000000007F0000000001803F0000000001803F0000000001803F8000 000001801F8000000001801F8000000003000FC000000003000FC0000000030007E00000 00060007E0000000060003F0000000060001F00000000C0000F80000001800007C000000 1800003E0000003000001F0000006000000FC00001C0000003E0000380000001F8000E00 0000007F007C000000001FFFF00000000001FF000000313D7CBB39>67 D69 D<000001FF00008000001FFFE0018000007F007801800001F8000E03800003E00007 0780000FC000018780001F000000CF80003E0000006F80007C0000003F8000F80000003F 8001F00000001F8003F00000000F8007E00000000F8007C000000007800FC00000000780 0FC000000007801F8000000003801F8000000003803F8000000003803F0000000001803F 0000000001807F0000000001807F0000000001807E0000000000007E000000000000FE00 0000000000FE000000000000FE000000000000FE000000000000FE000000000000FE0000 00000000FE000000000000FE000000000000FE000000000000FE000000000000FE000000 0000007E0000007FFFFE7E0000007FFFFE7F000000007FC07F000000001F803F00000000 1F803F000000001F803F800000001F801F800000001F801F800000001F800FC00000001F 800FC00000001F8007E00000001F8007E00000001F8003F00000001F8001F00000001F80 00F80000001F80007C0000003F80003E0000003F80001F0000007F80000FC00000E78000 03E00001C7800001FC0007838000007F803E018000001FFFF80080000001FF800000373D 7CBB3F>71 D73 D76 D82 D<000FF00080007FFE018001F00F818003C001C380070000E3800E00 0037801C00003F803C00001F803800000F807800000F8070000007807000000780F00000 0380F000000380F000000380F000000380F000000180F800000180F800000180FC000001 807C000000007E000000003F000000003FC00000001FF00000000FFF0000000FFFF00000 03FFFE000001FFFFC00000FFFFF000003FFFFC000003FFFE0000003FFF00000003FF0000 00007F800000001FC00000000FC000000007E000000003E000000003E000000001F0C000 0001F0C0000001F0C0000000F0C0000000F0C0000000F0E0000000F0E0000000F0E00000 00E0F0000000E0F0000001E0F8000001C0F8000001C0FC00000380FE00000780F7000007 00E1C0001E00E0F0003C00C07E00F000C00FFFE0008001FF0000243D7CBB2C>I<003F80 000001C0F00000030038000004001C00000C001E000018000F00001C000F80003E000780 003F0007C0003F0007C0003F0007C0001E0007C000000007C000000007C000000007C000 00003FC000000FE7C000007E07C00001F007C00007E007C0000F8007C0001F0007C0003F 0007C0003E0007C0007E0007C0007C0007C060FC0007C060FC0007C060FC0007C060FC00 0FC060FC000FC0607C000FC0607E0017C0603E0023E0C01F0041F18007C180FF0000FE00 3E0023257CA427>97 D<03E0000000FFE0000000FFE000000007E000000003E000000003 E000000003E000000003E000000003E000000003E000000003E000000003E000000003E0 00000003E000000003E000000003E000000003E000000003E000000003E000000003E000 000003E000000003E000000003E000000003E03F800003E1C0F00003E3003C0003E4001E 0003E8000F0003F000078003F00003C003E00003E003E00001E003E00001F003E00001F0 03E00001F803E00000F803E00000F803E00000FC03E00000FC03E00000FC03E00000FC03 E00000FC03E00000FC03E00000FC03E00000FC03E00000FC03E00000F803E00000F803E0 0001F803E00001F003E00001F003E00003E003E00003C003F00007C003D000078003D800 0F0003CC001E000386003800038181E00003007F0000263C7DBB2C>I<0007F800003C0E 0000F0018001E000C003C00060078000300F0000701F0000F81F0001F83E0001F83E0001 F87E0000F07C0000007C000000FC000000FC000000FC000000FC000000FC000000FC0000 00FC000000FC000000FC0000007C0000007C0000007E0000003E0000003E00000C1F0000 0C1F0000180F8000180780003003C0006001E000C000F00180003C0E000007F8001E257D A423>I<0000001F00000007FF00000007FF000000003F000000001F000000001F000000 001F000000001F000000001F000000001F000000001F000000001F000000001F00000000 1F000000001F000000001F000000001F000000001F000000001F000000001F000000001F 000000001F000000001F000003F81F00001E061F000070019F0001E000DF0003C0007F00 0780003F000F80003F000F00001F001F00001F003E00001F003E00001F007E00001F007C 00001F007C00001F00FC00001F00FC00001F00FC00001F00FC00001F00FC00001F00FC00 001F00FC00001F00FC00001F00FC00001F007C00001F007C00001F007E00001F003E0000 1F003E00001F001E00001F001F00001F000F00003F000780007F0003C0005F0001E0009F 0000F0031F80003C0E1FFC0007F01FFC263C7DBB2C>I<000FF00000383C0000E00F0001 C00780038003C0078001E00F0001F01F0000F01E0000F83E0000F83E0000F87C00007C7C 00007C7C00007CFC00007CFC00007CFFFFFFFCFC000000FC000000FC000000FC000000FC 000000FC0000007C0000007C0000007E0000003E0000003E00000C1E00000C1F0000180F 0000180780003003C0006001E000C000F00180003C0E000007F8001E257DA423>I<0000 0007C0000FE01860003838207000F01E40F001E00F80F003C007806007C007C0000F8003 E0000F8003E0000F8003E0001F8003F0001F8003F0001F8003F0001F8003F0001F8003F0 001F8003F0000F8003E0000F8003E0000F8003E00007C007C00003C007800001E00F0000 03F01E00000238380000060FE00000040000000004000000000C000000000C000000000E 000000000E000000000700000000078000000003FFFF000003FFFFF00001FFFFFC0000FF FFFE00078000FF000E00001F801C000007C038000003C078000003C070000001E0F00000 01E0F0000001E0F0000001E0F0000001E0F0000001E078000003C038000003803C000007 801E00000F000700001C0003C0007800007803C000000FFE000024387EA527>103 D<03E0000000FFE0000000FFE000000007E000000003E000000003E000000003E0000000 03E000000003E000000003E000000003E000000003E000000003E000000003E000000003 E000000003E000000003E000000003E000000003E000000003E000000003E000000003E0 00000003E000000003E03FC00003E0C0F00003E100780003E2003C0003E4003E0003E800 1E0003E8001E0003F0001F0003F0001F0003F0001F0003E0001F0003E0001F0003E0001F 0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F00 03E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003 E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0007F0 003F80FFFF87FFFCFFFF87FFFC263C7DBB2C>I<038007C00FE00FE00FE007C003800000 00000000000000000000000000000000000000000000000003E0FFE0FFE007E003E003E0 03E003E003E003E003E003E003E003E003E003E003E003E003E003E003E003E003E003E0 03E003E003E003E003E003E003E003E003E003E007F0FFFFFFFF10397EB815>I<03E000 FFE000FFE00007E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E00003E000 03E00003E00003E00003E00003E00003E00003E00003E00007F000FFFF80FFFF80113C7E BB15>108 D<03E01FC0007F0000FFE060780181E000FFE1803C0600F00007E2001E0800 780003E4001F10007C0003E4001F10007C0003E8000F20003C0003F0000FC0003E0003F0 000FC0003E0003F0000FC0003E0003E0000F80003E0003E0000F80003E0003E0000F8000 3E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0 000F80003E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0000F8000 3E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0 000F80003E0003E0000F80003E0003E0000F80003E0003E0000F80003E0003E0000F8000 3E0003E0000F80003E0003E0000F80003E0003E0000F80003E0007F0001FC0007F00FFFF 83FFFE0FFFF8FFFF83FFFE0FFFF83D257DA443>I<03E03FC000FFE0C0F000FFE1007800 07E2003C0003E4003E0003E8001E0003E8001E0003F0001F0003F0001F0003F0001F0003 E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0 001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E000 1F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F 0003E0001F0003E0001F0007F0003F80FFFF87FFFCFFFF87FFFC26257DA42C>I<0007F0 0000003C1E000000F007800001C001C000038000E000078000F0000F000078001E00003C 001E00003C003E00003E003E00003E007C00001F007C00001F007C00001F00FC00001F80 FC00001F80FC00001F80FC00001F80FC00001F80FC00001F80FC00001F80FC00001F80FC 00001F807C00001F007C00001F007C00001F003E00003E003E00003E001E00003C001F00 007C000F00007800078000F00003C001E00001C001C00000F0078000003C1E00000007F0 000021257DA427>I<03E03F8000FFE1C0F000FFE3003C0007E4001E0003E8000F0003F0 00078003F00007C003E00003E003E00003E003E00001F003E00001F003E00001F803E000 01F803E00000F803E00000FC03E00000FC03E00000FC03E00000FC03E00000FC03E00000 FC03E00000FC03E00000FC03E00000FC03E00000F803E00001F803E00001F803E00001F0 03E00003F003E00003E003E00007C003F00007C003F0000F8003F8001F0003EC001E0003 E600780003E181E00003E07F000003E000000003E000000003E000000003E000000003E0 00000003E000000003E000000003E000000003E000000003E000000003E000000003E000 000003E000000003E000000007F0000000FFFF800000FFFF80000026367DA42C>I<0003 F80300001E060700007803070001E000870003E000CF0007C0006F000F80002F000F8000 3F001F00003F003F00001F003E00001F007E00001F007E00001F007C00001F00FC00001F 00FC00001F00FC00001F00FC00001F00FC00001F00FC00001F00FC00001F00FC00001F00 FC00001F007C00001F007E00001F007E00001F003E00001F003E00001F001F00001F001F 00003F000F80003F000780007F0003C000DF0001E0019F0000F0031F00003C0C1F000007 F01F000000001F000000001F000000001F000000001F000000001F000000001F00000000 1F000000001F000000001F000000001F000000001F000000001F000000001F000000001F 000000003F80000007FFFC000007FFFC26367DA42A>I<03E0FC00FFE10600FFE20F0007 E41F8003E81F8003E81F8003F00F0003F0060003F0000003F0000003E0000003E0000003 E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003 E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003E0000003 E0000003E0000003E0000003E0000007F00000FFFFC000FFFFC00019257DA41E>I<00FF 02000700C6000C002E0010001E0030001E0060000E0060000E00E0000600E0000600E000 0600F0000600F8000600FC0000007F0000003FF000003FFF80001FFFE00007FFF00001FF FC00003FFE000001FE0000003F00C0001F00C0000F80C0000780E0000380E0000380E000 0380E0000380F0000300F0000300F8000700F8000600E4000C00E2001800C1807000807F 800019257DA41F>I<001800000018000000180000001800000018000000380000003800 000038000000380000007800000078000000F8000000F8000001F8000003F8000007F800 001FFFFE00FFFFFE0000F8000000F8000000F8000000F8000000F8000000F8000000F800 0000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F8000000F800 0000F8000000F8000000F8000000F8018000F8018000F8018000F8018000F8018000F801 8000F8018000F8018000F801800078018000780300007C0300003C0200001E0600000F0C 000003F00019357FB41E>I<03E0001F00FFE007FF00FFE007FF0007E0003F0003E0001F 0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F00 03E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003 E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0001F0003E0 001F0003E0001F0003E0003F0003E0003F0001E0003F0001E0005F0001F0009F0000F000 9F000078011F80001C061FFC0007F81FFC26257DA42C>I 120 DI<3FFFFFF83F8001F0 3E0003F03C0003E0380007C030000FC030000F8070001F0060003F0060003E0060007C00 6000FC006000F8000001F0000003F0000003E0000007C0000007C000000F8000001F0000 001F0000003E0000007E000C007C000C00F8000C01F8000C01F0000C03E0001C07E0001C 07C000180F8000181F8000381F0000383E0000787E0000F87C0007F8FFFFFFF81E257EA4 23>II E end %%EndProlog %%BeginSetup %%Feature: *Resolution 300dpi TeXDict begin %%EndSetup %%Page: 18 1 18 0 bop 30 28 a Fn(List)27 b(of)g(Figures)103 150 y Fo(1)88 b(A)14 b(p)q(erio)q(dic)g(ev)o(en)f(solution)i(that)g(is)f (unstable)g(to)h(o)q(dd)g(p)q(erturbations,)g(leading)f(to)215 222 y(nonsymmetric)g(c)o(haos.)23 b(P)o(arameter)15 b(v)m(alues:)23 b Fl(R)15 b Fo(=)g(16)p Fl(;)8 b(\026)15 b Fo(=)g(1)p Fl(;)8 b(\027)18 b Fo(=)d Fj(\000)p Fo(7.)23 b(A)16 b(small)215 295 y(o)q(dd)i(p)q(erturbation)e(is)g(in)o(tro)q(duced)g(at)h Fl(t)c Fo(=)h(2.)48 b(.)25 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.) h(.)f(.)h(.)f(.)g(.)h(.)54 b(19)103 367 y(2)88 b(A)16 b(c)o(haotic)g(ev)o(en)f(solution)h(that)h(is)f(unstable)g(to)h(o)q(dd) g(p)q(erturbations,)f(leading)g(to)215 439 y(nonsymmetric)e(c)o(haos.) 23 b(P)o(arameter)15 b(v)m(alues:)23 b Fl(R)15 b Fo(=)g(16)p Fl(;)8 b(\026)15 b Fo(=)g(5)p Fl(;)8 b(\027)18 b Fo(=)d Fj(\000)p Fo(7.)23 b(A)16 b(small)215 511 y(o)q(dd)i(p)q(erturbation)e (is)g(in)o(tro)q(duced)g(at)h Fl(t)c Fo(=)h(2.)48 b(.)25 b(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(20)103 583 y(3)88 b(A)14 b(c)o(haotic)g(o)q(dd)i(solution)f(that)f (is)h(unstable)f(to)h(ev)o(en)f(p)q(erturbations.)21 b(P)o(arameter)215 656 y(v)m(alues:)h Fl(R)14 b Fo(=)g(36,)j Fl(\027)g Fo(=)d Fj(\000)p Fo(10,)i Fl(\026)f Fo(=)f(12.)22 b(A)16 b(small)e(ev)o(en)i(p)q(erturbation)g(is)h(added)f(at)215 728 y Fl(t)e Fo(=)g(0)p Fl(:)p Fo(5.)g(.)24 b(.)g(.)h(.)f(.)h(.)f(.)h (.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.) h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(21)103 800 y(4)88 b(A)19 b(c)o(haotic)f(solution)h(that)g(is)f(o)q(dd)i(ab)q (out)g(0)f(and)h(ev)o(en)d(ab)q(out)j Fl(\031)r(=)p Fo(2.)29 b(P)o(arameter)215 872 y(v)m(alues:)20 b Fl(R)14 b Fo(=)g(16,)g Fl(\027)j Fo(=)d Fj(\000)p Fo(5,)f Fl(\026)i Fo(=)e(12.)21 b(A)13 b(small)e(o)q(dd)j(p)q(erturbation)g(that)g(is)f(neither)215 945 y(o)q(dd)18 b(ab)q(out)f(0)g(nor)f(ev)o(en)g(ab)q(out)h Fl(\031)r(=)p Fo(2)g(has)g(b)q(een)f(added)g(at)h Fl(t)c Fo(=)h(1.)h(.)25 b(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(22)103 1017 y(5)88 b(Dominan)o(t)15 b(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)g(for)f (a)h(solution)g(in)g(Fix)o(\(\006)1376 1024 y Fi(3)1396 1017 y Fo(\))f(with)h Fl(R)e Fo(=)g(16.)22 b(1:)215 1089 y(p)q(erturbations)f(in)e Fl(W)630 1096 y Fi(1)669 1089 y Fo(\(=)h(Fix)o(\(\006)871 1096 y Fi(3)891 1089 y Fo(\)\),)g(2:)28 b(p)q(erturbations)20 b(in)g Fl(W)1444 1096 y Fi(2)1463 1089 y Fo(,)g(3:)29 b(p)q(erturba-)215 1161 y(tions)17 b(in)f Fl(W)438 1168 y Fi(3)458 1161 y Fo(,)f(4:)22 b(p)q(erturbations) 17 b(in)f Fl(W)955 1168 y Fi(4)975 1161 y Fo(.)f(.)24 b(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f (.)g(.)h(.)54 b(23)103 1233 y(6)88 b(Gro)o(wth)21 b(and)f(deca)o(y)g (of)g(functions)g Fl(A)945 1240 y Fi(2)964 1233 y Fo(,)h Fl(A)1036 1240 y Fi(4)1075 1233 y Fo(and)g Fl(A)1211 1240 y Fi(3)1250 1233 y Fo(o)o(v)o(er)e(time)f(with)h Fl(R)i Fo(=)f(16,)215 1306 y Fl(\026)k Fo(=)g Fj(\000)p Fl(\027)j Fo(=)c(2)p Fl(:)p Fo(9.)39 b(In)22 b(eac)o(h)g(case)g(a)g (small)e(p)q(erturbation)j(has)g(b)q(een)f(added)g(at)215 1378 y Fl(t)14 b Fo(=)g(1)p Fl(:)p Fo(5)i(to)h(a)f(solution)h(in)f(Fix) o(\(\006)844 1385 y Fi(3)864 1378 y Fo(\).)30 b(.)24 b(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f (.)h(.)f(.)g(.)h(.)54 b(24)103 1450 y(7)88 b(Dominan)o(t)18 b(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)f(for)h(an)g(ev)o(en)e(solution)i (\(solid)f(line\))g(with)g(an)215 1522 y(o)q(dd)g(p)q(erturbation)e (\(dashed)h(line\))e(for)i Fl(R)d Fo(=)g(1)p Fl(:)p Fo(05.)28 b(.)d(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(25)103 1595 y(8)88 b(An)12 b(stable)f(c)o(haotic)g(solution)h(that)g (is)f(ev)o(en)f(ab)q(out)j(the)e(origin.)20 b(P)o(arameter)10 b(v)m(alues:)215 1667 y Fl(R)15 b Fo(=)e(1)p Fl(:)p Fo(05,)k Fl(\027)g Fo(=)d(4,)i Fl(\026)e Fo(=)g Fj(\000)p Fo(4.)19 b(.)25 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)f(.) h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)g(.)h(.)54 b(26)103 1739 y(9)88 b(Demonstration)20 b(of)g(the)f(orbital)h(stabilit)o(y)e(of)j (an)f(ev)o(en)e(c)o(haotic)i(solution.)32 b(The)215 1811 y(dark)22 b(line)f(is)g(one)h(of)g(the)g(t)o(w)o(o)f(p)q(oin)o(ts)h(ab) q(out)h(whic)o(h)e(the)h(\014rst)g(mo)q(de)f(is)g(ev)o(en,)215 1884 y(for)g(b)q(oth)h(the)f(real)f(and)i(imaginary)d(parts)j(of)f Fl(A)p Fo(.)34 b(See)21 b(text)f(for)h(more)f(details.)215 1956 y(P)o(arameter)15 b(v)m(alues:)21 b Fl(R)15 b Fo(=)e(1)p Fl(:)p Fo(05,)k Fl(\027)g Fo(=)d(4,)i Fl(\026)e Fo(=)g Fj(\000)p Fo(4.)40 b(.)25 b(.)f(.)g(.)h(.)f(.)h(.)f(.)h(.)f(.)h(.)f(.)h (.)f(.)g(.)h(.)54 b(27)936 2893 y(18)p eop %%Page: 17 2 17 1 bop 30 28 a Fo([28])24 b(E.)d(Ott)g(and)g(J.)f(Sommerer.)e(Blo)o (w)o(out)i(bifurcations:)31 b(the)20 b(o)q(ccurrence)g(of)h(riddled)f (basins)130 101 y(and)d(on-o\013)h(in)o(termittenc)o(y)l(,)13 b Fm(Phys.)k(L)n(ett.)g(A)g Fd(188)p Fo(,)e(39-47,)j(1994.)30 214 y([29])24 b(L.M.)c(P)o(ecora)h(and)h(T.C.)e(Carroll.)h(Sync)o (hronisation)g(in)f(c)o(haotic)g(oscillators,)i Fm(Phys.)f(R)n(ev.)130 287 y(L)n(ett.)16 b Fd(64)p Fo(,)g(821-824,)j(1990.)30 400 y([30])24 b(K.)18 b(Stew)o(artson)h(and)f(J.T.)g(Stuart.)g(A)g (non{linear)g(instabilit)o(y)f(theory)g(for)i(a)f(w)o(a)o(v)o(e)f (system)130 473 y(in)f(plane)g(P)o(oiseulle)f(\015o)o(w.)h Fm(J.)h(Fluid)h(Me)n(ch.)e Fd(48)p Fo(,)g(529-545,)i(1971.)30 586 y([31])24 b(J.T.)15 b(Stuart)i(and)f(R.C.)f(DiPrima.)f(The)i(Ec)o (khaus)g(and)g(Benjamin{F)l(eir)e(resonance)i(mec)o(ha-)130 659 y(nisms.)f Fm(Pr)n(o)n(c.)h(R)n(oy.)g(So)n(c.)i(L)n(ond.)e(A)h Fd(362)p Fo(,)f(27-41,)h(1978.)30 772 y([32])24 b(B.)19 b(W)l(erner.)g(Eigen)o(v)m(alue)h(problems)e(with)i(the)g(symmetry)c (of)21 b(a)f(group)h(and)g(bifurcations,)130 845 y(in:)30 b Fm(Continuation)22 b(and)g(Bifur)n(c)n(ations:)30 b(Numeric)n(al)22 b(T)l(e)n(chniques)h(and)e(Applic)n(ations)g Fo(\(eds.)130 917 y(D.)16 b(Ro)q(ose,)h(B.)e(De)i(Dier)e(and)i(A.)f(Sp)q(ence\),)f (NA)l(TO)g(ASI)h(Series)f(C,)h Fd(313)p Fo(,)g(71-88,)i(1990.)30 1031 y([33])24 b(H.)c(Y)l(amada)f(and)i(T.)f(F)l(ujisak)m(a.)h (Stabilit)o(y)d(theory)i(of)h(sync)o(hronised)f(motion)f(in)h(coupled-) 130 1103 y(oscillator)c(systems,)f Fm(Pr)n(o)n(g.)h(The)n(or.)h(Phys.)e Fd(69)p Fo(,)h(32-47,)i(1983.)936 2893 y(17)p eop %%Page: 16 3 16 2 bop 30 28 a Fo([14])24 b(C.R.)13 b(Do)q(ering,)h(J.D.)f(Gibb)q (on,)h(D.D.)g(Holm)d(and)j(B.)f(Nicolaenk)o(o.)e(Exact)j(Ly)o(apuno)o (v)g(dimen-)130 101 y(sion)22 b(of)f(the)g(univ)o(eral)f(attractor)i (for)g(the)f(complex)e(Ginsburg{Landau)24 b(equation,)e Fm(Phys.)130 173 y(R)n(ev.)c(L)n(ett.)e Fd(59)p Fo(,)f(2911-2914)q(,)k (1987.)30 284 y([15])24 b(C.R.)c(Do)q(ering,)h(J.D.)f(Gibb)q(on,)i (D.D.)e(Holm)e(and)j(B.)e(Nicolaenk)o(o.)g(Lo)o(w-dimensional)h(b)q(e-) 130 356 y(ha)o(viour)c(in)g(the)g(complex)e(Ginzburg-Landau)19 b(equation,)c Fm(Nonline)n(arity)i Fd(1)p Fo(,)f(279-309,)i(1988.)30 467 y([16])24 b(C.R.)16 b(Do)q(ering,)g(J.D.)f(Gibb)q(on)i(and)g(C.D.)e (Lev)o(ermore.)f(W)l(eak)i(and)g(strong)h(solutions)g(of)f(the)130 539 y(complex)e(Ginsburg{Landau)19 b(equation,)d Fm(Physic)n(a)h(D)f Fd(71)p Fo(,)g(285-318,)i(1994.)30 650 y([17])24 b(J.Q.)f(Duan,)k(E.S.) c(Titi)h(and)h(P)l(.)e(Holmes.)f(Regularit)o(y)l(,)j(app)q(o)o (ximation)e(and)i(asypm)o(totic)130 722 y(dynamics)c(for)i(a)f (generalized)f(Ginzburg{Landau)k(equation,)e Fm(Nonline)n(arity)g Fd(6)p Fo(,)g(915-933,)130 794 y(1993.)30 905 y([18])h(J.P)l(.)16 b(Ec)o(kmann)f(and)i(D.)g(Ruelle.)d(Ergo)q(dic)j(theory)g(of)g(c)o (haos)g(and)g(strange)g(attractors,)g Fm(R)n(ev.)130 977 y(Mo)n(d.)g(Phys.)e Fd(57)p Fo(,)h(617-656,)j(1985.)30 1088 y([19])24 b(M.)15 b(Golubitsky)l(,)f(I.)h(Stew)o(art)g(and)g(D.G.) g(Sc)o(hae\013er.)g Fm(Singularities)j(and)e(Gr)n(oups)g(in)g(Bifur)n (c)n(a-)130 1160 y(tion)i(The)n(ory.)e(V)l(ol.)j(II.)c Fo(Applied)g(Math.)h(Sciences)f Fd(69)p Fo(,)h(Springer{V)l(erlag,)f (1988.)30 1271 y([20])24 b(T.)16 b(Healey)l(.)e(Numerical)f (bifurcation)i(with)h(symmetr)o(y:)i(Diagnosis)f(&)f(computation)f(of)h (sin-)130 1343 y(gular)f(p)q(oin)o(ts,)g(in)f Fm(Bifur)n(c)n(ation)i (The)n(ory)e(and)i(its)g(Numeric)n(al)g(A)o(nalysis)f Fo(\(eds.)f(Li)h(Kaitai,)f(J.E.)130 1415 y(Marsden,)22 b(M.)d(Golubitsky)i(and)g(G.)f(Io)q(oss\),)i(Xi'an)e(Jiatong)h(Univ)o (ersit)o(y)d(Press,)k(218-227,)130 1488 y(1989.)30 1598 y([21])i(L.R.)d(Keefe.)f(Dynamics)h(of)h(p)q(erturb)q(ed)g(w)o(a)o(v)o (etrain)e(solutions)j(to)f(the)f(Ginzburg-Landau)130 1671 y(equation,)16 b Fm(Stud.)i(Appl.)g(Math.)e Fd(73)p Fo(,)g(91-153,)i(1985.)30 1781 y([22])24 b(L.)14 b(Keefe.)e(Prop)q (erties)h(of)h(Ginzburg-Landau)i(attractors)e(asso)q(ciated)h(with)e (their)g(Ly)o(apuno)o(v)130 1854 y(v)o(ectors)j(and)h(sp)q(ectra,)f Fm(Phys.)h(L)n(ett.)g(A)f Fd(140)p Fo(,)g(317-322,)j(1989.)30 1964 y([23])24 b(Y.)14 b(Kuramoto)g(and)h(S.)g(Koga.)g(T)l(urbulized)e (rotating)j(c)o(hemical)11 b(w)o(a)o(v)o(es,)j Fm(Pr)n(o)n(g.)h(The)n (or.)g(Phys.)130 2037 y(Suppl.)i Fd(66)p Fo(,)f(1081-1085,)j(1981.)30 2147 y([24])24 b(The)35 b(MathW)l(orks,)40 b(Inc.)34 b(24)h(Prime)e(P)o(ark)i(W)l(a)o(y)g(Natic)o(k,)i(MA)d(01760-150)q(0,) 42 b(USA.)130 2220 y Fa(http://www.)o(ma)o(thw)o(ork)o(s.c)o(om)o(/)30 2330 y Fo([25])24 b(A.C.)13 b(New)o(ell)f(and)i(J.A.)f(Whitehead.)g (Finite)g(bandwidth,)h(\014nite)f(amplitude)f(con)o(v)o(ection.)g Fm(J.)130 2403 y(Fluid)18 b(Me)n(ch.)e Fd(38)p Fo(,)g(279-303,)i(1969.) 30 2513 y([26])24 b(M.J.)17 b(Ogorzalek.)f(T)l(aming)h(c)o(haos)h({)g (P)o(art)g(I:)e(Sync)o(hronisation,)i Fm(IEEE)g(T)l(r)n(ans.)g(Cir)n (c.)g(Syst.)130 2586 y(I)e Fo(\()p Fm(F)l(und.)i(Thy)f(Appls)p Fo(\))g Fd(40)p Fo(,)f(693-699,)i(1993.)30 2696 y([27])24 b(V.I.)16 b(Oseledec.)f(A)i(m)o(ultiplicativ)n(e)d(ergo)q(dic)k (theorem.)d(Ly)o(apuno)o(v)j(c)o(haracteristic)e(n)o(um)o(b)q(ers)130 2769 y(for)h(dynamical)d(systems,)h Fm(T)l(rudy)i(Mosk.)g(Mat.)g(Obsc.) g Fd(19)p Fo(,)f(197-231,)j(1968.)936 2893 y(16)p eop %%Page: 15 4 15 3 bop 30 28 a Fn(References)54 150 y Fo([1])24 b(J.C.)17 b(Alexander,)e(I.)h(Kan,)h(J.A.)f(Y)l(ork)o(e)g(and)h(Z.)g(Y)l(ou.)f (Riddled)g(basins,)h Fm(Int.)i(J.)e(Bif.)h(Chaos)130 222 y Fd(2)p Fo(,)e(795-813,)i(1992.)54 336 y([2])24 b(P)l(.)15 b(Ash)o(win,)f(J.)g(Buescu)g(and)h(I.)f(Stew)o(art.)h (Bubbling)f(of)h(attractors)g(and)h(sync)o(hronisation)f(of)130 408 y(c)o(haotic)h(oscillators,)g Fm(Phys.)h(L)n(ett.)g(A)f Fd(193)p Fo(,)g(126-139,)i(1994.)54 522 y([3])24 b(P)l(.)c(Ash)o(win,)f (J.)g(Buescu)g(and)i(I.)d(Stew)o(art.)i(F)l(rom)e(attractor)i(to)g(c)o (haotic)f(saddle:)28 b(a)20 b(tale)g(of)130 594 y(transv)o(erse)c (instabilit)o(y)l(,)e Fm(Nonline)n(arity)j Fd(9)p Fo(,)f(703-737,)i (1996.)54 708 y([4])24 b(P)l(.)d(Ash)o(win,)f(P)l(.J.)g(Aston)h(and)g (M.)f(Nicol.)f(On)i(the)f(unfolding)h(of)g(a)g(blo)o(w)o(out)g (bifurcation.)130 780 y Fm(Physic)n(a)c(D)f Fd(111)p Fo(,)g(81-95,)i(1998.)54 894 y([5])24 b(P)l(.J.)12 b(Aston.)f(Analysis) h(and)g(computation)g(of)g(symmetry-breaki)o(ng)e(bifurcation)i(and)g (scaling)130 966 y(la)o(ws)17 b(using)f(group)h(theoretic)f(metho)q (ds.)f Fm(SIAM)i(J.)g(Math.)g(A)o(nal.)g Fd(22)p Fo(,)f(181-212,)j (1991.)54 1080 y([6])24 b(P)l(.J.)16 b(Aston)i(and)f(M.)f(Dellnitz.)f (Symmetry)e(breaking)k(bifurcations)g(of)g(c)o(haotic)g(attractors.)130 1152 y Fm(Int.)h(J.)f(Bif.)h(Chaos)d Fd(5)p Fo(,)h(1643-1676,)j(1995.) 54 1266 y([7])24 b(P)l(.J.)15 b(Aston)h(and)g(C.R.)f(Laing.)i(Symme)o (try)c(and)j(c)o(haos)g(in)g(the)f(complex)e(Ginsburg-Landau)130 1338 y(equation.)j(I)q(I)g(T)l(ranslational)h(symmetri)o(es.)c(In)j (preparation.)54 1452 y([8])24 b(M.V.)e(Bartuccelli,)h(P)l(.)g(Constan) o(tin,)j(C.R.)c(Do)q(ering,)k(J.D.)d(Gibb)q(on)h(and)g(M.)f (Gisselfalt.)130 1524 y(On)e(the)g(p)q(ossibilit)o(y)f(of)h(soft)g(and) g(hard)h(turbulence)e(in)g(the)h(complex)d(Ginsburg{Landau)130 1596 y(equation,)e Fm(Physic)n(a)h(D)f Fd(44)p Fo(,)g(421-444,)i(1990.) 54 1710 y([9])24 b(M.V.)d(Bartuccelli,)h(J.D.)g(Gibb)q(on)i(and)f(M.)f (Oliv)o(er.)e(Length)k(scales)f(in)f(solutions)h(of)g(the)130 1782 y(complex)14 b(Ginzburg{Landau)19 b(equation,)c Fm(Physic)n(a)i(D)f Fd(89)p Fo(,)g(267-286,)i(1996.)30 1896 y([10])24 b(C.)14 b(Can)o(uto,)g(M.Y.)e(Hussaini,)h(A.)g (Quarteroni)h(and)g(T.A.)e(Zang.)i Fm(Sp)n(e)n(ctr)n(al)h(Metho)n(ds)f (in)i(Fluid)130 1968 y(Dynamics)p Fo(,)g(Springer{V)l(erlag,)f(1988.)30 2082 y([11])24 b(P)l(.)16 b(Constan)o(tin,)h(??,)f(1989.)30 2196 y([12])24 b(E.)11 b(Co)o(v)m(as,)i(P)l(.)d(Ash)o(win)h(and)h(R.)e (T)l(a)o(v)m(ak)o(ol.)h(Non-normal)f(parameter)g(blo)o(w)o(out)h (bifurcation)g(in)g(a)130 2268 y(truncated)i(mean)e(\014eld)h(dynamo,)g (T)l(ec)o(hnical)f(Rep)q(ort)i(97/15,)h(Departmen)o(t)d(of)i (Mathematics)130 2340 y(and)k(Statistics,)f(Univ)o(ersit)o(y)d(of)k (Surrey)l(,)e(1997.)30 2454 y([13])24 b(E.)16 b(Co)o(v)m(as,)g(R.)f(T)l (a)o(v)m(ak)o(ol,)g(P)l(.Ash)o(win,)f(A.)h(Tw)o(ork)o(o)o(wski)g(and)h (J.M.)f(Bro)q(ok)o(e.)g(In-out)h(in)o(termit-)130 2526 y(tency)21 b(in)h(PDE)g(and)g(ODE)h(mo)q(dels)d(of)i(axisymmetri)o(c)d (mean-\014eld)h(dynamos,)i(T)l(ec)o(hnical)130 2598 y(Rep)q(ort,)16 b(Departmen)o(t)f(of)i(Mathematics)d(and)j(Statistics,)e(Univ)o(ersit)o (y)f(of)j(Surrey)l(,)e(1998.)936 2893 y(15)p eop %%Page: 14 5 14 4 bop 30 28 a Fo(p)q(erturbation)17 b(w)o(as)h(added)f(at)g Fl(t)d Fo(=)h(10)i(and)h(w)o(e)e(see)g(that)i(the)e(solution)h(quic)o (kly)e(settled)h(to)h(b)q(eing)30 101 y(ev)o(en)e(ab)q(out)j(an)f Fl(x)f Fo(v)m(alue)h(of)g(appro)o(ximately)d(0)p Fl(:)p Fo(87)e Fj(\002)f Fo(2)p Fl(\031)r Fo(.)22 b(Note)16 b(that)h(during)g(the)f(transien)o(ts)g(the)30 173 y(\014rst)g(mo)q (des)f(of)h(the)g(real)f(and)h(imaginary)f(parts)h(of)g Fl(A)g Fo(are)f(ev)o(en)g(ab)q(out)i(di\013eren)o(t)e(p)q(oin)o(ts,)h (but)g(on)30 245 y(the)f(attractor)i(they)e(are)g(ev)o(en)g(ab)q(out)i (the)e(same)f(p)q(oin)o(t,)i(as)g(they)f(m)o(ust)f(b)q(e)i(for)g(an)g (ev)o(en)e(solution.)30 317 y(See)i([19])g(for)g(more)f(details)h(on)h (orbital)f(stabilit)o(y)l(.)103 390 y(Despite)i(searc)o(hing,)g(w)o(e)f (could)h(not)h(\014nd)f(an)o(y)g(evidence)e(of)j(a)f(\\blo)o(w)o(out")h (bifurcation)f([4])f(in)30 462 y(whic)o(h)f(the)g(ev)o(en)f(solution)i (remains)e(c)o(haotic)h(while)g(the)g(dominan)o(t)f(Ly)o(apuno)o(v)i (exp)q(onen)o(t)f(in)g(the)30 534 y(normal)f(direction)g(c)o(hanges)h (from)f(zero)h(to)g(p)q(ositiv)o(e)f(as)i(a)f(parameter)f(is)g(v)m (aried.)21 b(The)16 b(reason)h(for)30 606 y(this)e(is)f(that,)h(as)g (sho)o(wn)h(in)e(Figure)g(7,)h(the)f(solution)h(in)g(the)f(ev)o(en)g (subspace)h(b)q(ecomes)e(p)q(erio)q(dic)i(or)30 679 y(quasip)q(erio)q (dic)h(b)q(efore)g(the)g(normal)g(Ly)o(apuno)o(v)g(exp)q(onen)o(t)g(b)q (ecomes)f(p)q(ositiv)o(e.)30 863 y Fn(7)81 b(Conclusions)30 985 y Fo(Our)14 b(n)o(umerical)e(results)i(sho)o(w)h(that)f(for)h(m)o (uc)o(h)d(of)i(the)g(parameter)f(space)i(for)f(the)g(CGL)i(equation,)30 1057 y(c)o(haotic)11 b(solutions)i(whic)o(h)e(ha)o(v)o(e)g(some)f(sort) j(of)f(re\015ectional)f(symmetry)d(are)k(unstable)g(to)g(p)q(erturba-) 30 1129 y(tions)k(whic)o(h)g(break)f(that)i(symmetry)-5 b(,)13 b(while)i(there)h(are)g(also)g(small)f(regions)h(of)g(parameter) f(space)30 1201 y(in)21 b(whic)o(h)f(there)h(are)g(c)o(haotic)f(ev)o (en)g(solutions)i(that)f(are)g(asymptotically)e(stable)i(with)g(resp)q (ect)30 1274 y(to)e(o)q(dd)h(p)q(erturbations.)30 b(Of)19 b(course)g(w)o(e)f(ha)o(v)o(e)g(not)h(in)o(v)o(estigated)f(all)g(of)h (the)g(three-dimensional)30 1346 y(parameter)14 b(space)i(and)h(there)e (ma)o(y)f(b)q(e)i(more)e(in)o(teresting)h(b)q(eha)o(viour)g(w)o(aiting) h(to)g(b)q(e)g(found.)21 b(P)o(e-)30 1418 y(rio)q(dic)13 b(solutions)h(with)g(symmetry)c(are)k(sometime)o(s)d(stable)j(with)f (resp)q(ect)g(to)h(symmetry-breaking)30 1490 y(p)q(erturbations)h(but)f (it)g(w)o(ould)g(app)q(ear)i(that)e(for)h(most)e(parameter)g(v)m(alues) h(for)h(the)f(CGL)h(equation)30 1563 y(that)j(for)f(arbitrary)g (initial)f(conditions,)h(if)g(the)g(\014nal)h(solution)f(is)g(c)o (haotic)g(then)g(it)g(will)f(ha)o(v)o(e)g(the)30 1635 y(minim)n(um)f(p)q(ossible)20 b(amoun)o(t)f(of)h(symmetry)-5 b(.)29 b(Clearly)18 b(these)h(ideas)h(apply)f(to)h(an)o(y)g(PDEs)g (with)30 1707 y(symmetry)9 b(and)14 b(are)e(not)i(restricted)d(to)i (the)g(CGL)g(equation.)20 b(Di\013eren)o(t)12 b(results)h(ma)o(y)e(b)q (e)i(obtained)30 1779 y(for)k(di\013eren)o(t)e(equations.)30 1905 y Fd(Ac)n(kno)n(wledgemen)n(ts)30 2032 y Fo(This)f(w)o(ork)h(w)o (as)g(supp)q(orted)g(b)o(y)f(the)g(EPSR)o(C)h(Applied)e(Nonlinear)g (Mathematics)g(Initiativ)o(e.)18 b(W)l(e)30 2104 y(thank)f(Mic)o(hele)c (Bartuccelli)h(for)j(helpful)e(discussions)i(regarding)f(the)g(CGL)i (equation.)936 2893 y(14)p eop %%Page: 13 6 13 5 bop 30 28 a Fo(whic)o(h)13 b(are,)h(for)h(Matlab,)f(t)o(ypically)e (smaller)g(than)i(the)g(quan)o(tit)o(y)f(b)q(eing)h(transformed)g(b)o (y)f(a)i(factor)30 101 y(of)j(10)135 83 y Fi(16)173 101 y Fo(.)27 b(Th)o(us,)18 b(for)g(example,)e(the)i(discrete)f(F)l(ourier) g(transform)g(of)i(a)f(real,)g(ev)o(en)e(function)i(will)30 173 y(not)i(b)q(e)f(purely)f(real,)h(but)h(will)e(ha)o(v)o(e)g(a)i (small)d(imaginary)h(comp)q(onen)o(t,)h(and)h(when)f(the)g(in)o(v)o (erse)30 245 y(transform)e(is)h(tak)o(en)f(the)h(result)f(will)f(ha)o (v)o(e)h(a)h(small)e(o)q(dd)j(comp)q(onen)o(t.)25 b(If)17 b(this)h(ev)o(en)e(solution)i(is)30 317 y(stable)f(with)g(resp)q(ect)g (to)g(o)q(dd)h(p)q(erturbations)g(this)f(will)f(not)h(matter,)f(but)h (for)g(solutions)h(suc)o(h)e(as)30 390 y(those)h(sho)o(wn)g(in)f (Figures)g(1{2)h(where)f(w)o(e)g(ha)o(v)o(e)g(instabilit)o(y)f(to)h(o)q (dd)i(p)q(erturbations)f(these)f(errors)30 462 y(will)f(gro)o(w)i(exp)q (onen)o(tially)e(in)h(time)e(and)j(ultimately)c(o)o(v)o(erwhelm)h(the)i (ev)o(en)f(solution.)103 534 y(If)h(w)o(e)g(w)o(an)o(t)h(the)f (solution)h(to)g(remain)e(within)h(an)h(in)o(v)m(arian)o(t)f(subspace)h (w)o(e)f(m)o(ust)f(mo)q(dify)g(the)30 606 y(n)o(umerical)e(sc)o(heme.) 19 b(F)l(or)d(the)f(ab)q(o)o(v)o(e)h(example,)e(this)h(is)h(simply)e (done)i(b)o(y)f(setting)h(the)f(imaginary)30 679 y(comp)q(onen)o(t)21 b(of)i(the)f(transformed)g(v)m(ariable)g(to)h(zero)f(imme)o(diately)d (after)j(it)g(is)g(calculated,)h(or)30 751 y(b)q(etter)e(still,)f(only) h(w)o(orking)g(with)g(the)g(real)f(part)i(of)f(the)g(transformed)f(v)m (ariable.)35 b(This)21 b(is)g(the)30 823 y(tec)o(hnique)16 b(used)i(for)g(Ly)o(apuno)o(v)h(exp)q(onen)o(t)e(calculations.)26 b(This)18 b(metho)q(d)g(cannot,)g(ho)o(w)o(ev)o(er,)f(b)q(e)30 895 y(used)j(in,)g(for)g(example,)f(the)g(calculations)h(for)g(Figure)f (1.)33 b(Here,)20 b(although)g(w)o(e)g(added)g(a)h(small)30 967 y(o)q(dd)c(p)q(erturbation)g(after)f(2)g(seconds)h(to)f (demonstrate)f(instabilit)o(y)g(in)h(this)f(isot)o(ypic)g(comp)q(onen)o (t,)30 1040 y(strictly)j(sp)q(eaking)i(this)g(w)o(as)g(not)g(necessary) f(as,)h(giv)o(en)f(su\016cien)o(t)f(time,)g(the)h(n)o(umerical)e (errors)30 1112 y(in)o(tro)q(duced)23 b(b)o(y)g(the)g(F)l(ourier)g (transform)g(w)o(ould)h(ha)o(v)o(e)e(gro)o(wn)j(large)e(enough)h(to)g (destro)o(y)f(the)30 1184 y(app)q(earance)17 b(of)g(ev)o(enness)e(|)h (all)g(w)o(e)g(did)g(w)o(as)g(hasten)h(the)f(onset)h(of)f(this)g (phenomenon.)30 1344 y Ff(6.1)66 b(Stable)24 b(ev)n(en)e(c)n(haos)30 1448 y Fo(Although)e(for)g(the)f(v)m(ast)i(ma)s(jorit)o(y)c(of)j (parameter)f(v)m(alues)g(w)o(e)h(examined,)d(solutions)k(that)f(w)o (ere)30 1520 y(c)o(haotic)f(when)h(restricted)e(to)j(lie)d(in)h(a)i (\014xed)e(p)q(oin)o(t)h(subspace)g(w)o(ere)f(unstable)h(with)f(resp)q (ect)h(to)30 1592 y(p)q(erturbations)g(normal)e(to)h(that)h(space,)f(w) o(e)g(did)g(\014nd)g(a)g(region)g(in)g(parameter)f(space)h(in)g(whic)o (h)30 1665 y(there)12 b(exist)f(c)o(haotic)h(solutions)h(in)f(the)g (subspace)g(of)h(ev)o(en)e(solutions)i(that)g(are)f(stable)g(with)g (resp)q(ect)30 1737 y(to)18 b(p)q(erturbations)g(in)f(the)f(o)q(dd)j (subspace.)24 b(W)l(e)17 b(plot)g(the)g(Ly)o(apuno)o(v)h(exp)q(onen)o (ts)f(in)g(part)h(of)f(this)30 1809 y(region)i(in)f(Figure)h(7)g(and)h (sho)o(w)f(a)g(t)o(ypical)f(example,)f(at)i(parameter)f(v)m(alues)g Fl(R)h Fo(=)f(1)p Fl(:)p Fo(05,)i Fl(\027)i Fo(=)c(4,)30 1881 y Fl(\026)d Fo(=)g Fj(\000)p Fo(4,)i(in)g(Figure)f(8,)h(where)g (this)g(particular)f(solution)h(is)g(ev)o(en)f(ab)q(out)i(the)f (origin.)23 b(W)l(e)17 b(migh)o(t)30 1954 y(describ)q(e)e(this)g(as)g (\\w)o(eak")h(c)o(haos)g(arising)f(from)f(the)h(bifurcation)g(of)g(a)h (p)q(erio)q(dic)f(or)g(quasip)q(erio)q(dic)30 2026 y(orbit)i(for)h (whic)o(h)e(the)h(dominan)o(t)f(Ly)o(apuno)o(v)i(exp)q(onen)o(t)e(in)h (the)g(normal)f(direction)h(remains)e(zero)30 2098 y(as)i(a)g (parameter)e(is)h(v)m(aried.)103 2170 y(The)f(c)o(haotic)g(ev)o(en)f (solution)h(at)h(these)f(parameter)f(v)m(alues)h(is)g Fm(orbital)r(ly)i(stable)p Fo(,)g(meaning)d(that)30 2242 y(there)g(is)h(a)g(con)o(tin)o(uous)f(family)f(of)i(suc)o(h)f (solutions)h(related)f(to)h(one)g(another)g(b)o(y)f(the)h(spatial)g (shift,)30 2315 y Fl(r)52 2322 y Fk(\013)77 2315 y Fo(,)h(and)i (\(assuming)e(there)g(are)h(no)g(co)q(existing)f(attractors\))i(an)f (arbitrary)g(initial)f(condition)g(will)30 2387 y(b)q(e)g(attracted)g (to)g(one)f(mem)o(b)q(er)e(of)j(this)f(family)f(i.e.)g(a)i(solution)g (that)g(is)f(ev)o(en)g(ab)q(out)i(some)d(p)q(oin)o(t.)30 2459 y(W)l(e)h(demonstrate)g(this)g(in)g(Figure)g(9)h(where)f(w)o(e)g (plot)h(a)g(solution)f(started)h(at)g(a)g(randomly)e(c)o(hosen)30 2531 y(initial)e(condition)g(together)h(with)g(one)g(of)g(the)g(t)o(w)o (o)g(p)q(oin)o(ts)g(in)f([0)p Fl(;)c Fo(2)p Fl(\031)r Fo(\))13 b(ab)q(out)h(whic)o(h)e(the)h(\014rst)g(mo)q(de)30 2604 y(of)h(the)g(solution)g(\(i.e.)f(that)h(describ)q(ed)f(b)o(y)h(a)g (linear)g(com)o(bination)e(of)i(sin)9 b Fl(x)k Fo(and)i(cos)8 b Fl(x)p Fo(\))14 b(is)g(ev)o(en,)f(for)30 2676 y(b)q(oth)j(the)e(real) h(and)g(imaginary)f(parts)i(of)f Fl(A)p Fo(.)20 b(W)l(e)15 b(see)f(that)i(after)e(3)i(seconds)f(the)g(solution)g(settled)30 2748 y(to)20 b(b)q(eing)h(ev)o(en)d(ab)q(out)k(an)e Fl(x)g Fo(v)m(alue)g(of)g(appro)o(ximately)e(0)p Fl(:)p Fo(1)c Fj(\002)f Fo(2)p Fl(\031)r Fo(.)33 b(A)19 b(large)h(randomly)f(c)o (hosen)936 2893 y(13)p eop %%Page: 12 7 12 6 bop 103 28 a Fo(Figure)13 b(3)g(sho)o(ws)h(a)f(c)o(haotic)f(o)q (dd)i(solution)f(that)h(is)e(unstable)h(to)g(ev)o(en)f(p)q (erturbations.)21 b(A)13 b(small)30 101 y(ev)o(en)h(p)q(erturbation)j (is)e(added)h(at)g Fl(t)d Fo(=)h(0)p Fl(:)p Fo(5)i(and)g(this)f(gro)o (ws)i(to)f(pro)q(duce)f(a)h(c)o(haotic)f(solution)h(with)30 173 y(no)h(symmetry)-5 b(.)19 b(P)o(arameter)14 b(v)m(alues)j(are)f Fl(R)e Fo(=)g(36,)j Fl(\027)g Fo(=)c Fj(\000)p Fo(10,)k Fl(\026)d Fo(=)g(12.)103 245 y(Figure)i(4)h(sho)o(ws)g(a)f(c)o(haotic)g (solution)h(in)e(Fix\(\006)1012 252 y Fi(3)1031 245 y Fo(\))i(that)f(is)g(unstable)h(to)f(p)q(erturbations)i(in)e(all)30 317 y(three)i(isot)o(ypic)f(comp)q(onen)o(ts,)i Fl(W)670 324 y Fi(2)689 317 y Fl(;)8 b(W)757 324 y Fi(3)777 317 y Fl(;)g(W)845 324 y Fi(4)865 317 y Fo(.)28 b(A)18 b(small)f(p)q (erturbation)i(not)g(in)g(Fix)o(\(\006)1654 324 y Fi(3)1674 317 y Fo(\))f(is)h(added)30 390 y(at)g Fl(t)d Fo(=)h(1)i(and)f(this)g (gro)o(ws)h(exp)q(onen)o(tially)e(in)h(time.)25 b(P)o(arameter)16 b(v)m(alues)i(are)h Fl(R)e Fo(=)g(16,)i Fl(\027)h Fo(=)d Fj(\000)p Fo(5,)30 462 y Fl(\026)d Fo(=)g(12.)103 534 y(In)j(Fig.)g(5)g(w)o(e)g(sho)o(w)h(the)f(dominan)o(t)f(Ly)o(apuno)o(v) i(exp)q(onen)o(ts)f(for)g(a)h(solution)g(in)e(Fix\(\006)1749 541 y Fi(3)1769 534 y Fo(\))h(as)h(a)30 606 y(function)e(of)g Fl(\026)h Fo(\(=)d Fj(\000)p Fl(\027)s Fo(\))i(for)g Fl(R)f Fo(=)e(16.)22 b(The)17 b(con)o(tin)o(uous)f(symmetri)o(es)d(men) o(tioned)h(ab)q(o)o(v)o(e)i(prev)o(en)o(t)30 679 y(the)h(dominan)o(t)e (Ly)o(apuno)o(v)j(exp)q(onen)o(t)e(asso)q(ciated)i(with)f Fl(W)1161 686 y Fi(1)1198 679 y Fo(and)g Fl(W)1339 686 y Fi(2)1376 679 y Fo(from)f(b)q(eing)h(negativ)o(e)f(but)30 751 y(put)j(no)g(restrictions)e(on)i(the)f(dominan)o(t)g(Ly)o(apuno)o (v)g(exp)q(onen)o(ts)h(asso)q(ciated)g(with)f Fl(W)1675 758 y Fi(3)1714 751 y Fo(and)h Fl(W)1857 758 y Fi(4)1876 751 y Fo(.)30 823 y(Note)12 b(that)g(the)g(dominan)o(t)f(Ly)o(apuno)o (v)i(exp)q(onen)o(t)f(asso)q(ciated)h(with)f Fl(W)1348 830 y Fi(1)1380 823 y Fo(is)g(alw)o(a)o(ys)g(zero)g(and)g(so)h(the)30 895 y(non-trivial)18 b(Ly)o(apuno)o(v)i(exp)q(onen)o(ts)f(will)e(all)i (b)q(e)g(negativ)o(e.)28 b(This)19 b(implies)e(that)i(the)g(underlying) 30 967 y(solution)h(is)f(either)f(p)q(erio)q(dic)h(or)h(quasip)q(erio)q (dic.)31 b(T)l(o)20 b(illustrate)e(the)h(stabilit)o(y)f(or)i (instabilit)o(y)e(of)30 1040 y(the)d(underlying)f(motion)h(with)g(resp) q(ect)f(to)i(the)f(di\013eren)o(t)f(p)q(erturbations,)i(w)o(e)f (de\014ne)f(pro)s(jections)30 1112 y(on)o(to)j(eac)o(h)e(of)i(the)f (\006)424 1119 y Fi(3)444 1112 y Fo(-isot)o(ypic)f(comp)q(onen)o(ts)h (as)298 1227 y Fl(A)335 1234 y Fi(1)354 1227 y Fo(\()p Fl(x;)8 b(t)p Fo(\))41 b(=)585 1194 y(1)p 585 1216 25 2 v 585 1262 a(4)615 1227 y([)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))h Fj(\000)i Fl(A)p Fo(\()p Fj(\000)p Fl(x;)d(t)p Fo(\))h Fj(\000)i Fl(A)p Fo(\()p Fl(x)g Fj(\000)f Fl(\031)r(;)e(t)p Fo(\))i(+)h Fl(A)p Fo(\()p Fl(\031)h Fj(\000)f Fl(x;)d(t)p Fo(\)])298 1344 y Fl(A)335 1351 y Fi(2)354 1344 y Fo(\()p Fl(x;)g(t)p Fo(\))41 b(=)585 1310 y(1)p 585 1333 V 585 1378 a(4)615 1344 y([)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))h(+)i Fl(A)p Fo(\()p Fj(\000)p Fl(x;)d(t)p Fo(\))i Fj(\000)g Fl(A)p Fo(\()p Fl(x)h Fj(\000)g Fl(\031)r(;)d(t)p Fo(\))i Fj(\000)g Fl(A)p Fo(\()p Fl(\031)i Fj(\000)f Fl(x;)d(t)p Fo(\)])298 1461 y Fl(A)335 1468 y Fi(3)354 1461 y Fo(\()p Fl(x;)g(t)p Fo(\))41 b(=)585 1427 y(1)p 585 1449 V 585 1495 a(4)615 1461 y([)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))h Fj(\000)i Fl(A)p Fo(\()p Fj(\000)p Fl(x;)d(t)p Fo(\))h(+)i Fl(A)p Fo(\()p Fl(x)g Fj(\000)g Fl(\031)r(;)d(t)p Fo(\))i Fj(\000)g Fl(A)p Fo(\()p Fl(\031)i Fj(\000)f Fl(x;)d(t)p Fo(\)])298 1577 y Fl(A)335 1584 y Fi(4)354 1577 y Fo(\()p Fl(x;)g(t)p Fo(\))41 b(=)585 1543 y(1)p 585 1566 V 585 1611 a(4)615 1577 y([)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))h(+)i Fl(A)p Fo(\()p Fj(\000)p Fl(x;)d(t)p Fo(\))i(+)h Fl(A)p Fo(\()p Fl(x)f Fj(\000)h Fl(\031)r(;)d(t)p Fo(\))i(+)h Fl(A)p Fo(\()p Fl(\031)h Fj(\000)f Fl(x;)d(t)p Fo(\)])p Fl(:)30 1685 y Fo(Note)17 b(that)h Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))14 b(=)i Fl(A)504 1692 y Fi(1)523 1685 y Fo(\()p Fl(x;)8 b(t)p Fo(\))j(+)g Fl(A)726 1692 y Fi(2)746 1685 y Fo(\()p Fl(x;)d(t)p Fo(\))j(+)g Fl(A)949 1692 y Fi(3)969 1685 y Fo(\()p Fl(x;)d(t)p Fo(\))j(+)g Fl(A)1172 1692 y Fi(4)1192 1685 y Fo(\()p Fl(x;)d(t)p Fo(\).)23 b(In)17 b(Figure)g(6)h(w)o(e)e(c)o(ho)q(ose)i (the)30 1757 y(parameter)d(v)m(alues)i Fl(R)e Fo(=)f(16,)j Fl(\026)e Fo(=)f(2)p Fl(:)p Fo(9)h(=)f Fj(\000)p Fl(\027)s Fo(.)22 b(W)l(e)16 b(start)h(with)g(an)g(initial)e(condition)h(in)g (Fix\(\006)1838 1764 y Fi(3)1857 1757 y Fo(\),)30 1830 y(let)g(this)g(ev)o(olv)o(e)f(un)o(til)g Fl(t)f Fo(=)g(1)p Fl(:)p Fo(5)j(and)g(then)f(add)h(a)g(small)e(p)q(erturbation)i(in)f (either)g Fl(W)1619 1837 y Fi(2)1638 1830 y Fo(,)h Fl(W)1715 1837 y Fi(3)1751 1830 y Fo(or)g Fl(W)1857 1837 y Fi(4)1876 1830 y Fo(.)30 1902 y(The)h(norm)f(of)h(the)f(pro)s(jection)g(of)h(the) g(solution)g(on)o(to)g(the)g(appropriate)g(isot)o(ypic)e(comp)q(onen)o (t)h(is)30 1974 y(then)h(plotted)f(as)i(a)f(function)f(of)i(time,)c (along)k(with)e(the)h(norm)f(of)h(the)f(solution)i(pro)s(jected)e(on)o (to)30 2046 y Fl(W)76 2053 y Fi(1)110 2046 y Fo(=)c(Fix)o(\(\006)286 2053 y Fi(3)306 2046 y Fo(\))k(on)f(a)h(logarithmic)d(scale.)103 2119 y(W)l(e)23 b(see)f(that)i(the)e(underlying)g(solution)h(is)g (unstable)g(with)g(resp)q(ect)f(to)i(p)q(erturbations)f(in)30 2191 y Fl(W)76 2198 y Fi(2)113 2191 y Fo(and)18 b Fl(W)255 2198 y Fi(4)292 2191 y Fo(but)g(is)f(stable)g(with)g(resp)q(ect)g(to)h (p)q(erturbations)g(in)f Fl(W)1326 2198 y Fi(3)1345 2191 y Fo(,)h(whic)o(h)e(is)h(consisten)o(t)g(with)30 2263 y(the)i(signs)g(of)g(the)g(corresp)q(onding)h(Ly)o(apuno)o(v)f(exp)q (onen)o(ts)g(sho)o(wn)g(in)g(Figure)f(5.)30 b(Note)18 b(that)h(the)30 2335 y(Ly)o(apuno)o(v)14 b(exp)q(onen)o(t)g(asso)q (ciated)g(with)g(p)q(erturbations)h(in)e Fl(W)1202 2342 y Fi(4)1236 2335 y Fo(is)g(larger)h(than)g(for)g(p)q(erturbations)30 2408 y(in)k Fl(W)135 2415 y Fi(2)173 2408 y Fo(and)h(also)g(that)f(the) g(solution)h(comp)q(onen)o(t)e(in)h Fl(W)1106 2415 y Fi(1)1144 2408 y Fo(ceases)g(to)h(b)q(e)f(p)q(erio)q(dic)g(when)g(the)g Fl(W)1870 2415 y Fi(4)30 2480 y Fo(comp)q(onen)o(t)f(b)q(ecomes)f (similar)f(in)i(magnitude)g(to)h(it.)24 b(This)18 b(is)f(the)g(p)q(oin) o(t)h(at)g(whic)o(h,)f(to)g(the)h(ey)o(e,)30 2552 y(the)e(solution)h(w) o(ould)f(no)h(longer)f(app)q(ear)h(to)g(ha)o(v)o(e)e(\006)1023 2559 y Fi(3)1059 2552 y Fo(symmetry)l(.)103 2624 y(This)i(t)o(yp)q(e)g (of)g(instabilit)o(y)f(has)h(implications)e(for)j(the)e(n)o(umerical)f (computation)h(of)h(solutions)30 2696 y(with)j(certain)f(symmetri)o (es.)30 b(The)20 b(source)g(of)g(the)g(problem)e(is)i(the)f(n)o (umerical)f(computation)h(of)30 2769 y(the)c(discrete)f(F)l(ourier)h (transform)g(and)h(its)f(in)o(v)o(erse.)20 b(Eac)o(h)15 b(of)h(these)f(pro)q(cesses)h(in)o(tro)q(duces)f(errors)936 2893 y(12)p eop %%Page: 11 8 11 7 bop 30 28 a Fo(\(This)13 b(is)g(just)g(the)g(in)o(v)o(erse)e (discrete)h(F)l(ourier)g(transform)h(of)g Fj(f)p Fl(X)1214 35 y Fk(k)1235 28 y Fo(\()p Fl(t)p Fo(\))p Fj(g)p Fo(.\))20 b(The)13 b(sp)q(ectral)g(co)q(e\016cien)o(ts,)30 101 y Fl(X)70 108 y Fk(k)92 101 y Fo(\()p Fl(t)p Fo(\),)i(are)h(obtained)h (from)e Fj(f)p Fl(A)636 108 y Fk(r)655 101 y Fo(\()p Fl(x)702 108 y Fk(n)725 101 y Fl(;)8 b(t)p Fo(\))p Fj(g)15 b Fo(via)h(the)g(discrete)f(F)l(ourier)h(transform)393 257 y Fl(X)433 264 y Fk(k)455 257 y Fo(\()p Fl(t)p Fo(\))d(=)596 195 y Fk(N)576 210 y Fg(X)579 314 y Fk(n)p Fi(=1)656 257 y Fl(A)693 264 y Fk(r)712 257 y Fo(\()p Fl(x)759 264 y Fk(n)782 257 y Fl(;)8 b(t)p Fo(\))g(exp)g(\()p Fj(\000)p Fl(ik)r(x)1062 264 y Fk(n)1084 257 y Fo(\))p Fl(;)83 b Fo(0)14 b Fj(\024)f Fl(k)j Fj(\024)e Fl(N)i Fj(\000)11 b Fo(1)p Fl(:)263 b Fo(\(5.2\))30 411 y(Since)15 b Fl(A)194 418 y Fk(r)213 411 y Fo(\()p Fl(x)260 418 y Fk(n)283 411 y Fl(;)8 b(t)p Fo(\))16 b(is)g(real,)526 533 y Fl(X)566 540 y Fk(k)588 533 y Fo(\()p Fl(t)p Fo(\))d(=)p 709 489 177 2 v 14 w Fl(X)749 540 y Fk(N)t Fe(\000)p Fk(k)829 533 y Fo(\()p Fl(t)p Fo(\))p Fl(;)73 b(k)16 b Fo(=)e(1)p Fl(;)8 b(:)g(:)g(:)16 b(;)8 b(N)q(=)p Fo(2)k Fj(\000)e Fo(1)p Fl(;)30 655 y Fo(and)18 b(b)q(oth)h Fl(X)283 662 y Fi(0)303 655 y Fo(\()p Fl(t)p Fo(\))e(and)h Fl(X)512 663 y Fk(N)q(=)p Fi(2)579 655 y Fo(\()p Fl(t)p Fo(\))f(are)g(real.)25 b(F)l(urthermore,)16 b(if)h Fl(A)1233 662 y Fk(r)1251 655 y Fo(\()p Fl(x)1298 662 y Fk(n)1321 655 y Fl(;)8 b(t)p Fo(\))17 b(is)h(ev)o(en)e(then)h Fj(f)p Fl(X)1737 662 y Fk(k)1759 655 y Fo(\()p Fl(t)p Fo(\))p Fj(g)g Fo(is)30 727 y(real,)e(and)i(if)f Fl(A)p Fo(\()p Fl(x)362 734 y Fk(n)385 727 y Fl(;)8 b(t)p Fo(\))15 b(is)h(o)q(dd)i (then)e Fj(f)p Fl(X)781 734 y Fk(k)802 727 y Fo(\()p Fl(t)p Fo(\))p Fj(g)g Fo(is)g(purely)g(imaginary)l(.)103 799 y(W)l(e)23 b(use)f Fj(f)p Fl(X)349 806 y Fk(k)371 799 y Fo(\()p Fl(t)p Fo(\))p Fj(g)g Fo(as)i(our)f(dep)q(enden)o(t)f(v)m (ariables,)i(i.e.)d(giv)o(en)h Fj(f)p Fl(X)1384 806 y Fk(k)1406 799 y Fo(\()p Fl(t)p Fo(\))p Fj(g)g Fo(w)o(e)g(w)o(an)o(t)h (to)g(kno)o(w)30 871 y Fj(f)74 859 y Fo(_)55 871 y Fl(X)95 878 y Fk(k)117 871 y Fo(\()p Fl(t)p Fo(\))p Fj(g)p Fo(.)d(Using)13 b(the)g(sp)q(ectral)h(represen)o(tation)f(of)h(our)g(solution,)g (\(5.2\),)g(ev)m(aluation)f(of)h(the)g(linear)30 944 y(terms)k Fl(RA)c Fo(+)g(\(1)f(+)h Fl(i\027)s Fo(\))p Fj(r)522 926 y Fi(2)541 944 y Fl(A)19 b Fo(in)h(\(1.1\))g(is)g(simple,) e(and)i(unlik)o(e)f(a)h(\014nite-di\013erence)e(sc)o(heme,)g(the)30 1016 y(spatial)c(di\013eren)o(tiation)f(is)h(exact.)20 b(T)l(o)15 b(calculate)d(the)i(nonlinear)g(term)e(\(1)6 b(+)g Fl(i\026)p Fo(\))p Fl(A)p Fj(j)p Fl(A)p Fj(j)1629 998 y Fi(2)1663 1016 y Fo(the)14 b(in)o(v)o(erse)30 1088 y(transform)i(\(5.1\))g(is)g(used)g(to)g(form)f Fj(f)p Fl(A)p Fo(\()p Fl(x)813 1095 y Fk(n)835 1088 y Fl(;)8 b(t)p Fo(\))p Fj(g)16 b Fo(from)f(its)g(real)h(and)g(imaginary)f (parts.)22 b(The)16 b(cubic)30 1160 y(term)g Fj(f)p Fl(A)p Fo(\()p Fl(x)257 1167 y Fk(n)279 1160 y Fl(;)8 b(t)p Fo(\))p Fj(j)p Fl(A)p Fo(\()p Fl(x)436 1167 y Fk(n)458 1160 y Fl(;)g(t)p Fo(\))p Fj(j)531 1142 y Fi(2)550 1160 y Fj(g)17 b Fo(is)g(then)h(calculated,)e(and)i(a)f(F)l(ourier)g (transform)g(of)h(the)f(form)f(\(5.2\))30 1233 y(is)21 b(used)g(to)g(ev)m(aluate)g(the)f(con)o(tribution)g(of)i(this)e (nonlinear)h(term)e(to)i Fj(f)1439 1220 y Fo(_)1419 1233 y Fl(X)1459 1240 y Fk(k)1481 1233 y Fo(\()p Fl(t)p Fo(\))p Fj(g)p Fo(.)35 b(An)o(ti{aliasing)30 1305 y(is)18 b(p)q(erformed)g(in)g (the)g(calculation)g(of)g(the)h(nonlinear)f(term)f(using)h(padding)i (and)f(truncation)f(as)30 1377 y(describ)q(ed)e(in)h([10])f(and)i Fl(N)k Fo(is)17 b(c)o(hosen)f(to)h(b)q(e)g(a)g(p)q(o)o(w)o(er)g(of)g(2) h(so)f(that)g(the)g(fast)g(F)l(ourier)f(transform)30 1449 y(and)h(its)f(in)o(v)o(erse)e(can)j(b)q(e)f(used.)30 1634 y Fn(6)81 b(Numerical)26 b(Results)30 1755 y Fo(In)14 b(Figure)g(1)h(w)o(e)f(sho)o(w)h(an)g(example)d(of)i(a)h(p)q(erio)q (dic)f(orbit)h(that)f(is)h(stable)f(within)g Fl(X)1599 1762 y Fk(e)1632 1755 y Fo(but)h(whic)o(h)e(is)30 1828 y(unstable)k(to)g(o)q(dd)g(p)q(erturbations.)23 b(W)l(e)16 b(start)h(from)f(a)h(randomly)e(c)o(hosen)i(ev)o(en)e(initial)g (condition)30 1900 y(whic)o(h)20 b(ev)o(olv)o(es)f(rapidly)h(to)g(a)h (p)q(erio)q(dic)f(state.)35 b(A)20 b(small)f(o)q(dd)i(p)q(erturbation)g (is)g(in)o(tro)q(duced)f(at)30 1972 y Fl(t)13 b Fo(=)h(2)j(and)f(this)g (gro)o(ws)h(exp)q(onen)o(tially)e(in)h(time)d(so)k(that)g(for)f Fl(t)d(>)h Fo(4)j(the)e(solution)i(is)f(c)o(haotic)f(with)30 2044 y(a)h(spatial)g(w)o(a)o(v)o(elength)f(of)h(2)p Fl(\031)r Fo(.)21 b(P)o(arameter)14 b(v)m(alues)i(are)f Fl(R)g Fo(=)e(16,)k Fl(\026)d Fo(=)g(1,)h Fl(\027)j Fo(=)13 b Fj(\000)p Fo(7.)21 b(The)16 b(real)g(part)30 2117 y(of)i Fl(A)g Fo(is)f(sho)o(wn)i(on)f(the)g(left)f(and)i(the)e(imaginary)g (part)h(on)h(the)e(righ)o(t.)26 b(In)18 b(Figures)f(1{4,)i(8{9)g(w)o(e) 30 2189 y(sho)o(w)g(con)o(tour)g(plots)g(of)g(the)g(real)f(and)i (imaginary)d(parts)j(of)f(the)f(solution)h(with)g(blac)o(k)f(con)o (tour)30 2261 y(lines)d(for)g(negativ)o(e)g(v)m(alues)g(and)h(grey)g (con)o(tour)f(lines)g(for)g(p)q(ositiv)o(e)g(v)m(alues.)21 b(The)15 b Fl(x)p Fo({axis)h(has)g(b)q(een)30 2333 y(rescaled)g(to)g ([0)p Fl(;)8 b Fo(1\))17 b(in)e(eac)o(h)h(of)h(these)f(Figures.)103 2406 y(Figure)d(2)h(sho)o(ws)h(an)f(example)d(of)j(a)g(c)o(haotic)f (solution)h(in)f Fl(X)1214 2413 y Fk(e)1247 2406 y Fo(that)h(is)f (unstable)h(to)g(o)q(dd)h(p)q(ertur-)30 2478 y(bations.)21 b(The)12 b(randomly)f(c)o(hosen)h(ev)o(en)g(initial)f(condition)h(ev)o (olv)o(es)e(to)j(an)g(ev)o(en)e(c)o(haotic)h(attractor.)30 2550 y(A)19 b(small)f(o)q(dd)j(p)q(erturbation)f(is)f(in)o(tro)q(duced) g(at)h Fl(t)f Fo(=)h(2)g(and)g(this)f(gro)o(ws)i(to)f(pro)q(duce)g(a)g (c)o(haotic)30 2622 y(solution)d(with)f(a)h(spatial)g(w)o(a)o(v)o (elength)e(of)h(2)p Fl(\031)r Fo(.)22 b(P)o(arameter)15 b(v)m(alues)i(are)f Fl(R)f Fo(=)f(16,)j Fl(\026)d Fo(=)g(5,)j Fl(\027)g Fo(=)d Fj(\000)p Fo(7.)30 2694 y(Clearly)e(there)g(are)g(t)o (w)o(o)h(co)q(existing)f(c)o(haotic)g(\\attractors")i(at)f(these)g (parameter)e(v)m(alues,)i(although)30 2767 y(the)j(ev)o(en)f(attractor) i(is)f(only)g(attracting)h(in)f(the)g(ev)o(en)f(subspace.)936 2893 y(11)p eop %%Page: 10 9 10 8 bop 30 28 a Fo(dominan)o(t)14 b(Ly)o(apuno)o(v)i(exp)q(onen)o(t)e (asso)q(ciated)j(with)e(the)f(attractor)i(in)f Fl(X)1405 35 y Fk(e)1439 28 y Fo(and)h(in)f Fl(X)k Fo(but)d(did)f(not)30 101 y(lo)q(ok)i(at)f(the)g(e\013ect)g(of)h(symmet)o(ry-breaking)d(p)q (erturbations.)103 173 y(The)j(group)h(\006)378 180 y Fi(3)415 173 y Fo(has)f(four)h(one-dimensional)d(irreducible)g (represen)o(tations)i(and)g(so)h(there)e(are)30 245 y(four)h(corresp)q (onding)g(isot)o(ypic)e(comp)q(onen)o(ts.)20 b(These)d(can)f(b)q(e)g (sp)q(eci\014ed)g(as)334 359 y Fl(W)380 366 y Fi(1)441 359 y Fo(=)42 b Fj(f)p Fl(A)13 b Fj(2)h Fl(X)k Fo(:)c Fl(A)p Fo(\(0)p Fl(;)8 b(t)p Fo(\))13 b(=)g(0)p Fl(;)25 b(A)1032 366 y Fk(x)1053 359 y Fo(\()p Fl(\031)r(=)p Fo(2)p Fl(;)8 b(t)p Fo(\))14 b(=)g(0)p Fj(g)30 b Fo(=)g(Fix)o(\(\006) 1547 366 y Fi(3)1567 359 y Fo(\))334 444 y Fl(W)380 451 y Fi(2)441 444 y Fo(=)42 b Fj(f)p Fl(A)13 b Fj(2)h Fl(X)k Fo(:)c Fl(A)766 451 y Fk(x)787 444 y Fo(\(0)p Fl(;)8 b(t)p Fo(\))14 b(=)f(0)p Fl(;)25 b(A)p Fo(\()p Fl(\031)r(=)p Fo(2)p Fl(;)8 b(t)p Fo(\))13 b(=)h(0)p Fj(g)334 529 y Fl(W)380 536 y Fi(3)441 529 y Fo(=)42 b Fj(f)p Fl(A)13 b Fj(2)h Fl(X)k Fo(:)c Fl(A)p Fo(\(0)p Fl(;)8 b(t)p Fo(\))13 b(=)g(0)p Fl(;)25 b(A)p Fo(\()p Fl(\031)r(=)p Fo(2)p Fl(;)8 b(t)p Fo(\))13 b(=)h(0)p Fj(g)334 613 y Fl(W)380 620 y Fi(4)441 613 y Fo(=)42 b Fj(f)p Fl(A)13 b Fj(2)h Fl(X)k Fo(:)c Fl(A)766 620 y Fk(x)787 613 y Fo(\(0)p Fl(;)8 b(t)p Fo(\))14 b(=)f(0)p Fl(;)25 b(A)1054 620 y Fk(x)1075 613 y Fo(\()p Fl(\031)r(=)p Fo(2)p Fl(;)8 b(t)p Fo(\))14 b(=)g(0)p Fj(g)p Fl(;)30 727 y Fo(and)22 b(corresp)q(ond)h(to)f(all)f(p)q(ossible)h(com)o(binations)f(of)h (homogeneous)f(Neumann)g(and)h(Diric)o(hlet)30 799 y(b)q(oundary)h (conditions)f(at)g Fl(x)h Fo(=)h(0)e(and)h Fl(x)g Fo(=)g Fl(\031)r(=)p Fo(2.)38 b(In)22 b(this)g(case,)h(there)e(are)h(again)h (t)o(w)o(o)e(zero)30 872 y(Ly)o(apuno)o(v)e(exp)q(onen)o(ts)g(asso)q (ciated)i(with)d Fl(W)885 879 y Fi(1)905 872 y Fo(.)30 b(No)o(w)19 b(if)f Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))18 b Fj(2)h Fo(Fix)o(\(\006)1448 879 y Fi(3)1468 872 y Fo(\),)g(then)g Fl(L)1667 879 y Fk(\013)1692 872 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))17 b(=)30 944 y Fl(A)67 951 y Fk(x)89 944 y Fo(\()p Fl(x;)8 b(t)p Fo(\))k Fj(2)j Fl(W)301 951 y Fi(2)337 944 y Fo(and)h(so)h(the)f(third)g(zero)g(Ly)o(apuno)o(v)h(exp) q(onen)o(t)f(is)g(asso)q(ciated)h(with)f Fl(W)1674 951 y Fi(2)1694 944 y Fo(.)103 1016 y(Expanding)k Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))19 b(as)h(a)g(F)l(ourier)f(series,)h(it)f (is)g(easy)h(to)g(iden)o(tify)e(the)h(mo)q(des)g(whic)o(h)g(m)o(ust)30 1088 y(o)q(ccur)d(for)h(eac)o(h)f(isot)o(ypic)f(comp)q(onen)o(t)g(as)43 1231 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k Fj(2)i Fl(W)291 1238 y Fi(1)353 1231 y Fo(=)-8 b Fj(\))41 b Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)699 1169 y Fe(1)681 1184 y Fg(X)685 1290 y Fk(k)q Fi(=1)761 1231 y Fl(b)782 1238 y Fk(k)803 1231 y Fo(\()p Fl(t)p Fo(\))c(sin)g(\(2)p Fl(k)14 b Fj(\000)d Fo(1\))p Fl(x)g Fo(+)g Fl(i)1223 1146 y Fg( )1280 1169 y Fe(1)1262 1184 y Fg(X)1266 1290 y Fk(k)q Fi(=1)1342 1231 y Fl(c)1363 1238 y Fk(k)1385 1231 y Fo(\()p Fl(t)p Fo(\))d(sin)g(\(2)p Fl(k)13 b Fj(\000)e Fo(1\))p Fl(x)1719 1146 y Fg(!)43 1399 y Fl(A)p Fo(\()p Fl(x;)d(t)p Fo(\))k Fj(2)i Fl(W)291 1406 y Fi(2)353 1399 y Fo(=)-8 b Fj(\))41 b Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)699 1336 y Fe(1)681 1351 y Fg(X)685 1457 y Fk(k)q Fi(=1)761 1399 y Fl(b)782 1406 y Fk(k)803 1399 y Fo(\()p Fl(t)p Fo(\))c(cos)h(\(2)p Fl(k)k Fj(\000)e Fo(1\))p Fl(x)g Fo(+)g Fl(i)1228 1313 y Fg( )1286 1336 y Fe(1)1268 1351 y Fg(X)1271 1457 y Fk(k)q Fi(=1)1348 1399 y Fl(c)1369 1406 y Fk(k)1390 1399 y Fo(\()p Fl(t)p Fo(\))d(cos)h(\(2)p Fl(k)k Fj(\000)e Fo(1\))p Fl(x)1730 1313 y Fg(!)43 1566 y Fl(A)p Fo(\()p Fl(x;)d(t)p Fo(\))k Fj(2)i Fl(W)291 1573 y Fi(3)353 1566 y Fo(=)-8 b Fj(\))41 b Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)699 1504 y Fe(1)681 1519 y Fg(X)685 1625 y Fk(k)q Fi(=1)761 1566 y Fl(b)782 1573 y Fk(k)803 1566 y Fo(\()p Fl(t)p Fo(\))c(sin)g(2)p Fl(k)r(x)k Fo(+)f Fl(i)1100 1481 y Fg( )1157 1504 y Fe(1)1139 1519 y Fg(X)1143 1625 y Fk(k)q Fi(=1)1219 1566 y Fl(c)1240 1573 y Fk(k)1261 1566 y Fo(\()p Fl(t)p Fo(\))d(sin)h(2)p Fl(k)r(x)1473 1481 y Fg(!)43 1734 y Fl(A)p Fo(\()p Fl(x;)f(t)p Fo(\))k Fj(2)i Fl(W)291 1741 y Fi(4)353 1734 y Fo(=)-8 b Fj(\))41 b Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)686 1700 y Fl(b)707 1707 y Fi(0)726 1700 y Fo(\()p Fl(t)p Fo(\))p 686 1722 97 2 v 722 1768 a(2)798 1734 y(+)865 1671 y Fe(1)847 1686 y Fg(X)851 1792 y Fk(k)q Fi(=1)927 1734 y Fl(b)948 1741 y Fk(k)969 1734 y Fo(\()p Fl(t)p Fo(\))c(cos)h(2)p Fl(k)r(x)i Fo(+)g Fl(i)1271 1648 y Fg( )1315 1700 y Fl(c)1336 1707 y Fi(0)1356 1700 y Fo(\()p Fl(t)p Fo(\))p 1315 1722 V 1351 1768 a(2)1428 1734 y(+)1495 1671 y Fe(1)1477 1686 y Fg(X)1480 1792 y Fk(k)q Fi(=1)1557 1734 y Fl(c)1578 1741 y Fk(k)1599 1734 y Fo(\()p Fl(t)p Fo(\))d(cos)h(2)p Fl(k)r(x)1816 1648 y Fg(!)1864 1734 y Fl(:)30 1925 y Fn(5)81 b(Numerical)26 b(Metho)r(d)30 2047 y Fo(In)11 b(this)h(section)f(w)o(e)h(brie\015y)f (describ)q(e)g(the)g(n)o(umerical)e(metho)q(d)i(used)h(for)g (calculating)f(the)g(solutions)30 2119 y(sho)o(wn)18 b(in)f(Section)g(6.)26 b(W)l(e)17 b(use)h(a)g(pseudo{sp)q(ectral)g (metho)q(d)f(co)q(ded)h(in)f(Matlab)g([24)q(])g(with)g(time)30 2191 y(in)o(tegration)i(p)q(erformed)g(b)o(y)g(a)h(v)m(ariable)f (step{size)g(Runge{Kutta)i(metho)q(d.)30 b(W)l(e)19 b(sho)o(w)i (details)30 2263 y(for)15 b(only)f(the)h(real)f(part)h(of)g Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\);)13 b(the)i(imaginary)e(part)i(is)f (dealt)h(with)f(similarly)l(.)k(W)l(e)c(write)g(the)30 2336 y(real)i(part)h(of)f Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))15 b(at)i(the)f(p)q(oin)o(ts)g Fj(f)p Fl(x)787 2343 y Fk(n)810 2336 y Fj(g)g Fo(as)422 2484 y Fl(A)459 2491 y Fk(r)477 2484 y Fo(\()p Fl(x)524 2491 y Fk(n)547 2484 y Fl(;)8 b(t)p Fo(\))14 b(=)686 2450 y(1)p 676 2472 45 2 v 676 2518 a Fl(N)734 2422 y Fk(N)t Fe(\000)p Fi(1)736 2437 y Fg(X)740 2543 y Fk(k)q Fi(=0)819 2484 y Fl(X)859 2491 y Fk(k)881 2484 y Fo(\()p Fl(t)p Fo(\))8 b(exp)f(\()p Fl(ik)r(x)1118 2491 y Fk(n)1141 2484 y Fo(\))p Fl(;)83 b Fo(1)14 b Fj(\024)g Fl(n)g Fj(\024)f Fl(N)r(;)292 b Fo(\(5.1\))30 2628 y(where)765 2746 y Fl(x)793 2753 y Fk(n)830 2746 y Fo(=)882 2676 y Fg(\022)923 2713 y Fl(n)12 b Fj(\000)e Fo(1)p 923 2735 115 2 v 958 2780 a Fl(N)1043 2676 y Fg(\023)1088 2746 y Fo(2)p Fl(\031)r(:)936 2893 y Fo(10)p eop %%Page: 9 10 9 9 bop 30 28 a Fo(pap)q(er,)21 b([7],)g(w)o(e)g(will)f(consider)g (\014nite)h(subgroups)h(of)g(the)e(space)h(translation)h(symmetry)c Fl(r)1796 35 y Fk(\013)1841 28 y Fo(|)30 101 y(this)k(corresp)q(onds)g (to)h(adding)f(p)q(erturbations)g(whic)o(h)g(ha)o(v)o(e)e(p)q(erio)q(d) j Fm(gr)n(e)n(ater)e Fo(than)h(that)g(of)g(the)30 173 y(underlying)16 b(solution.)103 245 y(W)l(e)g(de\014ne)g(the)g(t)o(w)o (o)g(subgroups)628 362 y(\006)663 369 y Fi(1)697 362 y Fo(=)e Fj(f)p Fl(I)t(;)8 b(s)845 369 y Fi(1)864 362 y Fj(g)p Fl(;)72 b Fo(\006)1010 369 y Fi(2)1044 362 y Fo(=)14 b Fj(f)p Fl(I)t(;)8 b(s)1192 369 y Fi(1)1211 362 y Fl(s)1234 369 y Fi(2)1253 362 y Fj(g)p Fl(:)30 479 y Fo(Clearly)l(,)21 b(these)h(t)o(w)o(o)f(subgroups)i(are)f(b)q (oth)g(isomorphic)e(to)i Fd(Z)1237 486 y Fi(2)1279 479 y Fo(and)g(their)f(\014xed)g(p)q(oin)o(t)h(spaces)30 551 y(are)17 b(resp)q(ectiv)o(ely)d(the)j(spaces)g(of)g(ev)o(en)f(and)h (o)q(dd)h(p)q(erio)q(dic)e(functions.)23 b(These)17 b(are)f(equiv)m (alen)o(t)g(to)30 623 y(homogeneous)21 b(Neumann)f(and)i(homogeneous)f (Diric)o(hlet)e(b)q(oundary)k(conditions)e(resp)q(ectiv)o(ely)l(.)30 695 y(A)f(third)f(symmetry)e(whic)o(h)i(w)o(e)h(could)g(consider)f(is)h (the)g(pro)q(duct)h(of)f(the)g(previous)g(t)o(w)o(o)g(and)g(is)30 768 y(simply)14 b Fl(s)209 775 y Fi(2)229 768 y Fo(.)21 b(Ho)o(w)o(ev)o(er,)14 b(only)i(the)g(zero)g(function)g(is)g(\014xed)g (b)o(y)g Fl(s)1217 775 y Fi(2)1253 768 y Fo(and)g(so)h(this)f(is)h(of)f (no)h(in)o(terest.)103 840 y(There)g(is)g(another)h(group)g(whic)o(h)f (is)g(of)g(in)o(terest)f(where)h(w)o(e)g(com)o(bine)e(one)i(of)h(the)f (re\015ections)30 912 y(with)f(a)h(space)f(translation.)22 b(This)16 b(is)g(giv)o(en)g(b)o(y)574 1029 y(\006)609 1036 y Fi(3)643 1029 y Fo(=)e Fj(f)p Fl(I)t(;)8 b(s)791 1036 y Fi(1)809 1029 y Fl(s)832 1036 y Fi(2)852 1029 y Fl(;)g(r)896 1036 y Fk(\031)920 1029 y Fl(s)943 1036 y Fi(1)962 1029 y Fl(;)g(r)1006 1036 y Fk(\031)1030 1029 y Fl(s)1053 1036 y Fi(2)1072 1029 y Fj(g)14 b(')g Fd(Z)1198 1036 y Fi(2)1229 1029 y Fj(\002)c Fd(Z)1312 1036 y Fi(2)1332 1029 y Fl(:)30 1146 y Fo(W)l(e)16 b(note)g(that)363 1263 y Fl(r)385 1270 y Fk(\031)409 1263 y Fl(s)432 1270 y Fi(1)452 1263 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)i Fl(A)p Fo(\()p Fl(\031)e Fj(\000)f Fl(x;)d(t)p Fo(\))p Fl(;)72 b(r)1000 1270 y Fk(\031)1023 1263 y Fl(s)1046 1270 y Fi(2)1066 1263 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)i Fj(\000)p Fl(A)p Fo(\()p Fl(x)c Fo(+)h Fl(\031)r(;)d(t)p Fo(\))p Fl(:)30 1379 y Fo(Th)o(us,)18 b(functions)h(\014xed)e(b)o(y)h Fl(s)594 1386 y Fi(1)614 1379 y Fl(s)637 1386 y Fi(2)674 1379 y Fo(are)h(o)q(dd,)g(functions)f(\014xed)g(b)o(y)f Fl(r)1293 1386 y Fk(\031)1317 1379 y Fl(s)1340 1386 y Fi(1)1378 1379 y Fo(are)h(ev)o(en)f(ab)q(out)i Fl(\031)r(=)p Fo(2)f(and)30 1452 y(functions)13 b(\014xed)g(b)o(y)g Fl(r)439 1459 y Fk(\031)463 1452 y Fl(s)486 1459 y Fi(2)519 1452 y Fo(satisfy)g Fl(A)p Fo(\()p Fl(x)5 b Fo(+)g Fl(\031)r(;)j(t)p Fo(\))13 b(=)g Fj(\000)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\).)19 b(Clearly)13 b(if)f(t)o(w)o(o)i(of)f(these)g(are)h (satis\014ed)30 1524 y(then)h(so)i(is)e(the)g(third.)21 b(Th)o(us)16 b(Fix)o(\(\006)720 1531 y Fi(3)740 1524 y Fo(\))f(consists)h(of)g(functions)g(whic)o(h)e(are)i(o)q(dd)g(\(ab)q (out)h(zero\))e(and)30 1596 y(ev)o(en)e(ab)q(out)j Fl(\031)r(=)p Fo(2)e(whic)o(h)g(is)g(equiv)m(alen)o(t)f(to)h(imp)q(osing)g(a)h(Diric) o(hlet)d(b)q(oundary)j(condition)g(at)f Fl(x)g Fo(=)g(0)30 1668 y(and)19 b(a)h(Neumann)d(b)q(oundary)j(condition)f(at)g Fl(x)f Fo(=)g Fl(\031)r(=)p Fo(2.)29 b(W)l(e)19 b(note)g(that)g(a)g (translation)g(in)g(space)30 1741 y(of)h(a)h(function)f(in)f(Fix\(\006) 515 1748 y Fi(3)534 1741 y Fo(\))i(b)o(y)e Fl(\031)r(=)p Fo(2)h(giv)o(es)g(a)g(function)g(whic)o(h)f(is)h(ev)o(en)f(ab)q(out)j Fl(x)e Fo(=)g(0)g(and)h(o)q(dd)30 1813 y(ab)q(out)e Fl(x)d Fo(=)g Fl(\031)r(=)p Fo(2.)25 b(Th)o(us,)18 b(in)o(terc)o(hanging)f (the)g(b)q(oundary)i(conditions)f(giv)o(es)e(rise)h(to)h(a)g(conjugate) 30 1885 y(solution.)103 1957 y(W)l(e)f(are)g(in)o(terested)f(in)h (determining)e(the)i(stabilit)o(y)f(of)i(c)o(haotic)e(solutions)i(whic) o(h)f(ha)o(v)o(e)f(some)30 2029 y(symmetry)11 b(to)k(symmetry)o (-breaking)d(p)q(erturbations.)21 b(F)l(or)15 b(the)f(\014rst)h(t)o(w)o (o)f(cases)h(of)f(the)g(subgroups)30 2102 y(\006)65 2109 y Fi(1)101 2102 y Fo(and)j(\006)231 2109 y Fi(2)251 2102 y Fo(,)f(w)o(e)g(decomp)q(ose)f(the)h(space)g Fl(X)21 b Fo(as)815 2219 y Fl(X)e Fo(=)13 b Fl(X)965 2226 y Fk(e)995 2219 y Fj(\010)e Fl(X)1085 2226 y Fk(o)30 2335 y Fo(where)k Fl(X)210 2342 y Fk(e)243 2335 y Fo(consists)h(of)f(all)f(ev)o(en)g (functions)h(of)g(p)q(erio)q(d)h(2)p Fl(\031)g Fo(and)g Fl(X)1268 2342 y Fk(o)1302 2335 y Fo(consists)g(of)f(all)f(o)q(dd)i (functions)30 2408 y(of)e(p)q(erio)q(d)h(2)p Fl(\031)r Fo(.)20 b(This)14 b(is)g(the)f(isot)o(ypic)g(decomp)q(osition)g(asso)q (ciated)i(with)f(b)q(oth)h(the)e(groups)j(\006)1778 2415 y Fi(1)1811 2408 y Fo(and)30 2480 y(\006)65 2487 y Fi(2)85 2480 y Fo(.)103 2552 y(F)l(or)h(solutions)f(with)h(symmetry)c(\006)769 2559 y Fi(1)789 2552 y Fo(,)i(w)o(e)h(require)f(solutions)i(in)f Fl(X)1355 2559 y Fk(e)1388 2552 y Fo(=)e(Fix)o(\(\006)1565 2559 y Fi(1)1585 2552 y Fo(\))i(and)h(w)o(e)f(com-)30 2624 y(pute)h(the)g(dominan)o(t)f(Ly)o(apuno)o(v)h(exp)q(onen)o(t)g (asso)q(ciated)h(with)f(p)q(erturbations)h(in)f Fl(X)1632 2631 y Fk(o)1652 2624 y Fo(.)23 b(Similarly)l(,)30 2696 y(for)d(\006)143 2703 y Fi(2)163 2696 y Fo(,)h(w)o(e)e(compute)g (solutions)i(in)e Fl(X)783 2703 y Fk(o)823 2696 y Fo(=)h(Fix)o(\(\006) 1006 2703 y Fi(2)1026 2696 y Fo(\))g(and)h(calculate)e(the)h(dominan)o (t)f(Ly)o(apuno)o(v)30 2769 y(exp)q(onen)o(t)f(asso)q(ciated)i(with)f (p)q(erturbations)g(in)f Fl(X)997 2776 y Fk(e)1016 2769 y Fo(.)29 b(W)l(e)18 b(note)h(that)g(Keefe)e([22)q(])h(computed)f(the) 948 2893 y(9)p eop %%Page: 8 11 8 10 bop 30 28 a Fo(orbit)16 b(at)h(the)f(origin,)g(w)o(e)f(ha)o(v)o(e) h(that)338 167 y Fl(L)371 174 y Fk(\022)391 167 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)608 133 y Fl(d)p Fo(\()p Fl(\022)q(A)p Fo(\()p Fl(x;)c(t)p Fo(\)\))p 608 156 230 2 v 698 201 a Fl(d\022)842 95 y Fg(\014)842 125 y(\014)842 155 y(\014)842 184 y(\014)859 216 y Fk(\022)q Fi(=0)937 167 y Fo(=)999 133 y Fl(d)p Fo(\()p Fl(e)1066 115 y Fk(i\022)1097 133 y Fl(A)p Fo(\()p Fl(x;)g(t)p Fo(\)\))p 999 156 260 2 v 1104 201 a Fl(d\022)1263 95 y Fg(\014)1263 125 y(\014)1263 155 y(\014)1263 184 y(\014)1280 216 y Fk(\022)q Fi(=0)1358 167 y Fo(=)14 b Fl(iA)p Fo(\()p Fl(x;)8 b(t)p Fo(\))p Fl(:)30 305 y Fo(Similarly)l(,)485 377 y Fl(L)518 384 y Fk(\013)543 377 y Fl(A)p Fo(\()p Fl(x;)g(t)p Fo(\))13 b(=)g Fl(A)787 384 y Fk(x)809 377 y Fo(\()p Fl(x;)8 b(t)p Fo(\))p Fl(;)72 b(L)1034 384 y Fk(\014)1058 377 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)i Fl(A)1302 384 y Fk(t)1316 377 y Fo(\()p Fl(x;)8 b(t)p Fo(\))p Fl(:)103 476 y Fo(Th)o(us,)17 b(an)o(y)g(c)o(haotic)f(attractor)h(whic)o(h)f(is)h(not)g(spatially)f (uniform)g(has)h(three)f(zero)h(Ly)o(apuno)o(v)30 549 y(exp)q(onen)o(ts)k(asso)q(ciated)h(with)g(it.)35 b(If)21 b(the)g(attractor)h(lies)e(in)h(Fix)o(\(\006\))h(for)f(some)g(subgroup) h(\006)g(of)30 621 y(\000,)h(then)f(the)g(zero)g(Ly)o(apuno)o(v)g(exp)q (onen)o(ts)g(ma)o(y)e(o)q(ccur)i(in)g(di\013eren)o(t)f(\006-isot)o (ypic)g(comp)q(onen)o(ts.)30 693 y(These)16 b(can)g(b)q(e)g(determined) d(b)o(y)i(\014nding)h(whic)o(h)f(isot)o(ypic)g(comp)q(onen)o(ts)g(con)o (tain)g(the)h(tra)s(jectories)30 765 y Fl(LA)p Fo(\()p Fl(x;)8 b(t)p Fo(\))22 b(for)h(eac)o(h)f Fl(L)j Fj(2)g(L)p Fo(.)42 b(Ho)o(w)o(ev)o(er,)22 b(for)h(an)o(y)g(symmetry)-5 b(,)21 b Fl(L)1307 772 y Fk(\022)1327 765 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))21 b(and)j Fl(L)1626 772 y Fk(\014)1650 765 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))21 b(will)30 838 y(alw)o(a)o(ys)15 b(ha)o(v)o(e)f(the)h(same)e(symmetry)f(as)j(the)g (solution)g(tra)s(jectory)g(and)g(so)h(o)q(ccur)f(in)f Fl(W)1660 845 y Fi(1)1694 838 y Fo(=)g(Fix)o(\(\006\))30 910 y(whereas)j(in)e(some)h(cases,)g Fl(L)563 917 y Fk(\013)588 910 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))15 b(ma)o(y)f(o)q(ccur)j(in)f (a)g(di\013eren)o(t)g(isot)o(ypic)f(comp)q(onen)o(t.)103 982 y(As)c(a)g(simple)e(example,)h(consider)g(the)h(CGL)h(equation)f (with)g(homogeneous)g(Neumann)f(b)q(ound-)30 1054 y(ary)26 b(conditions)g(whic)o(h)f(is)h(equiv)m(alen)o(t)e(to)j(considering)f (solutions)g(whic)o(h)f(are)h(in)o(v)m(arian)o(t)g(un-)30 1126 y(der)21 b(the)g(re\015ectional)f(symmetry)e Fl(s)718 1133 y Fi(1)738 1126 y Fo(.)36 b(Th)o(us,)22 b(the)f(symmetri)o(es)d (of)k(the)f(solution)g(are)g(giv)o(en)g(b)o(y)30 1199 y(\006)e(=)g Fj(f)p Fl(I)t(;)8 b(s)237 1206 y Fi(1)256 1199 y Fj(g)19 b(')g Fd(Z)392 1206 y Fi(2)412 1199 y Fo(.)30 b(The)20 b(\006-isot)o(ypic)e(comp)q(onen)o(ts)h(are)g Fl(W)1195 1206 y Fi(1)1234 1199 y Fo(=)g(Fix)o(\(\006\))h(whic)o(h)e (consists)i(of)g(all)30 1271 y(ev)o(en)14 b(p)q(erio)q(dic)h(functions) g(and)h Fl(W)675 1278 y Fi(2)710 1271 y Fo(whic)o(h)e(consists)i(of)f (all)g(o)q(dd)h(p)q(erio)q(dic)f(functions.)21 b(When)15 b(only)30 1343 y(a)k(re\015ectional)f(symmetry)e(is)j(in)o(v)o(olv)o (ed,)e(the)i(isot)o(ypic)f(comp)q(onen)o(ts)g Fl(W)1402 1350 y Fi(1)1441 1343 y Fo(and)h Fl(W)1584 1350 y Fi(2)1623 1343 y Fo(are)g(often)g(re-)30 1415 y(ferred)f(to)i(as)f(the)g (symmetri)o(c)d(and)k(an)o(tisymm)o(etric)15 b(spaces)20 b(resp)q(ectiv)o(ely)l(.)26 b(In)19 b(this)g(case)g(w)o(e)f(see)30 1488 y(that)e Fl(L)168 1495 y Fk(\022)188 1488 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))14 b(and)j Fl(L)473 1495 y Fk(\014)497 1488 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))14 b(are)i(symmetri)o(c)d(functions)j(while)f Fl(L)1342 1495 y Fk(\013)1367 1488 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))14 b(is)i(an)o(tisymme)o(tric)o(.)30 1560 y(Th)o(us,)g(there)f (will)g(b)q(e)h(t)o(w)o(o)g(zero)g(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)g (asso)q(ciated)h(with)e(the)h(motion)f(in)h(Fix)o(\(\006\))30 1632 y(arising)i(from)e(the)i(rotational)g(and)g(time)d(translation)k (symmetri)o(es)c(and)j(one)g(in)f(the)g(symmetry-)30 1704 y(breaking)k(an)o(tisymme)o(tric)c(direction)j(arising)h(from)f (the)g(space)h(translation)h(symme)o(try)l(.)32 b(Since)30 1777 y(w)o(e)18 b(n)o(umerically)d(calculate)j(only)g(the)g(largest)h (Ly)o(apuno)o(v)g(exp)q(onen)o(t,)f(this)g(means)g(that)h(neither)30 1849 y(of)c(the)f(dominan)o(t)g(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)g (asso)q(ciated)g(with)g(the)f(symmetric)d(and)k(an)o(tisymmetri)o(c)30 1921 y(spaces)i(can)f(b)q(e)g(negativ)o(e.)103 1993 y(These)i (theoretical)g(predictions)f(are)i(consisten)o(t)f(with)g(the)g(n)o (umerical)e(results)i(obtained)h(b)o(y)30 2065 y(Keefe)i([21])g(who)i (considered)e(the)h(CGL)g(equation)g(with)g(homogeneous)f(Neumann)g(b)q (oundary)30 2138 y(conditions)16 b(and)h(who)g(alw)o(a)o(ys)f(found)h (t)o(w)o(o)f(zero)g(Ly)o(apuno)o(v)h(exp)q(onen)o(ts.)30 2322 y Fn(4)81 b(Symmetric)26 b(Solutions)30 2444 y Fo(As)20 b(men)o(tioned)e(in)h(the)h(previous)g(section,)g(it)f(is)h(the)g (symmetri)o(es)d(of)j(the)g(solutions)h(whic)o(h)e(are)30 2516 y(imp)q(ortan)o(t)j(when)h(computing)f(Ly)o(apuno)o(v)h(exp)q (onen)o(ts,)h(rather)f(than)h(the)f(symmetri)o(es)d(of)j(the)30 2588 y(equation)e(itself.)36 b(In)22 b(this)f(pap)q(er,)i(w)o(e)e (restrict)f(atten)o(tion)i(to)g(the)f(re\015ectional)f(symmetries)e Fl(s)1870 2595 y Fi(1)30 2661 y Fo(and)24 b Fl(s)155 2668 y Fi(2)199 2661 y Fo(de\014ned)f(in)g(Section)h(2)g(and)g (consider)f(p)q(erturbations)i(whic)o(h)e(break)h(the)f(re\015ectional) 30 2733 y(symmetri)o(es)10 b(but)k(ha)o(v)o(e)e(the)h(same)f(spatial)h (p)q(erio)q(d)h(as)f(the)g(underlying)g(solution.)20 b(In)13 b(a)g(subsequen)o(t)948 2893 y(8)p eop %%Page: 7 12 7 11 bop 30 28 a Fo(exp)q(onen)o(t)14 b(asso)q(ciated)g(with)g(the)g (isot)o(ypic)e(comp)q(onen)o(t)h Fl(W)1127 35 y Fk(k)1163 28 y Fo(can)h(then)f(b)q(e)h(found)h(using)f(the)g Fm(ve)n(ctor)30 101 y Fo(form)h(of)i(the)f(v)m(ariational)g(equation)g(restricted)f(to) i Fl(W)1049 108 y Fk(k)1087 101 y Fo(giv)o(en)e(b)o(y)547 204 y(_)535 218 y Fl(\036)564 225 y Fk(k)599 218 y Fo(=)f Fl(D)691 225 y Fk(k)713 218 y Fl(f)5 b Fo(\()p Fl(x)p Fo(\()p Fl(t)p Fo(\)\))p Fl(\036)893 225 y Fk(k)914 218 y Fl(;)72 b(\036)1029 225 y Fk(k)1051 218 y Fo(\(0\))14 b(=)g Fl(w)1214 225 y Fk(k)1249 218 y Fj(2)g Fl(W)1342 225 y Fk(k)1363 218 y Fl(;)413 b Fo(\(3.4\))30 334 y(and)17 b(is)f(giv)o(en)f(b)o(y)701 458 y Fl(\025)729 465 y Fi(1)p Fk(;k)792 458 y Fo(=)22 b(lim)844 488 y Fk(t)p Fe(!1)941 425 y Fo(1)p 941 447 25 2 v 944 493 a Fl(t)978 458 y Fo(log)10 b Fj(k)p Fl(\036)1104 465 y Fk(k)1125 458 y Fo(\()p Fl(t)p Fo(\))p Fj(k)p Fl(:)30 575 y Fo(Note)i(that)g(only)g (the)g(v)o(ector)f(equation)h(needs)f(to)i(b)q(e)f(in)o(tegrated)f(to)i (\014nd)f(the)g(dominan)o(t)f(Ly)o(apuno)o(v)30 647 y(exp)q(onen)o(t)j (as)i(a)f(randomly)e(c)o(hosen)i(initial)e(condition)h Fl(\036)1087 654 y Fk(k)1108 647 y Fo(\(0\))h(=)e Fl(w)1271 654 y Fk(k)1307 647 y Fo(will)h(swiftly)f(line)h(up)h(with)f(the)30 720 y(direction)h(in)h(phase)h(space)f(of)h(greatest)f(expansion)h(or)f (least)h(con)o(traction.)103 792 y(These)24 b(ideas)g(generalise)f (naturally)h(to)g(the)g(ev)o(olution)f(of)h(partial)g(di\013eren)o (tial)f(equations)30 864 y(de\014ned)g(on)g(appropriate)h(Hilb)q(ert)e (spaces.)41 b(In)23 b(particular,)h(the)f(isot)o(ypic)f(decomp)q (osition)g(for)30 936 y(PDE's)17 b(is)g(w)o(ell)f(de\014ned)g(and)i(an) o(y)e(linear)h(partial)f(di\013eren)o(tial)g(op)q(erator)i(whic)o(h)e (comm)o(utes)e(with)30 1009 y(all)k(the)h(elemen)o(ts)d(of)j(a)g(group) g(lea)o(v)o(es)f(the)g(isot)o(ypic)g(comp)q(onen)o(ts)g(asso)q(ciated)h (with)g(that)g(group)30 1081 y(in)o(v)m(arian)o(t)e([5].)26 b(Th)o(us,)18 b(the)g(linear)f(systems)g(de\014ned)g(b)o(y)h(\(3.4\))g (are)g(also)g(w)o(ell-de\014ned)f(when)h Fl(f)23 b Fo(is)30 1153 y(a)c(partial)g(di\013eren)o(tial)e(op)q(erator.)30 b(Finally)l(,)18 b(the)h(dominan)o(t)e(Ly)o(apuno)o(v)i(exp)q(onen)o(t) g(for)g(a)g(partial)30 1225 y(di\013eren)o(tial)11 b(equation)h(can)g (b)q(e)g(de\014ned)g(in)f(terms)g(of)h(the)g(rate)g(of)g(asymptotic)f (exp)q(onen)o(tial)g(gro)o(wth)30 1298 y(of)17 b(the)f(length)g(of)g(a) h(v)o(ector)e([15])h(and)h(is)f(de\014ned)g(b)o(y)515 1424 y Fl(\025)543 1431 y Fi(1)p Fk(;k)606 1424 y Fo(=)e(lim)8 b(sup)691 1464 y Fk(t)p Fe(!1)821 1391 y Fo(1)p 821 1413 V 824 1459 a Fl(t)858 1424 y Fo(log)q Fj(f)p Fo(sup)964 1464 y Fk(x)984 1469 y Fc(0)1076 1424 y Fo(sup)1028 1467 y Fe(jj)p Fk(\036)1069 1473 y Fb(k)1087 1467 y Fi(\(0\))p Fe(jj\024)p Fi(1)1206 1424 y Fj(jj)p Fl(\036)1263 1431 y Fk(k)1283 1424 y Fo(\()p Fl(t)p Fo(\))p Fj(jjg)p Fl(;)30 1567 y Fo(where)13 b Fl(\036)197 1574 y Fk(k)218 1567 y Fo(\()p Fl(t)p Fo(\))f(is)h(the)g(solution)g(of)h(the)e(v)m (ariational)i(equation)f(\(3.4\))g(and)g Fl(x)1395 1574 y Fi(0)1428 1567 y Fo(is)g(the)g(initial)e(condition)30 1639 y(for)22 b(the)g(PDE.)g(Th)o(us,)h(the)f(whole)g(framew)o(ork)e (describ)q(ed)h(in)h(this)g(section)f(applies)h(to)g(partial)30 1711 y(di\013eren)o(tial)15 b(equations)h(as)h(w)o(ell)e(as)i(to)g (ordinary)f(di\013eren)o(tial)f(equations.)30 1870 y Ff(3.2)66 b(Con)n(tin)n(uous)23 b(symmetries)30 1974 y Fo(As)15 b(men)o(tioned)e(in)i(Section)f(2,)h(the)g(CGL)h(equation)f (has)h(three)f(con)o(tin)o(uous)g(symmetri)o(es,)d(namely)30 2046 y Fl(\022)q Fo(,)j Fl(r)105 2053 y Fk(\013)145 2046 y Fo(and)h Fl(\034)260 2053 y Fk(\014)284 2046 y Fo(.)21 b(It)15 b(w)o(as)h(sho)o(wn)g(in)f([6,)g(Theorem)f(2.15])i(that)f(con)o (tin)o(uous)h(symmetri)o(es)c(giv)o(e)j(rise)g(to)30 2119 y(zero)j(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)g(pro)o(vided)f(that)i (the)f(con)o(tin)o(uous)f(symmetry)e(acts)j(on)h(the)f(solution)30 2191 y(non-trivially)l(.)26 b(This)19 b(result)f(is)g(based)h(on)g(the) f(fact)g(that)h(if)f Fl(x)p Fo(\()p Fl(t)p Fo(\))f(is)h(a)h(tra)s (jectory)f(of)g(\(3.1\),)h(then)30 2263 y Fl(Lx)p Fo(\()p Fl(t)p Fo(\))14 b(is)g(a)h(solution)f(of)h(the)f(v)m(ariational)h (equation)f(for)h(all)f Fl(L)g Fj(2)g(L)h Fo(where)f Fj(L)g Fo(is)h(the)f(Lie)g(algebra)h(of)30 2335 y(\000.)23 b(If)16 b(a)h(one-parameter)e(subgroup)j(acts)f(trivially)e(on)i(the)f (tra)s(jectory)g Fl(x)p Fo(\()p Fl(t)p Fo(\))g(and)h(if)f Fl(L)e Fj(2)h(L)i Fo(is)f(the)30 2408 y(corresp)q(onding)k(elemen)o(t)c (of)j(the)f(Lie)h(algebra,)g(then)g Fl(Lx)p Fo(\()p Fl(t)p Fo(\))f(=)g(0)h(and)g(so)h(in)e(this)h(case,)g(there)f(is)30 2480 y(not)g(a)g(zero)f(Ly)o(apuno)o(v)h(exp)q(onen)o(t)g(asso)q (ciated)g(with)g(this)f(symmetry)l(.)22 b(This)c(is)f(a)h (generalisation)30 2552 y(of)c(the)g(w)o(ell)f(kno)o(wn)i(result)e (that)i(autonomous)g(di\013eren)o(tial)e(equations)h(ha)o(v)o(e)f(one)i (zero)f(Ly)o(apuno)o(v)30 2624 y(exp)q(onen)o(t)i(whic)o(h)g(is)g(due)g (to)g(the)g(con)o(tin)o(uous)g(time)f(translation)h(symmetry)l(.)103 2696 y(W)l(e)h(denote)g(b)o(y)f Fl(L)447 2703 y Fk(\022)467 2696 y Fo(,)h Fl(L)531 2703 y Fk(\013)573 2696 y Fo(and)g Fl(L)701 2703 y Fk(\014)742 2696 y Fo(the)g(elemen)o(ts)d(of)k(the)e (Lie)h(algebra)h(asso)q(ciated)g(with)f(the)f Fl(\022)q Fo(,)30 2769 y Fl(r)52 2776 y Fk(\013)97 2769 y Fo(and)21 b Fl(\034)217 2776 y Fk(\014)261 2769 y Fo(symmetri)o(es)c(resp)q (ectiv)o(ely)l(.)31 b(Since)19 b(the)h(Lie)g(algebra)h(is)f(the)g (tangen)o(t)g(to)h(the)f(group)948 2893 y(7)p eop %%Page: 6 13 6 12 bop 30 28 a Ff(3.1)66 b(Classi\014cation)23 b(of)f(Ly)n(apuno)n(v) i(exp)r(onen)n(ts)30 133 y Fo(F)l(or)16 b(a)h(general)f Fl(n)p Fo({dimensional)f(ODE)718 255 y(_)-22 b Fl(x)13 b Fo(=)h Fl(f)5 b Fo(\()p Fl(x)p Fo(\))p Fl(;)73 b(x)p Fo(\(0\))14 b(=)g Fl(x)1169 262 y Fi(0)1188 255 y Fl(;)588 b Fo(\(3.1\))30 377 y(w)o(e)16 b(\014nd)g(the)g(Ly)o(apuno)o(v)h(exp)q (onen)o(ts)f(b)o(y)g(in)o(tegrating)g(the)g(v)m(ariational)g(equation) 658 486 y(_)647 499 y(\010)e(=)g Fl(D)q(f)5 b Fo(\()p Fl(x)p Fo(\()p Fl(t)p Fo(\)\)\010)p Fl(;)73 b Fo(\010\(0\))15 b(=)f Fl(I)t(;)524 b Fo(\(3.2\))30 621 y(where)16 b Fl(D)q(f)5 b Fo(\()p Fl(x)p Fo(\()p Fl(t)p Fo(\)\))17 b(is)f(the)g(Jacobian)h(of)f Fl(f)22 b Fo(ev)m(aluated)16 b(at)g Fl(x)p Fo(\()p Fl(t)p Fo(\),)g(the)g(solution)g(of)h(\(3.1\).)k(W)l(e)16 b(de\014ne)706 743 y(\003)d(=)22 b(lim)805 773 y Fk(t)p Fe(!1)889 743 y Fo([\010\()p Fl(t)p Fo(\))994 723 y Fk(T)1020 743 y Fo(\010\()p Fl(t)p Fo(\)])1125 723 y Fi(1)p Fk(=)p Fi(2)p Fk(t)1192 743 y Fl(;)584 b Fo(\(3.3\))30 865 y(pro)o(vided)14 b(this)h(limit)c(exists,)j(where)h(the)f(sup)q(erscript)h(\\)p Fl(T)7 b Fo(")15 b(denotes)f(the)h(matrix)e(transp)q(ose.)22 b(The)30 937 y(Multiplicativ)o(e)15 b(Ergo)q(dic)j(Theorem)f(of)h (Oseledec)f([18,)h(27])g(states)g(that)h(this)f(limit)d(exists)i(for)h Fl(\026)p Fo(-)30 1010 y(almost)c(all)h Fl(x)281 1017 y Fi(0)301 1010 y Fo(,)g(where)g Fl(\026)g Fo(is)h(the)f(in)o(v)m (arian)o(t)f(measure)g(asso)q(ciated)j(with)e(the)g(attractor)h(of)f (\(3.1\).)22 b(If)30 1082 y(the)13 b(eigen)o(v)m(alues)f(of)i(\003)f (are)h Fl(m)582 1089 y Fk(i)595 1082 y Fo(,)g Fl(i)f Fo(=)h(1)p Fl(;)8 b Fo(2)p Fl(;)g(:)g(:)g(:)16 b(;)8 b(n)p Fo(,)14 b(then)f(the)g(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)f(of)h (the)f(solution)30 1154 y Fl(x)p Fo(\()p Fl(t)p Fo(\))j(are)636 1226 y Fl(\025)664 1233 y Fk(i)693 1226 y Fo(=)d(log)d Fj(j)p Fl(m)873 1233 y Fk(i)886 1226 y Fj(j)p Fl(;)57 b(i)13 b Fo(=)h(1)p Fl(;)8 b Fo(2)p Fl(;)g(:)g(:)g(:)16 b(;)8 b(n:)103 1326 y Fo(If)17 b(\(3.1\))h(is)g(equiv)m(arian)o(t)e (with)i(resp)q(ect)f(to)h(some)f(compact)f(Lie)h(group)i(\000,)f(then)f (w)o(e)g(ha)o(v)o(e)g(the)30 1398 y(follo)o(wing)f(result)g([6].)30 1524 y Fd(Lemma)g(3.1)24 b Fm(L)n(et)18 b Fl(S)j Fm(b)n(e)e(an)g (invariant)h(set)f(c)n(ontaine)n(d)g(in)g Fo(Fix)o(\(\006\))g Fm(for)f(some)h(sub)n(gr)n(oup)f Fo(\006)h Fm(of)g Fo(\000)p Fm(.)30 1596 y(F)l(or)f Fl(x)150 1603 y Fi(0)186 1596 y Fj(2)f Fl(S)s Fm(,)i(let)h Fl(x)p Fo(\()p Fl(t)p Fo(\))e Fm(b)n(e)h(the)h(solution)f(of)g(\(3.1\))g(with)g Fl(x)p Fo(\(0\))d(=)h Fl(x)1283 1603 y Fi(0)1302 1596 y Fm(.)27 b(Then)19 b(the)h(solution)g Fo(\010\()p Fl(t)p Fo(\))e Fm(of)30 1668 y(the)g(variational)g(e)n(quation)g(\(3.2\))f(c)n (ommutes)h(with)g(the)g(action)g(of)f Fo(\006)p Fm(,)h(i.e.)807 1790 y Fl(\033)r Fo(\010\()p Fl(t)p Fo(\))13 b(=)h(\010\()p Fl(t)p Fo(\))p Fl(\033)30 1912 y Fm(for)j(al)r(l)i Fl(\033)c Fj(2)f Fo(\006)k Fm(and)f Fl(t)d Fj(\025)f Fo(0)p Fm(.)103 2039 y Fo(A)21 b(corollary)g(of)h(this)f(is)g(that)h(the)f(matrix)e (\003)i(de\014ned)h(b)o(y)e(\(3.3\))i(also)g(comm)o(utes)c(with)j(the) 30 2111 y(action)e(of)g(\006.)30 b(Th)o(us,)19 b(\003)g(can)g(b)q(e)g (put)h(in)o(to)e(blo)q(c)o(k)h(diagonal)g(form)f(and)i(so)f(its)g (eigen)o(v)m(alues,)f(and)30 2183 y(th)o(us)e(the)g(Ly)o(apuno)o(v)h (exp)q(onen)o(ts,)f(can)g(b)q(e)g(asso)q(ciated)i(with)e(particular)g (isot)o(ypic)f(comp)q(onen)o(ts.)103 2255 y(The)d(dominan)o(t)g(\(most) f(p)q(ositiv)o(e\))g(Ly)o(apuno)o(v)i(exp)q(onen)o(t)f(asso)q(ciated)h (with)f(eac)o(h)g(isot)o(ypic)f(com-)30 2328 y(p)q(onen)o(t)k(is)g(the) g(most)f(imp)q(ortan)o(t)g(one)i(for)f(our)h(purp)q(oses)g(since)e(it)h (indicates)f(whether)h(the)g(in)o(v)m(ari-)30 2400 y(an)o(t)h(set)h Fl(S)i Fo(is)d(stable)g(with)h(resp)q(ect)f(to)g(p)q(erturbations)i (asso)q(ciated)f(with)f(the)g(particular)g(isot)o(ypic)30 2472 y(comp)q(onen)o(t)f(\(a)i(p)q(ositiv)o(e)e(dominan)o(t)g(Ly)o (apuno)o(v)i(exp)q(onen)o(t)e(implies)f(that)i Fl(S)k Fo(is)15 b(unstable)i(to)f(suc)o(h)30 2544 y(p)q(erturbations\).)22 b(Using)16 b(the)g(blo)q(c)o(k)g(diagonal)h(form)e(of)i(\003)f(these)g (are)g(easily)f(computed.)103 2616 y(F)l(or)d(a)g(particular)f (\006-isot)o(ypic)g(comp)q(onen)o(t)g Fl(W)959 2623 y Fk(k)992 2616 y Fo(w)o(e)g(kno)o(w)h(that)g Fl(D)q(f)5 b Fo(\()p Fl(x)p Fo(\()p Fl(t)p Fo(\)\))12 b(lea)o(v)o(es)f Fl(W)1669 2623 y Fk(k)1702 2616 y Fo(in)o(v)m(arian)o(t)30 2689 y(and)i(so)f(w)o(e)g(denote)g(its)f(restriction)g(to)i Fl(W)791 2696 y Fk(k)824 2689 y Fo(b)o(y)f Fl(D)928 2696 y Fk(k)950 2689 y Fl(f)5 b Fo(\()p Fl(x)p Fo(\()p Fl(t)p Fo(\)\))h(:)g Fl(W)1173 2696 y Fk(k)1206 2689 y Fj(!)14 b Fl(W)1316 2696 y Fk(k)1337 2689 y Fo(.)20 b(The)12 b(dominan)o(t)f(Ly)o(apuno)o(v)948 2893 y(6)p eop %%Page: 5 14 5 13 bop 103 28 a Fo(The)18 b(signi\014cance)e(of)i(this)f(isot)o(ypic) g(decomp)q(osition)f(is)h(that)h(for)g(a)g(linear)e(op)q(erator)j Fl(L)e Fo(satis-)30 101 y(fying)f(\(2.3\),)g(all)g(the)g(isot)o(ypic)f (comp)q(onen)o(ts)g(are)i(in)o(v)m(arian)o(t)e(under)h Fl(L)p Fo(,)g(that)h(is)810 218 y Fl(L)d Fo(:)f Fl(W)930 225 y Fk(k)966 218 y Fj(!)g Fl(W)1075 225 y Fk(k)1097 218 y Fl(:)30 335 y Fo(This)j(results)g(in)g(a)h(blo)q(c)o(k)f (diagonal)h(structure)f(to)g(the)g(linear)g(op)q(erator)h Fl(L)p Fo(.)103 407 y(W)l(e)i(assume)g(that)h Fl(W)517 414 y Fi(1)556 407 y Fo(is)g(asso)q(ciated)g(with)f(the)h(trivial)e (irreducible)f(represen)o(tation)i Fl(\015)j Fo(=)e Fl(I)30 480 y Fo(for)d(all)e Fl(\015)i Fj(2)d Fo(\000)j(and)g(so)g Fl(W)509 487 y Fi(1)542 480 y Fo(=)d(Fix)o(\(\000\).)103 552 y(This)h(is)f(relev)m(an)o(t)g(to)h(the)f(calculation)g(of)h(Ly)o (apuno)o(v)g(exp)q(onen)o(ts)g(since)f(the)g(v)m(ariational)h(equa-)30 624 y(tion)g(in)o(v)o(olv)o(es)e(the)i(linear)g(op)q(erator)h Fl(g)747 631 y Fk(A)776 624 y Fo(\()p Fl(A)p Fo(\).)k(It)15 b(is)g(w)o(ell)f(kno)o(wn)h(and)h(easily)f(v)o(eri\014ed)e(that)j(if)f (there)30 696 y(is)h(a)h(tra)s(jectory)e Fl(A)p Fo(\()p Fl(t)p Fo(\))h(of)g(\(2.1\))h(suc)o(h)f(that)g Fl(A)p Fo(\()p Fl(t)p Fo(\))d Fj(2)h Fo(Fix\(\006\))i(for)g(some)g(subgroup)h (\006)g(of)f(\000,)h(then)551 813 y Fl(\033)r(g)604 820 y Fk(A)632 813 y Fo(\()p Fl(A)p Fo(\()p Fl(t)p Fo(\)\))c(=)h Fl(g)851 820 y Fk(A)880 813 y Fo(\()p Fl(A)p Fo(\()p Fl(t)p Fo(\)\))p Fl(\033)49 b Fo(for)16 b(all)g Fl(\033)g Fj(2)e Fo(\006)p Fl(:)30 931 y Fo(Th)o(us,)i(the)f(linear)h(op)q (erator)h Fl(g)605 938 y Fk(A)649 931 y Fo(decomp)q(oses)f(on)g(the)g (\006-isot)o(ypic)f(comp)q(onen)o(ts)g(of)h(the)g(space)g Fl(X)t Fo(.)30 1003 y(Th)o(us,)j(the)f(imp)q(ortan)o(t)f(symmetry)e (group)k(in)f(this)g(case)h(is)f(not)h(the)f(symmetry)d(group)k(\000)g (of)f(the)30 1075 y(equation)e(but)g(the)g(subgroup)h(\006)f(of)g (symmetrie)o(s)d(of)k(the)e(particular)h(solution)g(b)q(eing)g (considered.)103 1147 y(W)l(e)22 b(no)o(w)h(consider)f(the)g(CGL)i (equation)e(\(1.1\))h(on)f(the)h(one{dimensional)e(domain)g([0)p Fl(;)8 b Fo(2)p Fl(\031)r Fo(\))30 1220 y(with)22 b(p)q(erio)q(dic)h(b) q(oundary)g(conditions.)41 b(This)22 b(equation)h(has)g(b)q(oth)h(con)o (tin)o(uous)e(and)h(discrete)30 1292 y(symmetri)o(es)14 b(whic)o(h)h(are)h(giv)o(en)g(b)o(y)571 1401 y Fl(\022)q(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))41 b(=)g Fl(e)881 1383 y Fk(i\022)912 1401 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))p Fl(;)77 b(\022)15 b Fj(2)f Fo([0)p Fl(;)8 b Fo(2)p Fl(\031)r Fo(\))549 1473 y Fl(r)571 1480 y Fk(\013)596 1473 y Fl(A)p Fo(\()p Fl(x;)g(t)p Fo(\))40 b(=)h Fl(A)p Fo(\()p Fl(x)11 b Fo(+)g Fl(\013;)d(t)p Fo(\))p Fl(;)40 b(\013)15 b Fj(2)f Fo([0)p Fl(;)8 b Fo(2)p Fl(\031)r Fo(\))551 1546 y Fl(\034)572 1553 y Fk(\014)596 1546 y Fl(A)p Fo(\()p Fl(x;)g(t)p Fo(\))40 b(=)h Fl(A)p Fo(\()p Fl(x;)8 b(t)i Fo(+)h Fl(\014)s Fo(\))p Fl(;)41 b(\014)16 b Fj(2)f Fh(R)553 1618 y Fl(s)576 1625 y Fi(1)596 1618 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))40 b(=)h Fl(A)p Fo(\()p Fj(\000)p Fl(x;)8 b(t)p Fo(\))p Fl(:)30 1723 y Fo(These)14 b(symmetri)o(es)d(corresp)q(ond)j(resp)q (ectiv)o(ely)e(to)i(a)g(rotation)g(of)g(the)g(complex)d(amplitude,)h (space)30 1795 y(translation,)k(time)e(translation)i(and)h(a)f(spatial) g(re\015ection.)k(W)l(e)c(note)f(that)i(a)f(sp)q(ecial)f(case)h(of)g (the)30 1868 y(rotation)j(o)q(ccurs)g(when)g Fl(\022)g Fo(=)f Fl(\031)i Fo(and)f(this)f(giv)o(es)g(another)h(symmetry)d(of)j (order)f(t)o(w)o(o.)28 b(Since)18 b(this)30 1940 y(will)d(b)q(e)i(imp)q (ortan)o(t)e(in)h(our)g(later)g(w)o(ork,)g(w)o(e)g(de\014ne)613 2057 y Fl(\031)r(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(:=)h Fl(s)886 2064 y Fi(2)906 2057 y Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))k(=)i Fj(\000)p Fl(A)p Fo(\()p Fl(x;)8 b(t)p Fo(\))p Fl(:)30 2174 y Fo(Relativ)o(e)19 b(equilibria)g(are)i(asso)q (ciated)h(with)f(con)o(tin)o(uous)g(symme)o(tries)d(and)j(in)g(this)g (case,)g(the)g Fl(\022)30 2247 y Fo(symmetry)e(giv)o(es)j(rise)g(to)h (suc)o(h)g(solutions)g(whic)o(h)f(are)g(often)h(referred)f(to)h(as)g (rotating)g(w)o(a)o(v)o(es.)30 2319 y(These)16 b(w)o(ere)g(studied)g (in)g(some)f(detail)g(in)h([15])g(whic)o(h)g(included)f(a)i(linear)e (stabilit)o(y)g(analysis.)30 2503 y Fn(3)81 b(Ly)n(apuno)n(v)26 b(Exp)r(onen)n(ts)h(and)f(Symmetry)30 2624 y Fo(The)12 b(w)o(a)o(y)f(that)i(symmetry)8 b(a\013ects)k(the)g(determination)e(of) i(Ly)o(apuno)o(v)g(exp)q(onen)o(ts)g(w)o(as)h(considered)30 2696 y(in)k([6])g(and)g(applied)g(to)h(systems)e(of)h(coupled)g (oscillators.)24 b(W)l(e)17 b(brie\015y)f(review)g(the)h(main)f (results)30 2769 y(of)h(that)f(w)o(ork)g(and)h(will)e(then)i(apply)f (the)g(ideas)g(to)g(the)g(CGL)i(equation.)948 2893 y(5)p eop %%Page: 4 15 4 14 bop 30 28 a Fn(2)81 b(Symmetries)26 b(of)h(the)g(CGL)h(Equation)30 150 y Fo(In)12 b(this)f(section)h(w)o(e)f(brie\015y)g(outline)g(some)g (of)h(the)g(theory)f(of)i(dynamical)d(systems)g(with)i(symmetry)-5 b(,)30 222 y(concen)o(trating)21 b(on)g(its)g(applicabilit)o(y)d(to)k (the)e(CGL)i(equation.)36 b(Group)21 b(theory)g(is)g(the)g(natural)30 295 y(language)k(with)f(whic)o(h)f(to)h(discuss)h(symmetr)o(y;)g(see)e ([19)q(])g(for)h(man)o(y)f(results)h(concerning)f(the)30 367 y(application)16 b(of)h(symmet)o(ry)c(to)k(dynamical)d(systems)h (and)i(their)f(bifurcations.)103 439 y(W)l(e)g(consider)g(a)h(general)f (ev)o(olution)f(equation)h(of)h(the)f(form)682 560 y Fl(A)719 567 y Fk(t)748 560 y Fo(=)d Fl(g)r Fo(\()p Fl(A)p Fo(\))p Fl(;)73 b(g)16 b Fo(:)d Fl(X)18 b Fj(!)c Fl(X)q(;)561 b Fo(\(2.1\))30 680 y(where)15 b Fl(g)j Fo(is)e(assumed)f(to)h(b)q(e)f (a)h(nonlinear)g(op)q(erator)h(in)o(v)o(olving)d(spatial)i(deriv)m (ativ)o(es)e(and)i Fl(X)k Fo(is)c(an)30 752 y(appropriate)g(Hilb)q(ert) e(space)h(whic)o(h)g(incorp)q(orates)h(the)f(b)q(oundary)i(conditions.) k(W)l(e)15 b(also)h(assume)30 825 y(that)h Fl(g)h Fo(satis\014es)f(the) f(equiv)m(ariance)f(condition)625 945 y Fl(\015)s(g)r Fo(\()p Fl(A)p Fo(\))f(=)g Fl(g)r Fo(\()p Fl(\015)s(A)p Fo(\))64 b(for)17 b(all)e Fl(\015)i Fj(2)d Fo(\000)p Fl(;)504 b Fo(\(2.2\))30 1066 y(where)19 b(\000)h(is)g(a)g(compact)e (Lie)h(group.)33 b(F)l(or)19 b(an)o(y)h(subgroup)h(\006)f(of)f(\000,)i (w)o(e)e(de\014ne)g(the)h(\014xed)f(p)q(oin)o(t)30 1138 y(space)506 1210 y(Fix)o(\(\006\))14 b(=)g Fj(f)p Fl(A)f Fj(2)h Fl(X)k Fo(:)c Fl(\033)r(A)e Fo(=)i Fl(A)i Fo(for)h(all)e Fl(\033)h Fj(2)e Fo(\006)p Fj(g)p Fl(:)30 1309 y Fo(It)i(is)g(easily)f (v)o(eri\014ed)g(that)i(if)e Fl(g)k Fo(satis\014es)d(the)g(equiv)m (ariance)g(condition)g(\(2.2\),)g(then)744 1430 y Fl(g)g Fo(:)d(Fix)o(\(\006\))h Fj(!)g Fo(Fix)o(\(\006\))30 1550 y(for)i(all)e(subgroups)j(\006)f(of)g(\000)f(and)h(this)f(implies)e (that)j(the)f(\014xed)g(p)q(oin)o(t)g(spaces)h(are)f(in)o(v)m(arian)o (t)g(under)30 1623 y(the)h(\015o)o(w)h(of)f(the)g(nonlinear)g(equation) g(\(2.1\).)103 1695 y(Supp)q(ose)h(that)g(a)g(linear)e(op)q(erator)j Fl(L)e Fo(comm)o(utes)d(with)k(\000)f(so)h(that)696 1815 y Fl(\015)s(L)d Fo(=)g Fl(L\015)68 b Fo(for)16 b(all)g Fl(\015)h Fj(2)d Fo(\000)p Fl(:)566 b Fo(\(2.3\))30 1936 y(In)18 b(this)h(case,)g(there)f(are)h(man)o(y)e(more)h(in)o(v)m(arian) o(t)g(subspaces)h(for)g(the)g(linear)f(op)q(erator)i Fl(L)f Fo(whic)o(h)30 2008 y(in)o(v)o(olv)o(e)14 b(the)i(isot)o(ypic)f (comp)q(onen)o(ts)h(of)g(the)g(space)g Fl(X)t Fo(,)h(whic)o(h)e(w)o(e)h (no)o(w)h(describ)q(e.)103 2080 y(The)f(space)h Fl(X)j Fo(can)d(b)q(e)f(decomp)q(osed)g(as)h(a)f(direct)g(sum)f(of)h (irreducible)f(subspaces)817 2201 y Fl(X)k Fo(=)927 2154 y Fg(X)957 2259 y Fk(i)1008 2201 y Fj(\010)p Fl(V)1075 2208 y Fk(i)1089 2201 y Fl(:)30 2351 y Fo(If)k(w)o(e)g(group)h (together)f(all)g(the)g Fl(V)703 2358 y Fk(i)741 2351 y Fo(on)h(whic)o(h)f(\000)g(acts)h(isomorphically)l(,)e(then)i(w)o(e)e (obtain)i(the)30 2423 y(\000)p Fm(-isotypic)18 b(de)n(c)n(omp)n (osition)812 2495 y Fl(X)g Fo(=)922 2448 y Fg(X)948 2554 y Fk(k)1002 2495 y Fj(\010)p Fl(W)1087 2502 y Fk(k)30 2624 y Fo(where)c(eac)o(h)g(isot)o(ypic)f(comp)q(onen)o(t)h Fl(W)748 2631 y Fk(i)776 2624 y Fo(is)h(the)f(sum)f(of)i(isomorphic)e (irreducible)g(subspaces)i(whic)o(h)30 2696 y(are)f(asso)q(ciated)i (with)e(one)h(of)f(the)g(irreducible)f(represen)o(tations)h(of)g(the)h (group)g(\000.)21 b(Moreo)o(v)o(er,)13 b(this)30 2769 y(decomp)q(osition)i(is)h(unique)g([5,)g(20,)g(32)q(].)948 2893 y(4)p eop %%Page: 3 16 3 15 bop 30 28 a Fo(if)15 b(this)g(motion)f(is)i(c)o(haotic,)e(then)h (w)o(e)g(can)h(determine)d(normal)h(Ly)o(apuno)o(v)i(exp)q(onen)o(ts)f (asso)q(ciated)30 101 y(with)j(di\013eren)o(t)g(normal)g(directions,)f (whic)o(h)h(can)h(b)q(e)f(determined)e(using)j(the)g(symmetr)o(y)l(.)25 b(Th)o(us,)30 173 y(w)o(e)20 b(are)g(returning)f(to)i(the)e(problem)g (whic)o(h)g(originally)g(motiv)m(ated)g(the)h(w)o(ork)g(of)g(Y)l(amada) f(and)30 245 y(F)l(ujisak)m(a)d([33])f(but)g(in)g(a)g(more)f(general)h (framew)o(ork.)k(Blo)o(w)o(out)c(bifurcations)g(in)g(PDE)h(mean)e (\014eld)30 317 y(dynamo)d(mo)q(dels)g(asso)q(ciated)h(with)g(a)g (simple)d(re\015ectional)i(symmetry)d(ha)o(v)o(e)j(also)h(b)q(een)g (considered)30 390 y(in)k([12,)g(13)q(].)103 462 y(The)22 b(equation)g(that)h(w)o(e)e(will)g(apply)h(these)g(ideas)g(to)g(is)g (the)g(complex)e(Ginzburg{Landau)30 534 y(\(CGL\))d(equation)517 656 y Fl(A)554 663 y Fk(t)582 656 y Fo(=)d Fl(RA)d Fo(+)g(\(1)g(+)g Fl(i\027)s Fo(\))p Fj(r)976 636 y Fi(2)995 656 y Fl(A)g Fj(\000)g Fo(\(1)g(+)g Fl(i\026)p Fo(\))p Fl(A)p Fj(j)p Fl(A)p Fj(j)1363 636 y Fi(2)1382 656 y Fl(;)394 b Fo(\(1.1\))30 778 y(with)13 b Fl(A)h Fj(2)g Fh(C)25 b Fo(and)14 b Fl(R;)8 b(\027;)g(\026)15 b Fj(2)f Fh(R)p Fo(.)j(This)d(equation)f(w)o(as)h (originally)e(deriv)o(ed)g(as)i(a)g(generic)e(amplitude)30 850 y(equation)22 b(whic)o(h)f(describ)q(es)g(the)g(motion)g(near)h(to) g(p)q(oin)o(ts)g(of)g(instabilit)o(y)e(in)h(equations)h(whic)o(h)30 923 y(mo)q(del)14 b(\015uid)h(dynamics)e([25)q(,)h(30)q(,)h(31)q(])f (and)i(c)o(hemical)c(turbulence)i([23)q(].)20 b(It)15 b(has)h(also)f(b)q(een)g(studied)30 995 y(widely)e(as)h(an)g(in)o (teresting)e(mo)q(del)h(partial)g(di\013eren)o(tial)g(equation)g(with)h (a)g(ric)o(h)e(solution)i(structure)30 1067 y(as)j(it)f(is)h(kno)o(wn)g (to)g(ha)o(v)o(e)f(a)h(\014nite-dimensional)d(attractor)k(and)f (inertial)e(manifolds)h([11,)g(15)q(,)g(17)q(].)30 1139 y(F)l(or)e(some)e(parameter)h(v)m(alues,)h(it)f(is)g(kno)o(wn)h(to)g (sho)o(w)g(complicated)e(spatio{temp)q(oral)h(b)q(eha)o(viour,)30 1212 y(whic)o(h)g(w)o(e)g(will)g(refer)g(to)h(as)g(\\c)o(haos".)22 b(W)l(e)13 b(consider)h(only)f(the)h(cubic)e(nonlinearit)o(y)h(and)h (one)g(space)30 1284 y(dimension)h(so)h(that)h(the)f(solutions)h(of)f (\(1.1\))h(exhibit)e(soft)i(turbulence)e([8].)103 1356 y(The)g(CGL)g(equation)g(\(1.1\))g(is)f(equiv)m(arian)o(t)g(with)g (resp)q(ect)g(to)h(v)m(arious)g(symmetry)d(groups)j(and)30 1428 y(its)j(solutions)g(\(c)o(haotic)f(or)h(otherwise\))g(ma)o(y)e(p)q (ossess)j(one)f(or)g(more)e(of)i(these)g(symmetri)o(es.)23 b(The)30 1500 y(aim)14 b(of)h(this)g(w)o(ork)g(is)g(to)g(n)o (umerically)c(in)o(v)o(estigate)j(solutions)i(of)f(the)g(CGL)h (equation)f(whic)o(h)f(ha)o(v)o(e)30 1573 y(v)m(arious)h(symmetries)d (and)j(study)g(their)f(stabilit)o(y)g(or)h(instabilit)o(y)e(with)i (resp)q(ect)f(to)h(p)q(erturbations)30 1645 y(whic)o(h)h(break)g(the)g (symmetry)d(of)j(the)g(solutions.)103 1717 y(The)g(Ly)o(apuno)o(v)g (exp)q(onen)o(ts)g(of)g(the)f(CGL)i(equation)e(are)h(also)g(of)g(in)o (terest)f(for)h(other)f(reasons.)30 1789 y(In)h(particular,)f(rigorous) i(b)q(ounds)g(on)g(the)e(dimension)g(of)h(the)g(attractor)g(ha)o(v)o(e) f(b)q(een)h(obtained)h(b)o(y)30 1862 y(b)q(ounding)j(the)e(Ly)o(apuno)o (v)h(dimension)f(whic)o(h)g(is)g(deriv)o(ed)f(from)h(the)h(Ly)o(apuno)o (v)g(exp)q(onen)o(ts)f(via)30 1934 y(the)g(Kaplan-Y)l(ork)o(e)g(form)o (ula)f([8,)g(14)q(,)h(15)q(].)26 b(F)l(urther)18 b(analytical)g (results)f(for)i(the)f(CGL)h(equation)30 2006 y(ha)o(v)o(e)c(b)q(een)i (obtained)f(in)g([9,)g(16)q(].)103 2078 y(In)j(Section)h(2,)g(w)o(e)f (describ)q(e)g(the)h(symme)o(tries)d(of)i(the)h(CGL)g(equation)g(while) f(in)g(Section)g(3,)30 2151 y(w)o(e)i(brie\015y)f(review)g(previous)h (results)g(regarding)g(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)f(and)h (symmetry)-5 b(.)33 b(W)l(e)30 2223 y(also)24 b(sho)o(w)g(that)h(the)e (CGL)i(equation)e(ma)o(y)f(ha)o(v)o(e)h(three)g(zero)g(Ly)o(apuno)o(v)h (exp)q(onen)o(ts)g(due)g(to)30 2295 y(the)c(con)o(tin)o(uous)g (symmetri)o(es)d(of)k(the)e(equation.)33 b(In)20 b(Section)f(4,)i(w)o (e)f(describ)q(e)f(the)h(re\015ectional)30 2367 y(symmetri)o(es)9 b(of)k(the)f(solutions)g(that)h(w)o(e)f(consider)f(in)h(this)g(pap)q (er)h(while)e(Sections)h(5)g(and)h(6)g(describ)q(e)30 2439 y(the)22 b(n)o(umerical)d(metho)q(ds)i(used)h(and)g(the)g(results) f(obtained.)38 b(Finally)l(,)21 b(some)g(conclusions)h(are)30 2512 y(dra)o(wn)17 b(in)f(Section)f(7.)948 2893 y(3)p eop %%Page: 2 17 2 16 bop 30 28 a Fn(1)81 b(In)n(tro)r(duction)30 150 y Fo(P)o(attern)21 b(formation)g(in)g(nonlinear)h(partial)f(di\013eren) o(tial)f(equations)i(is)f(a)h(m)o(uc)o(h)e(studied)h(topic.)30 222 y(One)16 b(common)e(problem)h(is)i(determining)d(the)i(spatial)h (patterns)g(whic)o(h)e(can)i(o)q(ccur)g(when)f(a)h(spa-)30 295 y(tially)j(uniform)g(\(steady\))h(state)g(loses)g(stabilit)o(y)l(.) 35 b(In)21 b(1983,)i(Y)l(amada)e(and)g(F)l(ujisak)m(a)h([33])f(w)o(ere) 30 367 y(in)o(terested)e(in)h(the)g(stabilit)o(y)f(of)h(spatially)g (uniform)f Fm(chaotic)i Fo(solutions)g(of)f(a)h(nonlinear)f(partial)30 439 y(di\013eren)o(tial)13 b(equation)i(to)f(p)q(erturbations)i(whic)o (h)e(are)g(not)h(spatially)f(uniform.)19 b(In)14 b(order)h(to)g(study) 30 511 y(this)h(problem,)e(they)i(considered)f(a)i(\014nite)e (di\013erence)g(discretisation)h(of)g(the)g(PDE)h(whic)o(h)e(ga)o(v)o (e)h(a)30 583 y(\014nite-dimensional)g(system)g(of)i(coupled)g (oscillators.)26 b(The)17 b(uniform)g(state)h(for)g(the)g(PDE)g(corre-) 30 656 y(sp)q(onds)d(to)g(a)f(sync)o(hronised)g(state)g(for)g(the)g (coupled)f(oscillators.)21 b(Stabilit)o(y)12 b(of)i(this)g(uniform)f (state)30 728 y(w)o(as)k(describ)q(ed)f(in)g(terms)e(of)j(what)g(w)o(e) f(no)o(w)g(call)g(normal)f(Ly)o(apuno)o(v)i(exp)q(onen)o(ts.)k(Ho)o(w)o (ev)o(er,)14 b(the)30 800 y(only)i(n)o(umerical)e(results)i(presen)o (ted)f(in)h(this)g(w)o(ork)g(w)o(ere)g(for)g(t)o(w)o(o)g(coupled)g (Lorenz)h(systems.)103 872 y(This)e(w)o(ork)f(w)o(en)o(t)g(largely)f (unnoticed)h(un)o(til)f(P)o(ecora)i(and)f(Carroll)h([29])f (demonstrated)g(that)g(in)30 945 y(some)f(circumstances)f(it)i(is)g(p)q (ossible)g(to)h(sync)o(hronise)e(t)o(w)o(o)h(iden)o(tical)f(c)o(haotic) g(systems)g(b)o(y)h(linking)30 1017 y(them)f(with)h(a)g(common)e (signal.)21 b(Since)14 b(that)g(time,)e(there)i(has)h(b)q(een)f(m)o(uc) o(h)e(in)o(terest)h(in)h(and)h(study)30 1089 y(of)21 b(sync)o(hronisation)g(in)f(systems)g(of)h(coupled)f(oscillators,)i (one)f(in)o(teresting)e(application)i(b)q(eing)30 1161 y(secure)16 b(comm)o(uni)o(cation)e([26].)103 1233 y(Mathematically)c (sp)q(eaking,)j(sync)o(hronisation)f(corresp)q(onds)h(to)g(motion)e(in) h(an)h(in)o(v)m(arian)o(t)e(sub-)30 1306 y(space)i(whic)o(h)f(is)g (stable)h(with)f(resp)q(ect)g(to)h(p)q(erturbations)h(normal)d(to)i (the)g(subspace.)20 b(If)12 b(the)h(largest)30 1378 y(normal)19 b(Ly)o(apuno)o(v)h(exp)q(onen)o(t,)g(asso)q(ciated)h(with)f(p)q (erturbations)h(normal)d(to)j(the)e(subspace,)i(is)30 1450 y(negativ)o(e,)16 b(then)g(the)h(sync)o(hronised)f(state)h(has)g (a)g(p)q(ositiv)o(e)f(measure)g(basin)h(of)g(attraction,)g(whic)o(h)30 1522 y(ma)o(y)11 b(ho)o(w)o(ev)o(er)h(b)q(e)h(riddled)e(so)j(that)f (there)f(is)h(a)g(dense)f(set)h(of)g(p)q(ositiv)o(e)f(measure)f(in)i (an)o(y)f(neigh)o(b)q(our-)30 1595 y(ho)q(o)q(d)20 b(of)g(the)e(in)o(v) m(arian)o(t)g(subspace)i(whic)o(h)e(is)g(in)h(the)f(basin)h(of)g (another)h(attractor)f([1,)g(2,)f(3)q(,)g(28)q(].)30 1667 y(If)f(the)f(largest)i(normal)e(Ly)o(apuno)o(v)h(exp)q(onen)o(t)g (c)o(hanges)g(sign)g(as)h(a)g(parameter)d(is)i(v)m(aried,)g(then)g(a)30 1739 y(blo)o(w)o(out)f(bifurcation)f(o)q(ccurs)h(whic)o(h)f(ma)o(y)f(b) q(e)i(either)f(sup)q(ercritical)f(or)i(sub)q(critical)f([2].)21 b(In)15 b(order)30 1811 y(to)i(gain)g(a)g(deep)q(er)g(understanding)g (of)g(these)f(phenomena,)g(mo)q(del)f(equations)i(of)g(lo)o(w)g (dimension)30 1884 y(are)f(often)h(studied)f([4].)103 1956 y(The)23 b(sync)o(hronised)e(state)i(of)g(coupled)f(iden)o(tical)e (oscillators)j(is)f(a)h(natural)g(setting)f(for)h(an)30 2028 y(in)o(v)m(arian)o(t)14 b(subspace.)21 b(Another)15 b(natural)g(w)o(a)o(y)f(of)h(generating)g(in)o(v)m(arian)o(t)f (subspaces)h(is)f(b)o(y)h(the)f(use)30 2100 y(of)21 b(symmetry)l(.)33 b(Fixed)19 b(p)q(oin)o(t)j(spaces)f(are)g(in)o(v)m(arian)o(t)g(under)g (the)f(dynamics)g(of)h(a)h(system)d(with)30 2172 y(symmetry)e(and)k (blo)o(w)o(out)f(bifurcations)g(can)h(b)q(e)f(considered)g(from)f (these)h(in)o(v)m(arian)o(t)f(subspaces)30 2245 y(also.)i(Rings)15 b(of)g(coupled)f(oscillators)h(with)f(symmetry)e(w)o(ere)h(considered)h (b)o(y)h(Aston)f(and)i(Dellnitz)30 2317 y([6])e(and)i(it)e(w)o(as)h (sho)o(wn)g(ho)o(w)h(the)e(normal)g(Ly)o(apuno)o(v)h(exp)q(onen)o(ts)g (can)g(b)q(e)f(classi\014ed)h(according)g(to)30 2389 y(the)j(symmetry)d(of)j(the)g(problem.)26 b(This)18 b(also)h(leads)f (to)h(more)e(e\016cien)o(t)f(metho)q(ds)i(of)g(computing)30 2461 y(Ly)o(apuno)o(v)13 b(exp)q(onen)o(ts)g(since)g(the)f (linearisation)h(of)g(the)g(system)e(whic)o(h)i(is)g(used)g(to)g (compute)f(them)30 2534 y(can)17 b(b)q(e)g(decomp)q(osed)f(on)i(the)e (isot)o(ypic)g(comp)q(onen)o(ts)g(whic)o(h)h(are)g(asso)q(ciated)g (with)g(the)g(di\013eren)o(t)30 2606 y(irreducible)d(represen)o (tations)i(of)h(the)f(group)h(action.)103 2678 y(W)l(e)h(no)o(w)h (extend)f(the)g(ideas)h(of)g(Aston)f(and)i(Dellnitz)c([6])i(to)h (partial)g(di\013eren)o(tial)e(equations)30 2750 y(with)d(symmetry)l(.) j(Again,)e(the)f(\015o)o(w)h(of)f(a)h(PDE)g(is)f(in)o(v)m(arian)o(t)g (on)h(v)m(arious)g(\014xed)f(p)q(oin)o(t)g(spaces)h(and,)948 2893 y(2)p eop %%Page: 1 18 1 17 bop 473 251 a Fv(Symmetry)25 b(and)i(Chaos)g(in)g(the)303 375 y(Complex)f(Ginzburg{Landau)i(Equation.)507 500 y(I:)e (Re\015ectional)i(Symmetries)497 662 y Fu(Philip)16 b(J.)k(Aston)888 641 y Ft(\003)927 662 y Fu(and)f(Carlo)g(R.)h(Laing)1424 641 y Ft(y)406 752 y Fu(Departmen)n(t)e(of)i(Mathematics)e(and)h (Statistics,)696 842 y(Univ)n(ersit)n(y)f(of)i(Surrey)-5 b(,)691 931 y(Guildford)17 b(GU2)i(5XH,)746 1021 y(United)f(Kingdom)763 1142 y(Marc)n(h)h(12,)i(1999)860 1391 y Fs(Abstract)220 1487 y Fr(The)e(complex)h(Ginzburg{Landau)f(\(CGL\))f(equation)h(on)g (a)f(1{dimensional)j(domain)152 1555 y(with)11 b(p)q(erio)q(dic)j(b)q (oundary)d(conditions)h(has)f(a)g(n)o(um)o(b)q(er)g(of)g(di\013eren)o (t)g(symmetries.)19 b(Solutions)152 1622 y(of)i(the)g(CGL)g(ma)o(y)g (or)f(ma)o(y)h(not)g(b)q(e)h(\014xed)g(b)o(y)f(the)g(action)h(of)e (these)i(symmetries.)38 b(W)l(e)152 1690 y(in)o(v)o(estigate)19 b(the)f(stabilit)o(y)i(of)e(c)o(haotic)h(solutions)g(with)g(some)g (re\015ectional)h(symmetry)e(to)152 1758 y(p)q(erturbations)11 b(whic)o(h)h(break)f(that)f(symmetry)l(.)18 b(This)11 b(can)g(b)q(e)h(ac)o(hiev)o(ed)g(b)o(y)e(considering)j(the)152 1826 y(isot)o(ypic)h(decomp)q(osition)i(of)d(the)h(space)g(and)g (\014nding)h(the)f(dominan)o(t)g(Ly)o(apuno)o(v)g(exp)q(onen)o(t)152 1893 y(asso)q(ciated)22 b(with)h(eac)o(h)f(isot)o(ypic)g(comp)q(onen)o (t.)41 b(Our)22 b(n)o(umerical)i(results)e(indicate)i(that)152 1961 y(for)19 b(most)g(parameter)g(v)m(alues,)j(c)o(haotic)e(solutions) h(that)e(ha)o(v)o(e)g(b)q(een)i(restricted)g(to)e(lie)i(in)152 2029 y(in)o(v)m(arian)o(t)c(subspaces)f(are)g(unstable)h(to)e(p)q (erturbations)i(out)e(of)h(these)g(subspaces,)h(leading)152 2097 y(us)k(to)g(conclude)h(that)f(for)f(these)i(parameter)e(v)m(alues) i(arbitrary)f(initial)i(conditions)f(will)152 2164 y(generically)15 b(ev)o(olv)o(e)f(to)e(a)h(solution)h(with)g(the)f(minim)o(um)i(amoun)o (t)d(of)h(symmetry)g(allo)o(w)o(able.)152 2232 y(W)l(e)k(ha)o(v)o(e)f (also)g(found)h(a)f(small)h(region)g(of)f(parameter)g(space)h(in)g (whic)o(h)h(c)o(haotic)e(solutions)152 2300 y(that)e(are)h(ev)o(en)h (are)f(stable)g(with)h(resp)q(ect)g(to)e(o)q(dd)i(p)q(erturbations.)p 30 2610 744 2 v 86 2641 a Fq(\003)105 2656 y Fp(Corresp)q(onding)e (author)87 2694 y Fq(y)105 2709 y Fp(Address)j(from)d(1)i(Septem)o(b)q (er)g(1998:)21 b(Departmen)o(t)15 b(of)g(Mathematics,)g(301)g(Thac)o(k) o(era)o(y)h(Hall,)f(Univ)o(ersit)o(y)h(of)30 2769 y(Pittsburgh,)e (Pittsburgh,)g(P)o(ennsylv)n(ania)f(15260,)f(USA.)948 2893 y Fo(1)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF