%!PS-Adobe-2.0 %%Creator: dvipsk 5.58f Copyright 1986, 1994 Radical Eye Software %%Title: summ.dvi %%Pages: 1 %%PageOrder: Ascend %%BoundingBox: 0 0 596 842 %%DocumentFonts: Times-Bold Times-Roman Symbol Times-Italic %%EndComments %DVIPSCommandLine: dvips -o temp summ %DVIPSParameters: dpi=600, comments removed %DVIPSSource: TeX output 1997.09.29:1051 %%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 4 get round 4 exch put dup dup 5 get round 5 exch put 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 add]{ 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 showpage userdict /eop-hook known{eop-hook}if}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 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 %%BeginFont: Times-Bold % @psencodingfile{ % author = "S. Rahtz, P. MacKay, Alan Jeffrey, B. Horn, K. Berry", % version = "0.6", % date = "14 April 1995", % filename = "8r.enc", % email = "kb@cs.umb.edu", % address = "135 Center Hill Rd. // Plymouth, MA 02360", % codetable = "ISO/ASCII", % checksum = "xx", % docstring = "Encoding for TrueType or Type 1 fonts to be used with TeX." % } % % Idea is to have all the characters normally included in Type 1 fonts % available for typesetting. This is effectively the characters in Adobe % Standard Encoding + ISO Latin 1 + extra characters from Lucida. % % Character code assignments were made as follows: % % (1) the Windows ANSI characters are almost all in their Windows ANSI % positions, because some Windows users cannot easily reencode the % fonts, and it makes no difference on other systems. The only Windows % ANSI characters not available are those that make no sense for % typesetting -- rubout (127 decimal), nobreakspace (160), softhyphen % (173). quotesingle and grave are moved just because it's such an % irritation not having them in TeX positions. % % (2) Remaining characters are assigned arbitrarily to the lower part % of the range, avoiding 0, 10 and 13 in case we meet dumb software. % % (3) Y&Y Lucida Bright includes some extra text characters; in the % hopes that other PostScript fonts, perhaps created for public % consumption, will include them, they are included starting at 0x12. % % (4) Remaining positions left undefined are for use in (hopefully) % upward-compatible revisions, if someday more characters are generally % available. % % (5) hyphen appears twice for compatibility with both ASCII and Windows. % /TeXBase1Encoding [ % 0x00 (encoded characters from Adobe Standard not in Windows 3.1) /.notdef /dotaccent /fi /fl /fraction /hungarumlaut /Lslash /lslash /ogonek /ring /.notdef /breve /minus /.notdef % These are the only two remaining unencoded characters, so may as % well include them. /Zcaron /zcaron % 0x10 /caron /dotlessi % (unusual TeX characters available in, e.g., Lucida Bright) /dotlessj /ff /ffi /ffl /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef /.notdef % very contentious; it's so painful not having quoteleft and quoteright % at 96 and 145 that we move the things normally found there down to here. /grave /quotesingle % 0x20 (ASCII begins) /space /exclam /quotedbl /numbersign /dollar /percent /ampersand /quoteright /parenleft /parenright /asterisk /plus /comma /hyphen /period /slash % 0x30 /zero /one /two /three /four /five /six /seven /eight /nine /colon /semicolon /less /equal /greater /question % 0x40 /at /A /B /C /D /E /F /G /H /I /J /K /L /M /N /O % 0x50 /P /Q /R /S /T /U /V /W /X /Y /Z /bracketleft /backslash /bracketright /asciicircum /underscore % 0x60 /quoteleft /a /b /c /d /e /f /g /h /i /j /k /l /m /n /o % 0x70 /p /q /r /s /t /u /v /w /x /y /z /braceleft /bar /braceright /asciitilde /.notdef % rubout; ASCII ends % 0x80 /.notdef /.notdef /quotesinglbase /florin /quotedblbase /ellipsis /dagger /daggerdbl /circumflex /perthousand /Scaron /guilsinglleft /OE /.notdef /.notdef /.notdef % 0x90 /.notdef /.notdef /.notdef /quotedblleft /quotedblright /bullet /endash /emdash /tilde /trademark /scaron /guilsinglright /oe /.notdef /.notdef /Ydieresis % 0xA0 /.notdef % nobreakspace /exclamdown /cent /sterling /currency /yen /brokenbar /section /dieresis /copyright /ordfeminine /guillemotleft /logicalnot /hyphen % Y&Y (also at 45); Windows' softhyphen /registered /macron % 0xD0 /degree /plusminus /twosuperior /threesuperior /acute /mu /paragraph /periodcentered /cedilla /onesuperior /ordmasculine /guillemotright /onequarter /onehalf /threequarters /questiondown % 0xC0 /Agrave /Aacute /Acircumflex /Atilde /Adieresis /Aring /AE /Ccedilla /Egrave /Eacute /Ecircumflex /Edieresis /Igrave /Iacute /Icircumflex /Idieresis % 0xD0 /Eth /Ntilde /Ograve /Oacute /Ocircumflex /Otilde /Odieresis /multiply /Oslash /Ugrave /Uacute /Ucircumflex /Udieresis /Yacute /Thorn /germandbls % 0xE0 /agrave /aacute /acircumflex /atilde /adieresis /aring /ae /ccedilla /egrave /eacute /ecircumflex /edieresis /igrave /iacute /icircumflex /idieresis % 0xF0 /eth /ntilde /ograve /oacute /ocircumflex /otilde /odieresis /divide /oslash /ugrave /uacute /ucircumflex /udieresis /yacute /thorn /ydieresis ] def %%EndFont %%BeginProcSet: texps.pro TeXDict begin /rf{findfont dup length 1 add dict begin{1 index /FID ne 2 index /UniqueID ne and{def}{pop pop}ifelse}forall[1 index 0 6 -1 roll exec 0 exch 5 -1 roll VResolution Resolution div mul neg 0 0]/Metrics exch def dict begin Encoding{exch dup type /integertype ne{pop pop 1 sub dup 0 le{pop}{[}ifelse}{FontMatrix 0 get div Metrics 0 get div def} ifelse}forall Metrics /Metrics currentdict end def[2 index currentdict end definefont 3 -1 roll makefont /setfont load]cvx def}def /ObliqueSlant{dup sin S cos div neg}B /SlantFont{4 index mul add}def /ExtendFont{3 -1 roll mul exch}def /ReEncodeFont{/Encoding exch def}def end %%EndProcSet TeXDict begin 39158280 55380996 1000 600 600 (summ.dvi) @start /Fa 3 44 df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b 74[50 70[50 31[66 78[{ TeXBase1Encoding ReEncodeFont}3 100.000000 /Times-Italic rf /Fc 256[{}0 100.000000 /Symbol rf /Fd 204[37 37 50[{ TeXBase1Encoding ReEncodeFont}2 75.000000 /Times-Roman rf /Fe 256[{}0 75.000000 /Symbol rf /Ff 165[66 21[72 68[{TeXBase1Encoding ReEncodeFont}2 100.000000 /Times-Bold rf /Fg 75[33 29[50 1[44 44 24[44 50 50 72 50 50 28 39 33 50 50 50 50 78 28 50 28 28 50 50 33 44 50 44 50 44 11[72 61 3[55 72 72 4[33 72 6[72 15[50 2[25 33 25 2[33 33 37[55 2[{TeXBase1Encoding ReEncodeFont}45 100.000000 /Times-Roman rf /Fh 107[60 60 25[60 1[86 1[66 40 47 53 1[66 60 66 100 33 66 1[33 66 60 40 53 66 53 66 60 13[66 86 1[73 93 86 1[80 7[86 86 20[30 46[{TeXBase1Encoding ReEncodeFont}32 119.999947 /Times-Bold rf end %%EndProlog %%BeginSetup %%Feature: *Resolution 600dpi TeXDict begin %%PaperSize: a4 %%EndSetup %%Page: 1 1 1 0 bop 376 107 a Fh(Summary)30 b(of)f(the)h(Ph.D)n(.)g(thesis)g (\223Coupled)h(Oscillator)f(Netw)o(orks\224)1307 294 y(by)g(Carlo)g(Richard)i(Laing)120 552 y Fg(In)26 b(this)f(thesis)f(I)i (study)f(the)h(ef)n(fects)g(of)f(forced)i(symmetry)d(breaking)h(on)h (the)g(beha)n(viour)f(of)h(a)g(ring)f(of)120 702 y(three)g(coupled)f (modi\002ed)h(v)n(an)f(der)h(Pol)g(oscillators.)266 853 y(I)j(start)f(by)g(e)o(xamining)f(the)h(beha)n(viour)g(of)g(the)h (fully\226symmetric)d(system,)i(which)g(has)g Ff(D)3452 868 y Fd(3)3521 853 y Fg(sym-)120 1003 y(metry)-6 b(,)23 b(using)f(techniques)h(from)g(equi)n(v)n(ariant)f(Hopf)h(bifurcation)g (theory)g(to)g(determine)g(the)g(types)g(of)120 1154 y(orbit)i(created)i(in)f(Hopf)g(bifurcations.)33 b(I)27 b(then)f(perform)g(centre)g(manifold)f(and)h(normal)g(form)g(calcu-)120 1304 y(lations)c(on)i(the)f(system)g(studied)g(to)g(determine)g(the)h (stability)d(of)j(these)g(orbits.)29 b(I)24 b(also)f(use)h(the)f (theory)120 1455 y(of)33 b(weakly)g(coupled)f(oscillators)f(to)i(sho)n (w)f(the)g(e)o(xistence)g(of)h(a)h(number)e(of)h(orbits)f(not)g (created)h(in)120 1605 y(Hopf)22 b(bifurcations,)g(and)h(determine)f (their)g(stability)-6 b(.)28 b(These)22 b(tw)o(o)g(techniques)g (predict)g(dif)n(ferent)g(sta-)120 1756 y(bility)27 b(types)h(for)h (the)f(branch)h(of)f(e)o(xactly)g(synchronised)g(oscillations.)39 b(Interpolating)28 b(between)g(the)120 1906 y(parameter)i(re)o(gimes)e (where)i(these)f(analyses)f(are)i(v)n(alid)e(I)i(infer)f(the)g(e)o (xistence)g(of)g(a)h Ff(D)3229 1921 y Fd(3)3299 1906 y Fg(symmetric)120 2057 y(transcritical)24 b(bifurcation)g(of)h (periodic)g(orbits.)266 2207 y(I)36 b(then)e(in)l(v)o(estigate)f(the)i (ef)n(fects)g(of)g(breaking)g(the)g Ff(D)2188 2222 y Fd(3)2265 2207 y Fg(symmetry)e(to)i Ff(Z)2877 2222 y Fd(2)2954 2207 y Fg(and)g(then)g(remo)o(ving)120 2358 y(it)h(altogether)g(on)g(the)g(Hopf)g(bifurcations)g(and)g(the)g Ff(D)2105 2373 y Fd(3)2183 2358 y Fg(symmetric)f(transcritical)h (bifurcation)f(by)120 2508 y(e)o(xamining)c(the)i(perturbed)f(normal)g (forms)h(of)g(these)f(bifurcations.)54 b(The)33 b(study)f(of)h(the)f (perturbed)120 2659 y(normal)j(forms,)k(as)d(done)f(by)h(a)g(number)g (of)g(authors,)i(does)e(not)f(e)o(xplain)g(all)h(the)g(bifurcation)f (be-)120 2809 y(ha)n(viour)l(,)e(and)f(details)f(are)i(completed)e (numerically)-6 b(.)51 b(Applications)30 b(to)i(the)g(modelling)e(of)i (animal)120 2960 y(gaits)20 b(is)g(discussed.)29 b(A)21 b(partial)f(in)l(v)o(estigation)e(into)i(the)h(non\226symmetric)e (system)h(is)h(also)f(undertak)o(en)120 3110 y(which)k(sho)n(ws)e (phase)j(synchronisation,)d(a)i(ne)n(w)g(type)g(of)g(\223switching)f (bifurcation\224,)g(and)h(successi)n(v)o(e)120 3261 y(saddle\226node)k (bifurcations)f(of)i Fb(n)p Fg(\226tori)e(on)i Fa(\()p Fb(n)15 b Fa(+)g Fg(1)p Fa(\))p Fg(\226tori)27 b(as)i(a)g(mechanism)e (for)i(the)f(transition)f(from)120 3411 y(motion)c(on)i(an)g Fb(N)6 b Fg(\226torus)24 b(to)h(synchronised)e(beha)n(viour)-5 b(.)266 3562 y(I)36 b(also)g(study)e(the)i(general)g(case)g(of)g Fb(N)42 b Fg(identical)35 b(oscillators)g(dif)n(fusi)n(v)o(ely)e (coupled)i(in)h(a)g(ring,)120 3712 y(each)g(of)g(which)g(when)g (uncoupled)f(under)n(go)g(a)h(supercritical)g(Hopf)f(bifurcation)h(as)g (a)g(parameter)120 3863 y Fb(\265)c Fg(is)f(increased.)52 b(I)32 b(pro)o(v)o(e)f(that)g(for)h(small)f(enough)g(coupling,)i(the)e (system)g(as)h(a)g(whole)f(under)n(goes)120 4013 y(a)36 b(supercritical)f(Hopf)g(bifurcation)g(to)g(a)h(stable)f(rotating)f(w)o (a)n(v)o(e)i(as)f Fb(\265)g Fg(is)g(increased,)k(and)c(that)g(the)120 4163 y(magnitude)19 b(of)i(the)f(phase)h(dif)n(ference)g(between)f (neighbouring)f(oscillators)h(on)g(this)g(rotating)f(w)o(a)n(v)o(e)i (is)120 4314 y(either)g(a)g(maximum)e(or)h(a)h(minimum)e(\(i.e.)h (zero\),)j(depending)c(on)i(the)f(sign)g(of)h(the)g(coupling)e (constant.)266 4464 y(Also)27 b(included)f(are)i(a)g(numerical)e(study) g(of)i(bifocal)e(homoclinic)g(bifurcations)g(in)h(which)g(some)120 4615 y(of)h(the)f(theoretical)h(conjectures)f(re)o(garding)g(these)g (bifurcations)g(are)i(con\002rmed,)f(and)g(an)g(introduc-)120 4765 y(tion)h(to)h(the)g(softw)o(are)g(package)g(A)-5 b(UT)n(O,)30 b(the)f(use)h(of)g(which)g(is)f(demonstrated)g(on)h(the)g (Nos)6 b(\264)-39 b(e)30 b(equa-)120 4916 y(tions.)p eop %%Trailer end userdict /end-hook known{end-hook}if %%EOF