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Localized Structures in Forced Oscillatory Systems by Yiping Ma A Localized Structures in Forced Oscillatory Systems by Yiping Ma A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Edgar Knobloch, Chair Professor Jonathan Wurtele Professor Andrew Szeri Fall 2011 Localized Structures in Forced Oscillatory Systems Copyright 2011 by Yiping Ma 1 Abstract Localized Structures in Forced Oscillatory Systems by Yiping Ma Doctor of Philosophy in Physics University of California, Berkeley Professor Edgar Knobloch, Chair This dissertation studies nonlinear partial differential equations (PDEs) describing pattern formation, using a combination of analytical and numerical techniques. A major part focuses on spatially localized states in the 1:1 forced complex Ginzburg-Landau equation (FCGLE) 2 2 At =(μ + iν)A − (1 + iβ)|A| A +(1+iα)∇ A + γ, A = U + iV ∈ C,U,V∈ R, which is the normal form for a 1:1 resonantly forced Hopf bifurcation in spatially extended systems. One-dimensional (1D) steady localized states can be fruitfully studied using tools from dynamical systems theory. In particular, the localized states consisting of a Turing pattern embedded in a background equilibrium are shown to grow along its bifurcation curve via a new mechanism called defect-mediated snaking (DMS). In this growth mech- anism new rolls are nucleated from the center of the wavetrain, in contrast to standard homoclinic snaking observed in the generalized Swift-Hohenberg equation. The temporal dynamics of localized states outside the snaking region are mediated by successive phase slips resulting from the Eckhaus instability. The spatial dynamics of DMS are explained by an asymptotic theory near a saddle-center bifurcation in a planar reversible map. Aside from DMS, a new class of steady localized states consisting of phase-winding states, namely spatially periodic states with U and V out of phase, are shown to grow via collapsed snaking mediated by a central pacemaker defect rather than standard homoclinic snaking. Param- eter regimes exhibiting localized spatiotemporal chaos (STC) are also identified, and weak STC is interpreted in terms of its underlying coherent structures. Two-dimensional (2D) spatially localized states that exist in the same parameter regimes as their 1D counterparts include localized ring patterns, planar and circular localized hexagons, as well as 2D localized STC. These solutions are studied in detail using numerical continuation and direct numerical simulations. The final chapter derives a nonlocal pattern equation for weakly nonlinear Rayleigh- B´enard convection with large aspect ratio using the Bogoliubov method coupled with a dia- grammatic technique. A Lyapunov functional for the nonlocal pattern equation is found and a necessary condition for finding stable localized states in variational PDEs is formulated. i To my parents. ii Contents List of Figures vi List of Tables xvii 1 Introduction 1 1.1 Oscillons: from experiment to theory . .................... 1 1.2Anoverviewofmethods............................. 5 1.2.1 Asymptoticmethods........................... 5 1.2.2 Numericalcontinuation......................... 6 1.2.3 NumericalsolutionofPDE....................... 8 1.3Abriefsummaryofresults............................ 9 2 Background 12 2.1Localbifurcations................................. 12 2.1.1 Centermanifoldreduction........................ 14 2.1.2 Normalformtheory........................... 15 2.1.3 Localbifurcationsinflows........................ 18 2.1.4 Localbifurcationsinmaps........................ 21 2.2Globalbifurcations................................ 22 2.2.1 Thesaddlecase.............................. 23 2.2.2 Thesaddle-focuscase.......................... 25 2.3Homoclinicorbitsinreversiblesystems..................... 28 2.3.1 Lineartheory............................... 28 2.3.2 Thesaddlecase.............................. 29 2.3.3 Thesaddle-focuscase.......................... 30 2.3.4 Thesaddle-centercase.......................... 31 2.3.5 Otherhomoclinicorbits......................... 32 2.4Homoclinicsnakingandspatiallylocalizedstates............... 32 2.4.1 Standardhomoclinicsnaking...................... 33 2.4.2 Depinninginvariationalsystems.................... 36 2.4.3 Recent work on localized states . .................... 40 iii 3 Defect-mediated snaking 43 3.1Introduction.................................... 43 3.2TheforcedcomplexGinzburg-Landauequation................ 45 3.3The1:1resonance................................ 47 3.3.1 Collapsedsnaking............................ 49 3.3.2 Two-limitdefect-mediatedsnaking................... 49 3.3.3 Three-limitdefect-mediatedsnaking.................. 54 3.4Timeevolution.................................. 58 3.4.1 Thetwo-limitcase:depinning...................... 58 3.4.2 Thethree-limitcase........................... 62 3.5Discussion..................................... 63 3.6Appendix:ThesupercriticalTuringbifurcation................ 65 4 Depinning 70 4.1Introduction.................................... 70 4.2DepinningofType-ILS............................. 71 4.2.1 Timeevolution.............................. 71 4.2.2 Theoreticalexplanation......................... 73 4.3DepinningofType-IILS............................. 74 4.3.1 Timeevolution.............................. 75 4.3.2 Generalanalysisofdepinning...................... 78 4.3.3 Dynamicsnearthecodimension-2points................ 81 4.4Discussion..................................... 89 4.5Appendix:MovementofeigenvaluesfortheRBbifurcation......... 89 5 Homoclinic snakes bounded by a saddle-center periodic orbit 91 5.1Introduction.................................... 91 5.2Mathematicalsetting............................... 92 5.3Codimension-onebifurcation.......................... 94 5.4Codimension-twobifurcation.......................... 96 5.4.1 Outertangencycase........................... 97 5.4.2 Innertangencycase........................... 97 5.4.3 Scalingofthetangencycurves..................... 99 5.5Applications:Example1............................. 102 5.5.1 TheextendedS-Hequation....................... 103 5.5.2 Snakesmating.............................. 104 5.5.3 Explanationusingspatialdynamics.................. 107 5.6Theoryfornon-conservativereversiblesystems................ 109 5.6.1 Example 2: The 1 : 1 forcedCGLequation.............. 111 5.7Discussion..................................... 113 6 Localized phase-winding states 116 6.1Introduction.................................... 116 6.2The1:1case.................................... 117 6.2.1 Phasekinksbifurcatingfromsaddle-nodes............... 117 iv 6.2.2 Interactionbetweenfrontandsaddle-node............... 118 6.2.3 Phasewindingstates(PWS)...................... 119 6.2.4 SimplelocalizedPWS.......................... 123 6.3Temporaldynamics................................ 127 6.3.1 Travelingwaves.............................. 127 6.3.2 Dynamicsofdefects........................... 128 6.4Discussion..................................... 130 7 Classification of 1D localized states 132 7.1Introduction.................................... 132 7.2TheforcedcomplexGinzburg-Landauequation................ 133 7.3Propertiesoftheuniformstates......................... 134 7.3.1 Bifurcationanalysis........................... 135 7.3.2 Spatialeigenvalues............................ 138 7.3.3 Temporal stability . .................... 145 7.4Heterocliniccycles................................ 148 7.4.1 Localized phase-winding states (LPWS) (α =4,β =1,μ = 1) . 149 7.4.2 Defect-mediated snaking (DMS) (α = −1.5, β =6,μ = −1)..... 152 7.5Discussion..................................... 156 7.6 Appendix: Weakly nonlinear analysis near γ = γSN .............. 157 7.7 Appendix: Weakly nonlinear analysis near γ = γT .............. 159 8 2D localized states 163 8.1Introduction.................................... 163 8.2 Planar localized hexagons . .................... 164 8.3Circulartravelingfronts............................. 166 8.4Localizedringpatterns.............................. 168 8.5 Circular localized hexagons . .................... 170 8.6Discussion..................................... 173 9 Localized spatiotemporal chaos 175 9.1Introduction.................................... 175 9.2 Instabilities of uniform oscillations . .................... 176 9.3 LSTC from a Hopf instability . .................... 177 9.4LSTCfromDMSbranches............................ 180 9.5Discussion..................................... 182 10 Derivation of pattern equations 184 10.1Introduction.................................... 184 10.2 Rayleigh-B´enardConvection........................... 187 10.2.1BoussinesqEquations.......................... 187 10.2.2LinearTheory.............................. 188 10.2.3NonlinearTerms............................. 190 10.3TheExpansionProcedure............................ 191 10.3.1TheBogoliubovApproach........................ 191 v 10.3.2DiagrammaticNotation......................... 192 10.3.3AnIllustrativeCalculation....................... 195 10.4AConvectivePatternEquation......................... 197 10.4.1Setup................................... 197 10.4.2SecondOrder............................... 198 10.4.3ThirdOrder................................ 199 10.4.4TheEvolutionEquation......................... 200 10.5TheVariationalStructureandItsApplications................ 201 10.5.1VariationalStructureoftheNonlocalPatternEquation....... 201 10.5.2GeneralizedClausiusTheorem....................
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