Bifurcationsandresonance分岔和共振

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1、BifurcationandResonanceSijboHoltmanOverviewDynamicalsystemsResonanceBifurcationtheoryBifurcationandresonanceConclusionDynamicalsystemsWikipedia“Mathematicalformalizationforafixed"rule"whichdescribesthetimedependenceofapoint'spositioninitsambientspace.”InterpretationHowtodescribemathematically

2、anyprocessinvolvingmotionand/orchanges.DynamicalsystemsExamplesMilkywaySolarsystemClimateonearthMagmaPopulationGrowthCognitivetheoryDynamicalsystemsEvolutionruleusuallygivenimplicitlybyhowasystemchangesatanytime(e.g.byadifferentialequation).DynamicalsystemsForsimplesystemsknowingtrajectoriesi

3、senoughMorecomplexsystemsStabilityTypeoforbit:e.g.periodicorchaoticResonanceTypesofdynamicsChaosTwopointsthatstartclosedonotstaycloseResonanceMarchingsoldiersonbridgeTwoClocksonwall(ChristiaanHuygens)Moon-earth1:1resonanceElectricalcircuitsEtc.BifurcationtheoryBifurcation:smallchangeofevoluti

4、onrulecausesbigchangeinqualitativebehaviourofthesystem.Bifurcation&ResonanceCoupletwooscillatorswithsomefrequencyResonanceifratiooffrequenciesisrationalnumberSolutionofoscillatorisacircle(S1)Solutionoftwooscillatorsisonatorus(S1XS1=T2)Bifurcation&resonanceResonanceiftrajectoryclosesBifurcatio

5、n&resonanceConclusionGivenadynamicalsystemdescribingsomeprocessConditionsforresonanceareknownCorrespondingbifurcationdiagramknown