# Lecture-9.pdf - Lecture 9: CHAOS 混沌 G. Ron Chen Copyrighted...

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Discrete Data
Time SeriesContinuousversusDiscreteSmoothingversusSampling
FromContinuousToDiscrete… in the studies ofCHAOStheory …
Poincare SectionCross section:Poincare sectionBy considering the Poincare section, it is possibleto characterize the original continuous orbitContinuous orbitDiscrete orbit
Poincare SectionImage you have a very complicate orbit, e.g.,Rössler attractora =0.15, b =0.20,c =10.0
Poincare SectionImage you have a very complicate orbit, e.g.,Rössler attractorWhat does it look like in thePoincare Section?
Poincare MapforIterationsPoincare Map:A mapping fromcontinuous orbit todiscrete orbit(maynothave anexplicit formula)DEMO 1DEMO 2DEMO 3
Poincare SectionPeriod-4 oscillationPeriod-2 oscillationPeriod-1 oscillationchaosPeriodic:Finite crossingpoints in thePoincare sectionChaos:Infinite points inPoincare section
Bifurcation DiagramBifurcation diagram:it shows possible long-termvalues (fixed points or periodic orbits) of a systemas a function of thebifurcation parameterBifurcation parameterValues onthe Poincaresection
Discrete Chaos:Logistic MapR.M. May,Nature261(5560): 45967, 1976R. M. May,”Science197(4302): 463-465, 1977)](1)[()1(kxkxkxRobert M. May(1938 -)DEMO
For Example:
Logistic MapSelf-similarityLogistic Maphasself-similarityFractalstructure
),......(),......,2(),1(),0(.........)]1(1)[1()2()]0(1)[0()1()](1)[()1(kxxxxxxxxxxkxkxkxChaotic Maps:Infinitely many periodic orbits
Courtesy ofKazu Aihara2010 TokyoFashion ShowMusicfromLogistic Map
Characteristics of ChaosDomino ShowSensitivity to Initial ConditionsforbothContinuousandDiscreteChaos
Characteristics of ChaosNot convergentNot divergentNot periodicAre these also true fordiscrete chaos?
DiscreteChaos3 Key FactorsGlobalNot convergent, Not divergent, Not periodicLocalExpansion, Contraction, Fold-backKick out[not coordinating]Kick inNot convergent[Not periodic]Not divergentThink about itHow will the ball be bouncing around?

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