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1、BasicConceptsinNonlinearDynamicsandChaos"Outofconfusioncomeschaos.Outofchaoscomesconfusionandfear.Thencomeslunch."AWorkshoppresentedattheSocietyforChaosTheoryinPsychologyandtheLifeSciencesmeeting,July31,1997atMarquetteUniversity,Miwaukee,Wisconsin.©KeithClaytonTableofC
2、ontents·IntroductiontoDynamicSystems·NonlinearDynamicSystems·BifurcationDiagram·SensitivitytoInitialConditions·SymptomsofChaos·Two-andThree-dimensionalDynamicSystems·FractalsandtheFractalDimension·NonlinearStatisticalTools·GlossaryIntroductiontoDynamicSystemsWhatisadyn
3、amicsystem?Adynamicsystemisasetoffunctions(rules,equations)thatspecifyhowvariableschangeovertime.Firstexample...Alice'sheightdiminishesbyhalfeveryminute...Secondexample...xnew=xold+yoldynew=xoldThesecondexampleillustratesasystemwithtwovariables,xandy.Variablexischanged
4、bytakingitsoldvalueandaddingthecurrentvalueofy.Andyischangedbybecomingx'soldvalue.Sillysystem?Perhaps.We'rejustshowingthatadynamicsystemisanywell-specifiedsetofrules.HerearesomeimportantDistinctions:·variables(dimensions)vs.parameters·discretevs.continuousvariables·sto
5、chasticvs.deterministicdynamicsystemsHowtheydiffer:·Variableschangeintime,parametersdonot.·Discretevariablesarerestrictedtointegervalues,continuousvariablearenot.·Stochasticsystemsareone-to-many;deterministicsystemsareone-to-oneThislastdistinctionwillbemadeclearerasweg
6、oalong...TermsThecurrentstateofadynamicsystemisspecifiedbythecurrentvalueofitsvariables,x,y,z,...Theprocessofcalculatingthenewstateofadiscretesystemiscallediteration.Toevaluatehowasystembehaves,weneedthefunctions,parametervaluesandinitialconditionsorstartingstate.Toill
7、ustrate...Consideraclassiclearningtheory,thealphamodel,whichspecifieshowqn,theprobabilityofmakinganerrorontrialn,changedfromonetrialtothenextqn+1=ßqnThenewerrorprobabilityisdiminishedbyß(whichislessthan1,greaterthan0).Forexample,letthetheprobabilityofanerrorontrial1equ
8、alto1,andßequal.9.Nowwecancalculatethedynamicsbyiteratingthefunction,andplottheresults.q1=1q2=ßq1=(.9)(1)=.9q3=(.9)q2
