Arnold Random dynamical system.pdf

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1、RandomDynamicalSystemsLudwigArnoldInstitutffirDynamischeSystemeUniversitgt,Postfach330440D-28334Bremen,GermanyemaiI:arnold@mathematik.uni-bremen,deContentsIntroduction.................................2Lecture1:Theconceptofarandomdynamicalsystem.........31.1Metric,topolo

2、gical,andsmoothdynamics.........31.2Thenotionofarandomdynamicalsystem.........5Lecture2:Generationofrandomdynamicalsystems..........72.1Discretetime:Productsofrandommappings.......82.2Continuoustime1:Randomdifferentialequations....92.3Continuoustime2:Stochasticdifferent

3、ialequations...12Lecture3:Invariantmeasures.......................133.1InvariantmeasuresformeasurableRDS..........133.2InvariantmeasuresforcontinuousRDS..........17Lecture4:Themultiplicativeergodictheorem..............194.1Introduction.TheconceptofLyapunovexponents....1

4、94.2TheFhrstenberg-Kestentheorem..............204.3Themultiplieativeergodictheorem.............224.4Examples...........................24Lecture5:Smoothrandomdynamicalsystems.............285.1ThemultiplicativeergodictheoremforRDSonmanifolds285.2Invariantmanifolds.Pesin'

5、sformula............305.3NormalformshContinuouscoordinatetransformations.315.4Normalforms2:Smoothcoordinatetransformations...325.5Topologicalclassificationoflinearcocycles.Structuralsta-bility..............................32Lecture6:Towardsatheoryofstochasticbifurcation

6、..........336.1Whatisstochasticbifurcationtheory?...........336.2Aone-dimensionalexample.................35Furtherproblems.Outlook........................37References..................................38IntroductionThetheoryofrandomdynamicalsystemscontinues,extends,andun

7、itesvariousdevelopmentsinprobabilitytheoryanddynamicalsystems.Roughlyspeaking,arandomdynamicalsystemisacombinationof9ameasure-preservingdynamicalsystem(~,9",I?,(O(t))tcv)inthesenseofergodictheory,and9asmooth(ortopological)dynamicalsystem,typicallygeneratedbyadif-feren

8、tialordifferenceequation~=f(x)orXn+l=~(x~).Bothpartshavebeenverywellinvestigated,butthesymbiosisofthemleadstoa