dynamical_systems

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1、Math118,Spring2,000ShlomoSternbergOctober10,20002Contents1Iterationsandxedpoints51.1Squareroots..............................51.2Newton'smethod...........................71.2.1Thegutsofthemethod....................71.2.2Avectorversion........................81.2.3Implementation.

2、.......................91.2.4Theexistencetheorem....................101.2.5Basinsofattraction......................131.3Theimplicitfunctiontheorem....................151.4Attractorsandrepellers.......................171.5Renormalizationgroup........................182Bifurcations

3、252.1Thelogisticfamily...........................252.1.10<1...........................252.1.21<2............................272.1.32<<3............................292.1.4=3..............................32p2.1.53<<1+6........................322.1.63:449499:::<<3:569946::::

4、................342.1.7Reprise.............................342.2Localbifurcations...........................352.2.1Thefold............................352.2.2Perioddoubling........................382.3Newton'smethodandFeigenbaum'sconstant...........432.4Feigenbaumrenormalization..

5、...................452.5Period3impliesallperiods.....................482.6Intermittency..............................513Conjugacy593.1Aneequivalence...........................593.2ConjugacyofTandL4.......................613.3Chaos.................................663.4Thesaw-toot

6、htransformationandtheshift............6734CONTENTS3.5Sensitivitytoinitialconditions...................713.6Conjugacyformonotonemaps...................723.7Sequencespaceandsymbolicdynamics...............754Spaceandtimeaverages854.1histogramsandinvariantdensities.................85

7、4.2thehistogramofL4.........................884.3Themeanergodictheorem.....................924.4thearcsinelaw...........................934.5TheBetadistributions........................1005Thecontractionxedpointtheorem1055.1Metricspaces.............................1055.2Complete

8、nessandcompletion.....................10