gilmore r. topological analysis of dynamical systems (rmp 70, 1998)(75s)

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1、TopologicalanalysisofchaoticdynamicalsystemsRobertGilmoreDepartmentofPhysics&AtmosphericScience,DrexelUniversity,Philadelphia,Pennsylvania19104Topologicalmethodshaverecentlybeendevelopedfortheanalysisofdissipativedynamicalsystemsthatoperateinthechaoticregime.Theywereoriginallyd

2、evelopedforthree-dimensionaldissipativedynamicalsystems,buttheyareapplicabletoall``low-dimensional''dynamicalsystems.Thesearesystemsforwhichthe¯owrapidlyrelaxestoathree-dimensionalsubspaceofphasespace.Equivalently,theassociatedattractorhasLyapunovdimensiondL,3.Topologicalmethod

3、ssupplementmethodspreviouslydevelopedtodeterminethevaluesofmetricanddynamicalinvariants.However,topologicalmethodspossessthreeadditionalfeatures:theydescribehowtomodelthedynamics;theyallowvalidationofthemodelssodeveloped;andthetopologicalinvariantsarerobustunderchangesincontrol

4、-parametervalues.Thetopological-analysisproceduredependsonidentifyingthestretchingandsqueezingmechanismsthatacttocreateastrangeattractorandorganizealltheunstableperiodicorbitsinthisattractorinauniqueway.Thestretchingandsqueezingmechanismsarerepresentedbyacaricature,abranchedman

5、ifold,whichisalsocalledatemplateoraknotholder.Thisturnsouttobeaversionofthedynamicalsysteminthelimitofin®nitedissipation.Thistopologicalstructureisidenti®edbyasetofintegerinvariants.Oneofthetrulyremarkableresultsofthetopological-analysisprocedureisthattheseintegerinvariantscanb

6、eextractedfromachaotictimeseries.Furthermore,self-consistencycheckscanbeusedtocon®rmtheintegervalues.Theseintegerscanbeusedtodeterminewhetherornottwodynamicalsystemsareequivalent;inparticular,theycandeterminewhetheramodeldevelopedfromtime-seriesdataisanaccuraterepresentationofa

7、physicalsystem.Conversely,theseintegerscanbeusedtoprovideamodelforthedynamicalmechanismsthatgeneratechaoticdata.Infact,theauthorhasconstructedadoublydiscreteclassi®cationofstrangeattractors.Theunderlyingbranchedmanifoldprovidesonediscreteclassi®cation.Eachbranchedmanifoldhasan`

8、`unfolding''orperturbationinwhichsomesubsetoforbitsisr