Structural Stability and Hyperbolicity Violation i

arXiv:nlin/0408011v1[nlin.CD]5Aug2004StructuralStabilityandHyperbolicityViolationinHigh-DimensionalDynamicalSystemsD.J.Albers1,2,∗andJ.C.Sprott1,†1PhysicsDepartment,UniversityofWisconsin,Madison,WI537062SantaFeInstitute,1399HydeParkRoad,SantaFe,NM87501(Dated:February8,2008)ThisreportinvestigatesthedynamicalstabilityconjecturesofPalisandSmale,andPughandShubfromthestandpointofnumericalobservationandlaysthefoundationforastabilityconjecture.Asthedimensionofadissipativedynamicalsystemisincreased,itisobservedthatthenumberofpositiveLyapunovexponentsincreasesmonotonically,theLyapunovexponentstendtowardscontinuouschangewithrespecttoparametervariation,thenumberofobservableperiodicwindowsdecreases(atleastbelownumericalprecision),andasubsetofparameterspaceexistssuchthattopologicalchangeisverycommonwithsmallparameterperturbation.However,thisseeminglyinevitabletopologicalvariationisnevercatastrophic(thedynamictypeispreserved)ifthedimensionofthesystemishighenough.PACSnumbers:05.45.-a,89.75.-k,05.45.Tp,89.70.+c,89.20.FfKeywords:Partialhyperbolicity,dynamicalsystems,structuralstability,stabilityconjecture,Lyapunovexponents,complexsystemsI.INTRODUCTIONMuchoftheworkinthefieldsofdynamicalsystemsanddifferentialequationshave,forthelasthundredyears,entailedtheclassificationandunderstandingofthequalitativefeaturesofthespaceofdifferentiablemap-pings.Aprimaryfocusistheclassificationoftopologicaldifferencesbetweendifferentsystems(e.g.structuralsta-bilitytheory).Ofcourseoneoftheprimarydifficultiesischoosinganotionofbehaviorthatisnotsostrictthatitdifferentiatesontootrivialalevel,yetisstrictenoughthatithassomemeaning(Palis-Smaleusedtopologicalequivalence,Pugh-Shubuseergodicity).ThepreviousstabilityconjecturesarewithrespecttoanyCr(r≥0variesfromconjecturetoconjecture)perturbationallow-ingforvariationofthemapping,bothofthefunctionalform(withrespecttotheWhitneyCrtopology)andofparametervariation.Wewillconcernourselveswiththelatterissue.Unlikemuchworkinvolvingstabilitycon-jectures,ourworkisnumerical,anditfocusesonobserv-ableasymptoticbehaviorsinhigh-dimensionalsystems.Ourchiefclaimisthatgenerally,forhigh-dimensionaldynamicalsystemsinourconstruction,thereexistlargeportionsofparameterspacesuchthattopologicalvaria-tioninevitablyaccompaniesparametervariation,yetthetopologicalvariationhappensina“smooth,”non-erraticmanner.Letusstateourresultswithoutrigor,notingthatwewillsavemorerigorousstatementsforsection(III).StatementofResults1(Informal)Givenourpar-ticularimpositions(sections(IIA4)and(IIA1))upon∗Electronicaddress:[e-mail]†Electronicaddress:[e-mail]-timemapsfromcompactsetstothemselves,andaninvariantmeasure(usedforcalculat-ingLyapunovexponents),inthelimitofhighdimension,thereexistsasubsetofparameterspacesuchthatstricthyperbolicityisviolatedonanearlydense(andhenceunavoidable),yetzero-measure(withrespecttoLebesguemeasure),subsetofparameterspace.Amorerefinedversionofthisstatementwillcontainallofourresults.Formathematicians,wenotethatalthoughthestabilityconjectureofPalisandSmale[1]isquitetrue(asprovedbyRobbin[2],Robinson[3],andMa˜n´e[4]),weshowthatinhighdimensions,thisstructuralstabilitymayoccuroversuchsmallsetsintheparameterspacethatitmayneverbeobservedinchaoticregimesofparameterspace.Nevertheless,thislackofobserv-ablestructuralstabilityhasverymildconsequencesforappliedscientists.A.OutlineAsthispaperisattemptingtoreachadiversereader-ship,wewillbrieflyoutlinetheworkforeaseofreading.Oftheremainingintroductionsections,section(IB)canbeskippedbyreadersfamiliarwiththestabilityconjec-tureofSmaleandPalisandthestableergodicityofPughandShub.Followingtheintroductionwewilladdressvariouspre-liminarytopicspertainingtothisreport.Beginninginsection(IIA1),wepresentthemathematicaljustifica-tionforthestudyoftime-delaymapsbeingsufficientforageneralstudyofd1dimensionaldynamicalsys-tems.Thissectionisfollowedwithadiscussionofneuralnetworks,beginningwiththeirdefinitionintheabstract(section(IIA2)).Followingthedefinitionofneuralnet-works,weexplainthemappingsneuralnetworksareable2toapproximate(section(IIA3)).Insection(IIA4)wegiveourspecificconstructionofneuralnetworks.Thoseuninterestedinthemathematicaljustificationsforourmodelsandonlyinterestedinourspecificformulationshouldskipsections(IIA1)thru(IIA3)andconcentrateonsection(IIA4).Thediscussionofoursetofmappingsisfollowedbyrelevantdefinitionsfromhyperbolicityandergodictheory(section(IIB)).ItisherewherewedefinetheLyapunovspectrum,hyperbolicmaps,anddiscussrelevantstabilityconjectures.Section(IIC)providesjus-tificationforouruseofLyapunovexponentcalculationsuponourspaceofmappings(theneuralnetworks).Read-ersfamiliarwithtopicsinhyperbolicityandergodicthe-orycanskipthissectionandrefertoitasisneededforanunderstandingoftheresults.Lastly,insection(IID),wemakeaseriesofdefinitionswewillneedforournu-mericalarguments.Withoutanunderstandingofthesedefinitions,itisdifficulttounderstandbothourconjec-turesofourarguments.Section(III)discussestheconjectureswewishtoin-vestigateformally.Forthoseinterestedinjusttheresultsofthisrepor