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1、7NonlinearSystemsInthischapterwebeginthestudyofnonlineardifferentialequations.Inlinear(constantcoefficient)systemswecanalwaysfindtheexplicitsolutionofanyinitialvalueproblem;however,thisisrarelythecasefornonlinearsystems.Infact,basicpropertiessuchastheexistenceanduniquenessofsolutions,whichwassoobv
2、iousinthelinearcase,nolongerholdfornonlinearsystems.Asweshallsee,somenonlinearsystemshavenosolutionswhatsoevertoagiveninitialvalueproblem.Ontheotherhand,thereareothersystemsthathaveinfinitelymanydiffer-entsuchsolutions.Evenifwedofindasolutionofsuchasystem,thissolutionneednotbedefinedforalltime;fore
3、xample,thesolutionmaytendto1infinitetime.Otherquestionsalsoarise:Forexample,whathappensifwevarytheinitialconditionofasystemeversoslightly?Doesthecorrespondingsolu-tionvarycontinuously?Allofthisisclearforlinearsystems,butnotatallclearinthenonlinearcase.Thismeansthattheunderlyingtheorybehindnonline
4、arsystemsofdifferentialequationsisquiteabitmorecomplicatedthanthatforlinearsystems.Inpractice,mostnonlinearsystemsthatariseare“nice”inthesensethatwedohaveexistenceanduniquenessofsolutions,aswellascontinuityofsolutionswheninitialconditionsarevariedandother“natural”properties.Thuswehaveachoice:Giv
5、enanonlinearsystem,wecouldsimplyplungeaheadandeitherhopethator,ifpossible,verifythat,ineachspecificcase,thesystem’ssolutionsbehavenicely.Alternatively,wecouldtakealongpauseatDifferentialEquations,DynamicalSystems,andanIntroductiontoChaos.DOI:10.1016/B978-0-12-382010-5.00007-5c2013ElsevierInc.Allr
6、ightsreserved.139140Chapter7NonlinearSystemsthisstagetodevelopthenecessaryhypothesesthatguaranteethatsolutionsofagivennonlinearsystembehavenicely.Inthisbookwepursueacompromiseroute.Inthischapter,wespelloutinprecisedetailmanyofthetheoreticalresultsthatgovernthebehaviorofsolu-tionsofdifferentialeq
7、uations.Wepresentexamplesofhowandwhentheseresultsfail,butwewillnotprovethesetheoremshere.Rather,wewillpostponeallofthetechnicalitiesuntilChapter17,primarilybecauseunderstandingthismaterialdemandsafirmandextensivebackgroundint
