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1、August7,201221:05c09Sheetnumber1Pagenumber495cyanblackCHAPTER9NonlinearDifferentialEquationsandStabilityTherearemanydifferentialequations,especiallynonlinearones,thatarenotsus-ceptibletoanalyticalsolutioninanyreasonablyconvenientmanner.Numericalapproximat
2、ionmethods,suchasthosediscussedintheprecedingchapter,provideonemeansofdealingwiththeseequations.Anotherapproach,presentedinthischapter,isgeometricalincharacterandleadstoaqualitativeunderstandingofthebehaviorofsolutionsratherthantodetailedquantitativeinfor
3、mation.9.1ThePhasePlane:LinearSystemsSincemanydifferentialequationscannotbesolvedconvenientlybyanalyticalmeth-ods,itisimportanttoconsiderwhatqualitative1informationcanbeobtainedabouttheirsolutionswithoutactuallysolvingtheequations.Thequestionsthatweconsid
4、erinthischapterareassociatedwiththeideaofstabilityofasolution,andthemethodsthatweemployarebasicallygeometrical.Boththeconceptofstabilityandtheuse1ThequalitativetheoryofdifferentialequationswascreatedbyHenriPoincaré(1854–1912)inseveralmajorpapersbetween188
5、0and1886.PoincaréwasaprofessorattheUniversityofParisandisgenerallyconsideredtheleadingmathematicianofhistime.Hemadefundamentaldiscoveriesinseveraldiffer-entareasofmathematics,includingcomplexfunctiontheory,partialdifferentialequations,andcelestialmechanic
6、s.Inaseriesofpapersbeginningin1894,heinitiatedtheuseofmodernmethodsintopology.Indifferentialequationshewasapioneerintheuseofasymptoticseries,oneofthemostpowerfultoolsofcontemporaryappliedmathematics.Amongotherthings,heusedasymptoticexpansionstoobtainsolu-
7、tionsaboutirregularsingularpoints,therebyextendingtheworkofFuchsandFrobeniusdiscussedinChapter5.495August7,201221:05c09Sheetnumber2Pagenumber496cyanblack496Chapter9.NonlinearDifferentialEquationsandStabilityofgeometricalanalysiswereintroducedinChapter1and
8、usedinSection2.5forfirstorderautonomousequationsdy/dt=f(y).(1)Inthischapterwerefinetheideasandextendthediscussiontoautonomoussystemsofequations.Weareparticularlyinterestedinnonlinearsystemsbecausetheytypicallycannotbe
