Chapter 8 Nonlinear systems第8章非线系统

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1、Chapter8Nonlinearsystems8.1Linearization,criticalpoints,andequilibriaNote:1lecture,§6.1–§6.2in[EP],§9.2–§9.3in[BD]Exceptforafewbriefdetoursinchapter1,weconsideredmostlylinearequations.Linearequationssufficeinmanyapplications,butinrealitymostphenomenarequirenonlinearequations.Nonlinearequations,howev

2、er,arenotoriouslymoredifficulttounderstandthanlinearones,andmanystrangenewphenomenaappearwhenweallowourequationstobenonlinear.Nottoworry,wedidnotwasteallthistimestudyinglinearequations.Nonlinearequationscanoftenbeapproximatedbylinearonesifweonlyneedasolution“locally,”forexample,onlyforashortperiodof

3、time,oronlyforcertainparameters.Understandinglinearequationscanalsogiveusqualitativeunderstandingaboutamoregeneralnonlinearproblem.Theideaissimilartowhatyoudidincalculusintryingtoapproximateafunctionbyalinewiththerightslope.In§2.4welookedatthependulumoflengthL.Thegoalwastosolvefortheangleθ(t)asafu

4、nctionofthetimet.TheequationforthesetupisthenonlinearequationL��gθ+sinθ=0.θLInsteadofsolvingthisequation,wesolvedtherathereasierlinearequation��gθ+θ=0.LWhilethesolutiontothelinearequationisnotexactlywhatwewerelookingfor,itisratherclosetotheoriginal,aslongastheangleθissmallandthetimeperiodinvolvedi

5、sshort.Youmightask:Whydon’twejustsolvethenonlinearproblem?Well,itmightbeverydifficult,impractical,orimpossibletosolveanalytically,dependingontheequationinquestion.Wemaynotevenbeinterestedintheactualsolution,wemightonlybeinterestedinsomequalitativeideaofwhatthesolutionisdoing.Forexample,whathappensas

6、timegoestoinfinity?301302CHAPTER8.NONLINEARSYSTEMS8.1.1AutonomoussystemsandphaseplaneanalysisWerestrictourattentiontoatwodimensionalautonomoussystem��x=f(x,y),y=g(x,y),wheref(x,y)andg(x,y)arefunctionsoftwovariables,andthederivativesaretakenwithrespecttotimet.Solutionsarefunctionsx(t)andy(t)suchthat

7、������x(t)=fx(t),y(t),y(t)=gx(t),y(t).Thewaywewillanalyzethesystemisverysimilarto§1.6,wherewestudiedasingleautonomousequation.Theideasintwodimensionsarethesame,butthebehaviorcanbefarmorecomplicated.Itmaybebesttot