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1、文档1.Forthefollowingdynamicalsystems1）2）a)Findallfixedpointsandclassifythem.b)Sketchthephasespaceportrait.Solutionfor1）：Set.Then,theequationbecomesto，Setvectorvariablez,wecanwrite,whereThereisonlyfixedpointTheJacobianmatrixJacobianmatrixforlinearizedsystematthefixedpoint,Eigenvalu
2、esforthissystemare,sotheyhavezerorealpartandthemethodoflinearizationcannotdecideaboutthestability.文档Solutionfor2）：Jacobianmatrix：JacobianmatrixforlinearizedsystematthefixedpointisEigenvaluesforthissystemare,repellingnode,whichisunstable.文档1.GiventhesystemShowthattheequilibrium(0,
3、0)isgloballyasymptoticallystable.Solution：Set.Then,theequationbecomesto，Setvectorvariablez,wecanwrite,whereThereisonlyfixedpoint文档TheJacobianmatrixJacobianmatrixforlinearizedsystematthefixedpoint,Eigenvaluesforthissystemare,sotheyhavenegativerealparts.Thus,itisstable.1.Forarealnu
4、mberc,definetheone-parameterfamilyforwhatvaluesofcisthereabifurcationinthisfamily?Describethebifurcationsandlistthebifurcationpoints(a,x),andSketchthebifurcationdiagram.Solution:Suppose.Set,.When,thereisabifurcationin.文档Bifurcationpoints:文档1.Showthattheoneparametersystemundergoes
5、aHopfbifurcationatμ=0.Plotthephaseportraitsandsketchthebifurcationdiagram.Solution:Set,thecorrespondingstate-spaceequationsisSolvetheequationsFixedpointsareobtainedas(0,0).JacobianmatrixandEigenvaluesareWhen,thereisannodecenter.Thephasespaceportraitisshownnext文档Thebifurcationdiag
6、ramisshownnext文档1.ForHénonmap1)Findthepointsofperiod-1andperiod-2fortheHénonmap.2)InvestigatethebifurcationdiagramsfortheHénonmapbyplottingthevaluesasafunctionofawhenb=0.4.Solution:Forperiod-1,Suppose,andthenSolvetheequations,wegetForperiod-2,Andthen文档Solvetheequations,wegetOrthe
7、bifurcationdiagramisshownatb=0.4
