非线性分析作业第2次(硕士博士非线性分析)

1.Forthefollowingdynamicalsystems1）''30xxx2）''2(1),(1)3xxxxyyyyxya)Findallfixedpointsandclassifythem.b)Sketchthephasespaceportrait.Solutionfor1）：''30xxxSet121,yxyy.Then,theequationbecomesto，123211yyyyySetvectorvariablez,wecanwrite()zfz,where12yzy213211()yyfzfyyyThereisonlyfixedpoint00zTheJacobianmatrix2101310DfyJacobianmatrixforlinearizedsystematthefixedpoint,0110DfyEigenvaluesforthissystemare12i,sotheyhavezerorealpartandthemethodoflinearizationcannotdecideaboutthestability.Solutionfor2）：''2(1),(1)3xxxxyyyyxyJacobianmatrix：243123xyxAyyxJacobianmatrixforlinearizedsystematthefixedpoint00is2001Eigenvaluesforthissystemare122,1,repellingnode,whichisunstable.2.Giventhesystem'''30xxxxShowthattheequilibrium(0,0)isgloballyasymptoticallystable.Solution：Set121,yxyy.Then,theequationbecomesto，1232211yyyyyySetvectorvariablez,wecanwrite()zfz,where12yzy2132211()yyfzfyyyyThereisonlyfixedpoint00zTheJacobianmatrix2101311DfyJacobianmatrixforlinearizedsystematthefixedpoint,0111DfyEigenvaluesforthissystemare120.50.866i,sotheyhavenegativerealparts.Thus,itisstable.3.Forarealnumberc,definetheone-parameterfamily()()(23),afxxaxaxcforwhatvaluesofcisthereabifurcationinthisfamily?Describethebifurcationsandlistthebifurcationpoints(a,x),andSketchthebifurcationdiagram.Solution:Suppose0a.Set212fxx,2fxxc.120afxfxfxWhen18c,thereisabifurcationinafx.Bifurcationpoints:10,44.Showthattheoneparametersystem''2'2'()0xxxxxundergoesaHopfbifurcationatμ=0.Plotthephaseportraitsandsketchthebifurcationdiagram.Solution:Set12,xxxx,thecorrespondingstate-spaceequationsis2311222122xxxxxxxxSolvetheequations2311222200xxxxxxFixedpointsareobtainedas(0,0).JacobianmatrixandEigenvaluesare21,2,42011AWhen0,thereisannodecenter.ThephasespaceportraitisshownnextThebifurcationdiagramisshownnext5.ForHénonmap211,nnnnnxaxbyyx1)Findthepointsofperiod-1andperiod-2fortheHénonmap.2)InvestigatethebifurcationdiagramsfortheHénonmapbyplottingthenxvaluesasafunctionofawhenb=0.4.Solution:Forperiod-1,2111()nnnnnnnxaxbyfXXyxSuppose()nnXfX,andthen2nnnnnxaxbyyxSolvetheequations,weget2(1)(1)42,nnbbaxyForperiod-2,()nnXfXfAndthen222()nnnnnnnxaaxbyxyaxbySolvetheequations,weget2(1)(1)42,nnbbaxyOr2(1)43(1)2,nnbabxythebifurcationdiagramisshownatb=0.4