poker_differential_equations_and_dynamic_system

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1、1LinearSystemsThischapterpresentsastudyoflinearsystemsofordinarydifferentialequations:X=Ax(1)wherexER",AisannxnmatrixanddxX=dtHItisshownthatthesolutionofthelinearsystem(1)togetherwiththeinitialconditionx(O)=xoisgivenbyx(t)=eAtxowhereeAtisannxnmatrixfunctio

2、ndefinedbyitsTaylorseries.AgoodportionofthischapterisconcernedwiththecomputationofthematrixeAtintermsoftheeigenvaluesandeigenvectorsofthesquarematrixA.ThroughoutthisbookallvectorswillbewrittenascolumnvectorsandATwilldenotethetransposeofthematrixA.1.1Uncoup

3、ledLinearSystemsThemethodofseparationofvariablescanbeusedtosolvethefirst-orderlineardifferentialequationi=ax.Thegeneralsolutionisgivenbyx(t)=ceatwheretheconstantc=x(0),thevalueofthefunctionx(t)attimet=0.Nowconsidertheuncoupledlinearsystem$1=-xlx2=2x2.21.Li

4、nearSystemsThissystemcanbewritteninmatrixformas*=Ax(1)whereA=[02NotethatinthiscaseAisadiagonalmatrix,A=diag[-1,2],andingeneralwheneverAisadiagonalmatrix,thesystem(1)reducestoanuncoupledlinearsystem.Thegeneralsolutionoftheaboveuncoupledlinearsystemcanonceag

5、ainbefoundbythemethodofseparationofvariables.Itisgivenbyxi(t)=cle-t(2)x2(t)=c2e2torequivalentlyby{e_tx(t)_Ot]c(2')wherec=x(0).Notethatthesolutioncurves(2)lieonthealgebraiccurvesy=k/x2wheretheconstantk=cc2.Thesolution(2)or(2')definesamotionalongthesecurves;

6、i.e.,eachpointcER2movestothepointx(t)ER2givenby(2')aftertimet.Thismotioncanbedescribedgeometricallybydrawingthesolutioncurves(2)inthex1ix2plane,referredtoasthephaseplane,andbyusingarrowstoindicatethedirectionofthemotionalongthesecurveswithincreasingtimet;c

7、f.Figure1.Forcl=c2=0,xI(t)=0andx2(t)=0foralltERandtheoriginisreferredtoasanequilibriumpointinthisexample.Notethatsolutionsstartingonthexl-axisapproachtheoriginast-ooandthatsolutionsstartingonthex2-axisapproachtheoriginast-.-oo.Thephaseportraitofasystemofdi

8、fferentialequationssuchas(1)withxER'isthesetofallsolutioncurvesof(1)inthephasespaceR".Figure1givesageometricalrepresentationofthephaseportraitoftheuncoupledlinearsystemconsideredabove.Thedynamicalsystemdefine