Towards Theory of Piecewise Linear Dynamical Syste

arXiv:0803.0490v1[math.DS]4Mar2008TowardsTheoryofPiecewiseLinearDynamicalSystems⋆ValeryA.Gaiko1BelarusianStateUniversityofInformaticsandRadioelectronics,DepartmentofMathematics,L.BedaStr.6–4,Minsk220040,BelarusWimT.vanHorssenDelftUniversityofTechnology,DelftInstituteofAppliedMathematics,Mekelweg4,2628CDDelft,TheNetherlandsAbstractInthispaper,weconsideraplanardynamicalsystemwithapiecewiselinearfunctioncontaininganarbitrarynumber(butfinite)ofdroppingsectionsandapproximatingsomecontinuousnonlinearfunction.Studyingallpossiblelocalandglobalbifur-cationsofitslimitcycles,weprovethatsuchapiecewiselineardynamicalsystemwithkdroppingsectionsand2k+1singularpointscanhaveatmostk+2limitcycles,k+1ofwhichsurroundthefocionebyoneandthelast,(k+2)-th,limitcyclesurroundsallofthesingularpointsofthissystem.Keywords:piecewiselineardynamicalsystem;fieldrotationparameter;bifurcation;limitcycle1IntroductionThepaperisbasedontheapplicationsofBifurcationTheorydevelopedbyAn-dronov,Arnold,Thom,Whitney,Zeemanetal.andcanbeusedformodeling⋆ThisworkissupportedbytheNetherlandsOrganizationforScientificResearch(NWO).Emailaddress:[e-mail](ValeryA.Gaiko).1ThefirstauthorisgratefultotheDelftInstituteofAppliedMathematicsofTUDelftforhospitalityduringhisstayattheUniversityintheperiodofOctober,2007–March,2008.PreprintsubmittedtoElsevier4March2008problems,wheresystemparametersplayacertainroleinvariousbifurcations.Thetheoreticalstudiesofbifurcationsdealwithso-calleduniversalproblems.Thismeansthatsufficientlymanyparametersareavailableforuniversalityofgenericfamiliesofdynamicalsystemsinthecontextathand,underarelevantequivalencerelation.Thishasledtotheclassificationofgeneric,localbifur-cations.Inmanyapplications,modelshaveagivennumberofparameters.Moreover,thebifurcationanalysis,takingplaceintheproductofphasespaceandparameterspace,isnotrestrictedtolocalfeaturesonly.Onthecontrary,oftentheinterestistheglobalorganizationoftheparameterspaceregardingbifurcationswhichcanbebothlocalandglobal.Thispaperdealswithso-calledseweddynamicalsystems,i.e.,withsystemsforwhichthedomainofdefinitionisdividedintosub-domainswheredifferentanalyticalsystemsaredefined.Thetrajectoriesofthesepartialsystemsaresewedinonewayoranotherontheboundariesofthesub-domains.Suchsys-temshavesometypicalfeatures,namely:1)thesystemsewingisimmediatefromthephysicalmeaningoftheproblemunderconsideration;2)thesystemispiecewiselinear,i.e.,thepartialsystemsfromwhichitissewedarelinearsys-tems;3)onthelineofsewing,apointmap(afirstreturnfunction)isdefined,whatallowstodeterminethecharacterofthesystemunderconsideration.Piecewiselineardynamicalsystemsalwayscontainsomeparametersand,un-derthevariationoftheseparameters,thequalitativebehaviorofthesystemscanobviouslychange.Wewillconsiderthesimplestbifurcationspossibleinthesewedsystemswhenthesewinglinesareunchangedundertheparametervariations.Itisnaturaltoconsiderthefollowingbifurcationswhicharesimilartothesimplestbifurcationsofcontinuousdynamicalsystems:1)thebifurca-tionofasingularpointoffocustype;2)thebifurcationofanimmovablepointoffocustype,aquasi-focus;3)thebifurcationofasewedlimitcycle;4)thebifurcationofasewedseparatrixgoingfromasaddletoanothersaddle(thesaddlescanbebothsewedandunsewed);5)thebifurcationofaseparatrixofasaddle-shapedsingularpoint(sewedorunsewed)goingfromasaddle-shapedsingularpointtoanothersuchpointortoasaddle(sewedorunsewed);6)thebifurcationofasewedsaddle-node;7)thebifurcationofasewedseparatrixofasaddle-node(sewedorunsewed)goingoutofthesaddle-nodeandgoingbacktoit.Besides,somespecificbifurcationscanoccurinsewedsystems.Sinceinsuchsystems,forexample,archesofattractionorrepulsioncomposedofimmovablepointscanbesimilartosingularpoints,somebifurcationswhicharesimilartothegenerationofalimitcyclefromafocuscanoccurinthecorrespondingconstructions.Piecewiselinearsystemshavemanyapplicationsinscienceandengineering.Specialcasesofsuchsystemsprovidemathematicalmodelsformechanicalsys-temswithCoulombfriction,forvalveoscillatorswithadiscontinuouschar-acteristic,fordirectcontrolsystemswithatwo-pointrelaymechanism,for2planardynamicalsystemsmodelingneuralactivity,etc.Despitetheirsim-plestructureandrelevancetotheapplications,thereis,tothebestofourknowledge,nocompletestudyoftheirdynamicalproperties.Inmostexist-ingpapers,whichdealwithplanardynamicalsystemswithpiecewiselinearright-handsides,eitherthesystemsconsideredarecontinuousoronlypar-ticularcasesareinvestigated.ThefirstanalyticalresultsonsuchsystemsgobacktoAndronov,Vitt,andKhaikininthe1930s(see[1]).Theexistenceandnon-existenceofanasymptoticallystableperiodicsolution(limitcycle)ofapiecewiselinearsystemcanbecomparativelyeasilyproved.However,forexample,periodicsolutionswithslidingmotionareofgreatimportancetotheapplications.Inparticular,theydescribetheso-calledstick-sliposcillationswhichappearinmechanicalsystemswithdryfriction.Themainobjectiveofthepresentpaperistoprovideacompleteanalysisofthedynamicalpropertiesofpiecewiselinearsystems,theirdependenceonthesystemparametersstudying,firstofall,theirlimitcyclebifurcations.Thereareseveralwaystoinve