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1、September1,20128:47WSPC/S0218-12741250197InternationalJournalofBifurcationandChaos,Vol.22,No.8(2012)1250197(16pages)cWorldScientificPublishingCompanyDOI:10.1142/S0218127412501970GENERALIZEDHOPFBIFURCATIONINAFREQUENCYDOMAINFORMULATIONA.M.TORRESIDepartamentodeMatem´atica,UniversidadNac
2、ionaldelSur,Bah´ıaBlanca,B8000CPB,Argentinaatorresi@criba.edu.arG.L.CALANDRINIInstitutodeInvestigacionesenIngenier´ıaEl´ectrica,IIIE(UNS-CONICET),DepartamentodeIngenier´ıaEl´ectricaydeComputadoras,UniversidadNacionaldelSur,Bah´ıaBlanca,B8000CPB,Argentinacalandri@criba.edu.arP.A.BONFI
3、LIDepartamentodeMatem´atica,UniversidadNacionalSanJuanBosco,Trelew,B9100CPB,Argentinapbonfili@hotmail.comJ.L.MOIOLAInstitutodeInvestigacionesenIngenier´ıaEl´ectrica,IIIE(UNS-CONICET),DepartamentodeIngenier´ıaEl´ectricaydeComputadoras,UniversidadNacionaldelSur,Bah´ıaBlanca,B8000CPB,Arg
4、entinajmoiola@criba.edu.arReceivedMay17,2011Int.J.BifurcationChaos2012.22.Downloadedfromwww.worldscientific.combyHARVARDUNIVERSITYLIBRARYon11/10/12.Forpersonaluseonly.Themultiplicityproblemoflimitcyclesarisingfromaweakfocusisaddressed.Theproposedmethodologyisacombinationofthefrequenc
5、ydomainmethodtohandlesomedegenerateHopfbifurcationswiththepowerfultoolsofthesingularitytheory.Thefrequencydomainapproachusestheharmonicbalancemethodtostudytheexistenceofperiodicsolutions.Ontheotherhand,thesingularitytheoryprovidestheconditionsandformulasfortheclassificationproblemofth
6、eunfoldingofthesingularityintermsofthedistinguishedandauxiliaryparameters.AclassicalexampleintroducedbyBautinisshowninwhichthemultiplicityoflimitcyclesisrecoveredbyusingthistypeofhybridmethodologyandstandardsoftwareinthecontinuationofperiodicsolutions(LOCBIFandXPPAUT).Forsmallamplitu
7、delimitcycles,theproposedmethodologygivesaccurateresults.Keywords:Limitcycles;Hopfbifurcation;singularitytheory;frequencydomain;harmonicbalancemethod.1250197-1September1,20128:47WSPC/S0218-12741250197A.M.Torresietal.1.Introductionstableperiodicsolutionimmediatelyaftertheequi-libriump
8、ointlosesits
