POINCARE BIFURCATIONS IN POLYNOMIAL DIFFERENTIAL SYSTEMS

1.IntroductionForagivenunperturbedplanarsystemwithafamilyofclosedorbits,considerthecreationofisolatedclosedorbits--limitcyclesbyperturbationofthesystem.ThatiscalledthePoillcar6bifurcation.Therearewell-knowntheoreticalresultsaboutPoincar6bifllrcati()11,seeforexam-Ple1],2].InthecasewheretheunperturbedsystemisHamiltonian,thepertllrbedsystemcanbewrittenasinwhichweassumethatp(x,y,0)=Q(x,U,(i)~0,and(1)A--ohasafamilyofclosedorbitsgivenbytheHamiltonianintegralnfH(x,y)=h,0

POINCARE BIFURCATIONS IN POLYNOMIAL DIFFERENTIAL SYSTEMS

The Poincare bifurcations for polynomial differential systems are considered in this paper. Usually, the Pontryagin's method of perturbed Hamiltonian systems are used to deal with such problem by studying the number and multiplicity of the zero points for certain Abelian integrals, and many results have been gived for concrete polynomial systems. But, the method is inapplicable to the case where the unperturbed system has complicated Hamiltonian function, or is integrable but non-Hamiltonian. We now start from a different angle to avoid the complicated calculations of the Abelian integrals and try to study the Poincare bifurcation from the Hopf bifurcations of all possible orders for the system, and give a complete result for the Poincare bifurcations of quadratic system in Bautin's form.