On The Limit Cycles Of A Class Of Planar Quintic Z₃-equivariant Near-Hamiltonian Differential System

In this paper,the number and distributions of limit cycles of a planar quinticZ₃equivariant near-Hamiltonian system are studied.First of all,we give the phase portraits of a planar quinticZ₃-equivariant Hamiltonian polynomial system,then we use the bifurcation theory to study the existence of the homoclinic loops and heteroclinic loop of planar quinticZ₃ equivariant Hamiltonian polynomial system after perturbation.Secondly,the stabilities of the homoclinic loops and heteroclinic loop are analyzed by some specical methods.Lastly,by using the method of changing the stability of the homoclinic loops and heteroclinic loop and Poincaré-Bendixson theorem,we get the number and distributions of limit cycles of the planar quinticZ₃ equivariant near-Hamiltonian polynomial system.The paper is divided into three parts.The first chapter is the introduction,which mainly introduces the background and the present situation of research,the main methods of our paper also be discussed in this part.The second chapter is the preliminary knowledge,which gives some basic conception and theory.The last chapter is the main results of this paper,it studies the number and distributions of limit cycles of a planar quinticZ₃-equivariant Hamiltonian system with 19 singularities and 6 homoclinic loops and 3 heteroclinic loops which under quintic polynomial perturbation.The innovations of the paper:?1?The planar quintic near-Hamiltonian polynomial system of our paper is new.?2?The number and distributions of limit cycles of the planar quinticZ₃-equivariant near-Hamiltonian polynomial system is new.