The Singular Closed Orbit Bifurcation Of Cubic Hamiltonian System With Five Perturbations

The problem about the number of the limit cycles which are bifurcated by the singular cycle is one of the important topics of bifurcation theory.A kind of cubic Hamilton system with four hyperbolic saddle points and five central singular points is discussed. This system consists of a singular cycle S(4)’ with four saddle points and heteroclinic orbits connecting them, and four singular cycles S(2) with two saddle points and heteroclinic orbits connecting them respectively. By using the method of the qualitative analysis and bifurcation theory.the singular cycle bifurcation of such cubic Hamilton system under five-order perturbation is analyzed by means of computersIn the first chapter.the research present situation of singular cycle and the main con-clusion of this article are introduced. The main conclusion is that the following system at least exists fourteen limit cycle under suitable perturbation and their distributions are given.where Pij, qij, are real parameters.In the second chapter.the preparation knowledge is given.Firstly consider the perturbed system of the Hamilton system under homogeneous polynomial of degree five.after under-going the translation.the perturbed polynomial of degree five is changed into: P(x.y) Then the Melnikov function of every singular orbit and the trace of the focus are calculated.In the third chapter.firstly a suitable small perturbation δ1,is given to the system. According to the same stability of the three singular cycle§2and the focus.by the the-orem of Poincare-Bendixon.the system at least produces one limit cycle outside three fo-cus respectively: A suitable smaller perturbation δ2.is given to the system again where0<||δ2||<<||δ1||.the system can have one double hoinoclinic loop and one singular cy cle S’3which connects three saddle points.and the stability of the cycles and saddle points are the same.by the theorem of Poincare-Bendixon.the system at least produces one limit cycle inside the neighborhood of the singular cycle respectively: A suitable smaller pertur-bation δ3.is given to the system again.where0<||δ3||<<||δ2||<<||δ2||<<||δ1||,the singular cycle S3through three points breaks up the1singular cycle S2which is through two points are produced and the stability of this cycle is changed, at the same time one hoinoclinic loop is produced too and the stability of the loop is the same to the focus,by the theorem of Poincare-Bendixon, the system at least produces one limit cycle inside the neighborhood of the singular cycle respectively: A suitable smaller perturbation δ4,is given to the system again.where0<||δ4||||δ3||?||δ2||<<||δ1||.the singular cycle through two points breaks up. homoclinic loop is produced and the stability of the loop is changed by the theorem of Poincare-Bendixon.the system at least produces one limit cycle inside the neighborhood of the singular cycle respectively:Lastly a suitable much smaller perturbation δ5is given, where0<||δ5||<<||δ4||<<||δ3||<<||δ2||<<||δ1||,all the homoclinic loops break up, by the theorem of Poincare-Bendixon.the system at least produces one limit cycle inside the neighborhood of the homoclinic loops respectively.Then the main conclusion of the article is proved completelyIn the fourth chapter, there exists an example which at least produces fourteen limit cycles.