The Research Of Bifurcation Of Limit Cycle For Planar Cubic Systems

The study of the existence and stability of critical points and limit cycles has important significance in the qualitative theory of planar differential systems.For a given planar differen-tial system,by studying the singularity of the system and the position distribution of the limit cycle.,the linear state of the orbit on the plane can be analyzed.For a parametric diffe.rential system,when a certain parameter changes in the system,the stability of the system trajectory may change substantially as the parameter changes.This is the main content of the qualitative structure and bifurcation theory of differential systems.The bifurcation problem is one of the important research directions of the qualitative the-ory of differential equations.There are many good theorems about the bifurcation problem of planar quadratic systems.The research on the bifurcation of limit cycle for planar cubic system has some basic results.For the study about bifurcation problem of limit cycle,many scholars such as Zhang Jinyan,Ye Yanqian,etc also gave a lot of conclusions.In 1999,Chavarriga stud-ied a new type of bifurcation of limit cycle for planar cubic systems.The critical point types are discussed in detail according to the changes of parameters,and the condition of existing a limit cycle and the two centers simultaneously is obtained.On the basis of this research,the article studies the bifurcation problem of limit cycle for another class of planar cubic systems,and extends the bifurcation problem of the limit cycle to a more general planar cubic system.The conclusions obtained are also applicable to the system that Chavarriga studied.The article mainly does three aspects of work.firstly,we find out the critical points of system through calculation,and analyze the type of critical points.Secondly,by constructing the first integral,the qualitative structure of the system trajectory is studied.Furthermore,through the discussion of the parameter symbols in the system,the relationship between the limit cycle and the critical points is analyzed,and the new type of bifurcation of limit cycle for planar cubic system is pointed out.This type includes all the results of Chavarriga.Meanwhile,The article also further studied the Hopf bifurcation problem of system.