In this paper, we study the perturbation of certain of cubic system. By using the method of multi-parameter perturbation theory and qualitative analysis, we infer that the system under consideration can have five limit cycles.
Consider a four-dimensional system having a two-dimensional invariant surface. By analyzing the solutions of bifurcation equations, this paper studied the bifurcation phenomena of a k multiple closed orbit in the invariant surface. Sufficient conditions for the existence of periodic orbits generated by the k multiple closed orbit were given.
This paper deals with a kind of fourth degree systems with perturbations. By using the method of multi-parameter perturbation theory and qualitative analysis, it is proved that the system can have six limit cycles.
Some global behavior for a slowly varying oscillator was investigated. Based on a series of transformations and the theory of periodic orbits and integral manifold, the bifurcations of subharmonic solutions and invariant tori generated from a semistable limit cycle in the fast dynamics were discussed.
In this paper a bifurcation theorem on the existence of integral manifolds is obtained by using contracting principle. As an application, sufficient conditions for a higher dimensional system to have an integral manifold are given. Especially the existence and uniqueness of a 3-dimensional invariant torus appearing in a 4-dimensional autonomous system with singularity of codimension two are proved.
