The Study of Bifurcation Theory of Limit Cycles and Homoclinic Orbits

Abstract

In this research we study the existence of limit cycle for polynomial planar system. We present a proof of a result on the existence of limit cycle of the Quadratic System. The aims of this research is to study the existence of limit cycle for Liénard system. We followed the historical analytical mathematical method to present a proof of a result on the existence of limit cycle for Liénard system. Deals with the study of two types of non Liénard cubic systems. We proof the uniqueness of limit cycle in most cases. For the second system, we find an example to show that the center conditions of quadratic systems. We study various types of bifurcations that occur in 𝐶1 – systems 𝑥̇=𝑓(𝑥,𝜇). In particular, we study bifurcations at nonhyperbolic equilibrium points and periodic orbits. We have introduced a method for the study of limit cycles of the quadratic and Liénard system, and a sequence approximation to the bifurcation set of the system. We present a variant of the method that gives very important qualitative and quantitative improvement