Limit Cycles Bifurcate From Generalized Homoclinic Loops In Piecewise Smooth Near-Hamiltonian Systems With Saddle-Fold Points

In this paper limit cycle bifurcations of a piecewise smooth near-Hamiltonian system with a generalized homoclinic loop are considered. The singular point of the homoclinic loop belongs to Saddle-Fold type at the origin. And the piecewise sys-tem has x-axis as the separation line. Under a non-degenerate condition six kinds of Saddle-Fold singular points and twelve kinds of the homoclinic loops are classified.Using theory of normal forms, the expansion of the first Melnikov function corre-sponding to the period annuluses near the homoclinic loops are derived. Applying the first six coefficients of the expansion, a sufficient condition is given to ensure that k (1 < k < 5) limit cycles can appear near the homoclinic loops. This paper includes the following three chapters:In the first chapter we state significance of limit cycles in dynamical system and some known results on bifurcation of limit cycles. Also, some definitions needed in this paper are presented.In the second chapter three main theorems of the paper are presented. Besides,8 preliminary lemmas and the proofs of the lemmas and main results are given.In the third chapter two examples of application are considered including convex and concave cases. It deserve to be mentioned that in the first example we research a piecewise quadratic system and find 5 limit cycles near the generalized homoclinic loop with a Saddle-Fold point. From [21, 22] we know that there are at most 2 limit cycles near a homoclinic loop of quadratic polynomial planar smooth systems.Compared with this result, we find 3 more limit cycles in non-smooth cases.