The Numbers Of Limit Cycles For Planar Piecewise Smooth Differential System

In this paper we study the following planar piecewise smooth differential systemOur main work is as follows:1. In some conditions, if the origin of the above system is a center, then there exists a piecewise smooth first integral H(x, y) of this system;2. We generalize the classical Poncare-Pontryagin Theorem, that is, we give an estimation of the number of limit cycles which bifurcate from an integrable system under polynomial perturbations;3. As an application, we consider two classes of piecewise smooth quadratic sys-tems, and give the lower bounds of the number of limit cycles of them, The bounds are 10 and 9 respective.