Bifurcation of Limit Cycles for a Kind of Piecewise Smooth Differential Systems with an Elementary Center of Focus-Focus Type

Abstract

In this paper, we study the number of limit cycles H(n) bifurcating from the piecewise smooth system formed by the quadratic reversible system (r22) for \(y\ge 0\) and the cubic system \({\dot{x}} =y\bigl (1+{{\bar{x}}}^2+y^2\bigr )\), \({\dot{y}} =-{\bar{x}}\bigl (1+{{\bar{x}}}^2+y^2\bigr )\) for \(y<0\) under the perturbations of polynomials with degree n, where \({{\bar{x}}}=x-1\). By using the first-order Melnikov function, it is proved that \(2n+3\le H(n)\le 2n+ 7\) for \(n\ge 3\) and the results are sharp for \(n=0,1,2\).