On the Study and Application of Limit Cycles of a Kind of Piecewise Smooth Equation

Abstract

In this paper we devote to study the piecewise smooth equation of the form: $$\begin{aligned} \frac{dx}{dt}=S(t,x)=\left\{ \begin{array}{ll} S_1(t,x)=a_1(t)x^m+b(t), &{} \quad \hbox {if }\quad x\ge {0}, \\ S_2(t,x)=a_2(t)x^m+b(t), &{} \quad \hbox {if }\quad x 0$$). In this study we pay more attention to the examples in which the equation has limit cycle(s) crossing the separation straight line $$x=0$$. In the end, we apply this result on a kind of piecewise smooth planar system which has a separation curve $$x^2+y^2=1$$.