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

Yuye Jin, Jianfeng Huang

• In this paper we devote to study the piecewise smooth equation of the form: dxdt=S(t,x)=S1(t,x)=a1(t)xm+b(t),ifx≥0,S2(t,x)=a2(t)xm+b(t),ifx<0,where (t,x)∈[0,2π]×R,m∈Z+ and a1(t),a2(t),b(t) are 2π-periodic smooth functions. A solution of the equation satisfying x(0)=x(2π) is called a periodic solution. Moreover, such solution is called a limit cycle if and only if it is isolated. We obtain that the maximum number of limit cycles for this equation is 1 (resp. 2) if (-1)ma1(t)·a2(t)<0 (resp. (-1)ma1(t)·a2(t)>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 x2+y2=1.