On the Poincaré–Bendixson Index Theorem for a Class of Piecewise Linear Differential Systems

Abstract

Poincaré–Bendixson index theorem showed that: if planar smooth differential systems have only finitely number of singular points inside a limit cycle, then the sum of the indices at these singular points is 1. Hence there is at least one singular point lie inside a limit cycle. These results are useful in practice since its provide information about the existence and location of limit cycles. It is also well known that planar smooth linear system cannot have limit cycles. While in a recent paper Llibre and Teixeira (Nonl Dyn 88:157–164, 2017), Llibre and Teixeira constructed a planar piecewise linear differential systems formed by two linear differential systems separated by the straight line $\Sigma :\{(x,y)|x=0\}$, such that both linear differential have no singular points, neither real nor virtual, but it can have a limit cycle. This paper revisit these piecewise linear differential systems by regularization process. Our results show that there are three $\Sigma -$singular points inside the limit cycle, and the sum of the indices at these singular points is 1. Thus the Poincaré-Bendixson index theorem is also valid for such piecewise linear differential systems.