Necessary condition for the basin of attraction of a periodic orbit in non-smooth periodic systems

We study a time-periodic non-smooth differential equation (x)over dot = f(t,x), x is an element of R. In [4] we have presented a sufficient condition for existence, uniqueness, stability and the basin of attraction of a periodic orbit in such a system, which is a generalized Borg's condition. In this paper we prove that this condition is necessary. The proof involves a generalization of Floquet exponents for periodic orbits of non-smooth differential equations.