The discontinuous matching of two globally asymptotically stable crossing piecewise smooth systems in the plane do not produce in general a piecewise differential system globally asymptotically stable

A differential system in the plane R 2 is globally asymptotically stable if it has an equilibrium point p and all the other orbits of the system tend to p in forward time. In other words if the basin of attraction of p is R 2 . The problem of determining the basin of attraction of an equilibrium point is one of the main problems in the qualitative theory of differential equations. We prove that planar crossing piecewise smooth systems with two zones formed by two globally asymptotically stable differential systems sharing the same equilibrium point localized in the separation line are not necessarily globally asymptotically stable.