The matching of two Markus-Yamabe piecewise smooth systems in the plane

Abstract

A Markus-Yamabe vector field is a smooth vector field in $R^n$ having only one equilibrium point and such that the spectrum of its Jacobian matrix at any point of $R^n$ is on the left of the imaginary axis in the complex plane. A vector field is globally asymptotically stable if it has a globally asymptotically stable equilibrium point $p$: all the orbits tend to $p$ in forward time. One of the great results of the Qualitative Theory of Differential Equations establishes that a planar Markus-Yamabe vector field is globally asymptotically stable, but a Markus-Yamabe vector field defined in $R^n$, $n⩾3$, does not have in general this property. We prove that planar crossing piecewise smooth vector fields defined in two zones formed by two Markus-Yamabe vector fields sharing the same equilibrium point located on the separation straight line are not necessarily globally asymptotically stable.