The Markus–Yamabe Conjecture Does not Hold for Discontinuous Piecewise Linear Differential Systems Separated by One Straight Line

Abstract

The Markus–Yamabe conjecture is a conjecture on global asymptotic stability. The conjecture states that if a differentiable system $$\dot{x}=f(x)$$ has a singularity and the Jacobian matrix Df(x) has everywhere eigenvalues with negative real part, then the singularity is a global attractor. In this paper we consider discontinuous piecewise linear differential systems in $$\mathbb {R}^2$$ separated by one straight line $$\Sigma $$ such that the unique singularity of the system is at $$\Sigma $$ and the Jacobian matrix of the system has everywhere eigenvalues with negative real part. We prove that these discontinuous piecewise linear differential systems can have one crossing limit cycle and consequently these systems do not satisfy the Markus–Yamabe conjecture.