Markus–Yamabe conjecture for planar piecewise linear refracting system

The Markus–Yamabe conjecture states that if a differential system  \(\dot{\varvec{z}}=\varvec{F(z)}\) $\dot{\mathit{z}}=\mathit{F}\mathbf{(}\mathit{z}\mathbf{)}$ , where  \(\varvec{F} \in C^1(\mathbb {R}^n)\) $\mathit{F}\in C^1({\mathbb{R}}^n)$ , has a unique equilibrium point and all eigenvalues of the Jacobian matrix of  \(\varvec{F(z)}\) $\mathit{F}\mathbf{(}\mathit{z}\mathbf{)}$ have negative real parts for any \(\varvec{z} \in \mathbb {R}^n\) $\mathit{z}\in {\mathbb{R}}^n$ , then the equilibrium point is globally asymptotically stable. It has been shown that the Markus–Yamabe conjecture holds for  \(n=2\) $n=2$ , but it fails for  \(n>2\) $n>2$ . Recently, the conjecture has been extended to piecewise linear systems. Previous literature indicates that the Markus–Yamabe conjecture holds for planar piecewise linear continuous systems but fails for piecewise linear discontinuous systems. In this paper, we focus on planar piecewise linear refracting systems, which are a type of discontinuous system without sliding. Our results reveal that the Markus–Yamabe conjecture holds for planar piecewise linear refracting systems with two zones separated by a straight line, but fails for planar piecewise linear refracting systems with three zones separated by two parallel straight lines.