The Markus–Yamabe Conjecture for Discontinuous Piecewise Linear Differential Systems in $$\mathbb {R}^n$$ Separated by a Conic$$\times \mathbb {R}^{n-2}$$

Abstract

In 1960 Markus and Yamabe made the conjecture that if a \(C^1\) differential system \(\dot{x} = F(x)\) in \(\mathbb {R}^n\) has a unique equilibrium point and DF(x) is Hurwitz for all \(x \in \mathbb {R}^n\), then the equilibrium point is a global attractor. This conjecture was completely solved in 1997 and it turned out to be true in \(\mathbb {R}^2\) and false in \(\mathbb {R}^n\) for all \(n \ge 3\). In (The Markus–Yamabe conjecture for continuous and discontinuous piecewise linear differential systems, 2020) the authors extended the Markus–Yamabe conjecture to continuous and discontinuous piecewise linear differential systems in \(\mathbb {R}^n\) separated by a hyperplane, they proved for the continuous systems that the extended conjecture is true in \(\mathbb {R}^2\) and false in \(\mathbb {R}^n\) for all \(n \ge 3\), but for discontinuous systems they proved that the conjecture is false in \(\mathbb {R}^n\) for all \(n \ge 2\). In this paper first we show that there are no continuous piecewise linear differential systems separated by a conic\(\times \mathbb {R}^{n-2}\) except the linear differential systems in \(\mathbb {R}^n\). And after we prove that the extended Markus–Yamabe conjecture to discontinuous piecewise linear differential systems in \(\mathbb {R}^n\) separated by a conic\(\times \mathbb {R}^{n-2}\) is false in \(\mathbb {R}^n\) for all \(n \ge 2\).