The Markus-Yamabe conjecture does not hold for discontinuous piecewise linear differential systems separated by one straight line
The Markus-Yamabe conjecture [\textit{L. Markus} and \textit{H. Yamabe}, Osaka Math. J. 12, 305--317 (1960; Zbl 0096.28802)] states that if a differentiable system \(\dot{x}=f(x)\) has a singularity \(x^*\in\mathbb{R}^n\), \(f(x^*)=0\), and all eigenvalues of the Jacobian matrix \(Df(x)\) have negative real part for any \(x\in\mathbb{R}^n\), then \(x^*\) is a global attractor. The conjecture is true for dimension \(n=1,2\) and for linear systems. There are counterexamples for polynomial vector fields in \(\mathbb{R}^n\) with \(n>2\). The authors consider discontinuous piecewise linear differential systems of the form \[ \dot{z}= \begin{cases} X(z),& \text{if } x>0, \\ Y(z),& \text{if } x<0, \end{cases} \] where \(z=(x,y)\in\mathbb{R}^2\), \(X\) and \(Y\) are linear vector fields, the real part of the eigenvalues of \(DX(z)\) and \(DY(z)\) are negative, \(Y(0)=0\), and the singularity of \(X\) leaves in the half-plane \(\{x<0\}\). The authors call this kind of piecewise linear systems as \textit{piecewise linear Markus-Yamabe system} and prove that the Markus-Yamabe conjecture does not hold for them.
discontinuous piecewise linear differential system
scientific article; zbMATH DE number 7348188
scientific article