Abstract:

In 1960 Markus and Yamabe made the following conjecture: If a $C^1$ differential system $dot {bf x}=F({bf x})$ in $R^n$ has a unique equilibrium point and the Jacobian matrix of $F({bf x})$ for all ${bf x}in R^n$ has all its eigenvalues with negative real part, then the equilibrium point is a global attractor. Until 1997 we do not have the complete answer to this conjecture. It is true in $R^2$, but it is false in $R^n$ for all $n>2$.

Here we extend the conjecture of Markus and Yamabe to continuous and discontinuous piecewise linear differential systems in $R^n$ separated by a hyperplane, and we prove that for the continuous piecewise linear differential systems it is true in $R^2$, but it is false in $R^n$ for all $n>2$. But for discontinuous piecewise linear differential systems it is false in $R^n$ for all $nge 2$.

This is a joint work with Xiang Zhang.