THE MARKUS-YAMABE CONJECTURE FOR CONTINUOUS AND DISCONTINUOUS PIECEWISE LINEAR DIFFERENTIAL SYSTEMS

In 1960 Markus and Yamabe made the following conjecture: If a C I- differential system (x) over dot = F(x) in R-n has a unique equilibrium point and the Jacobian matrix of F(x) for all x is an element of 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 for all n >= 2.