The Markus-Yamabe conjecture for continuous and discontinuous piecewise linear differential systems [ Back ]

Date:
02.03.20   
Times:
15:30
Place:
CRM - Auditori
Speaker:
Jaume Llibre
University:
Universitat Autònoma de Barcelona

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 $n\ge 2$.

 

This is a joint work with Xiang Zhang.