Lyapunov density for coupled systems

Abstract

We prove a necessary and sufficient condition for the existence of Lyapunov density for a system of coupled autonomous ordinary differential equations. In particular, we characterize the kinds of couplings that preserve almost everywhere uniform stability of the origin provided the isolated systems have an almost everywhere uniformly stable equilibrium point at the origin.

Keywords:

• almost everywhere uniform stability
• advection equation
• density function

AMS Subject Classifications:

• Primary: 93D05
• Secondary: 93B07