Fixed points imply chaos for a class of differential inclusions that arise in economic models

Abstract:

We consider multi-valued dynamical systems with continuous time of the form $\dot {x}\in F(x)$, where $F(x)$ is a set-valued function. Such models have been studied recently in mathematical economics. We provide a definition for chaos, $\omega$-chaos and topological entropy for these differential inclusions that is in terms of the natural $\mathbb {R}$-action on the space of all solutions of the model. By considering this more complicated topological space and its $\mathbb {R}$-action we show that chaos is the ‘typical’ behavior in these models by showing that near any hyperbolic fixed point there is a region where the system is chaotic, $\omega$-chaotic, and has infinite topological entropy.