On chaotic sets of solutions for a class of differential inclusions on $\mathbb{R}^2$

Abstract

We deal with a set of solutions of the continuous multi-valued dynamical systems on $\mathbb{R}^2$ of the form $\dot x \in F(x)$ where $F(x)$ is a set-valued function and $F=\{f_1,f_2\}$. Such dynamical systems are frequently used in mathematical economics. We rectify the sufficient conditions for a set of solutions of this system to exhibit Devaney chaos, $\omega$-chaos and infinite topological entropy from: B.R. Raines, D.R. Stockman, Fixed points imply chaos for a class of differential inclusions that arise in economic models, Trans. American Math. Society 364 (5) (2012), 2479--2492. We significantly improve their results. At the end, we illustrate these problems on our own macroeconomic model.