On Chaotic Sets of Solutions for a Class of Differential Inclusions in $\mathbb{R}^{2}$

Abstract We investigate a continuous multi-valued dynamical system in $\mathbb{R}^{2}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>R</mml:mi> <mml:mn>2</mml:mn> </mml:msup> </mml:math> of the form $\dot{x} \in F(x)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>x</mml:mi> <mml:mo>˙</mml:mo> </mml:mover> <mml:mo>∈</mml:mo> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> , where $F(x)$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> <mml:mo>(</mml:mo> <mml:mi>x</mml:mi> <mml:mo>)</mml:mo> </mml:math> is a set-valued function and $F=\{f_{1},f_{2}\}$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> <mml:mo>=</mml:mo> <mml:mo>{</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>1</mml:mn> </mml:msub> <mml:mo>,</mml:mo> <mml:msub> <mml:mi>f</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mo>}</mml:mo> </mml:math> . Such dynamical systems have applications in various fields, such as mathematical economics. We accurately establish the sufficient conditions for a set of solutions to such a system to exhibit Devaney chaos, $\omega $ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> -chaos, and infinite topological entropy. Finally, we illustrate these issues with our macroeconomic model, and we highlight the broader applicability of the proposed framework to systems with regime switching.