Existence of Chaos in Plane $\mathbb{R}^2$ and its Application in Macroeconomics

Abstract

The Devaney, Li-Yorke and distributional chaos in plane $\mathbb{R}^2$ can occur in the continuous dynamical system generated by Euler equation branching. Euler equation branching is a type of differential inclusion $\dot x \in \{f(x),g(x) \} $, where $f,g:X \subset \mathbb{R}^n \rightarrow \mathbb{R}^n$ are continuous and $f(x) \neq g(x)$ in every point $x \in X$. Stockman and Raines (Stockman, D. R.; Raines, B. R.: Chaotic sets and Euler equation branching, Journal of Mathematical Economics, 2010, Volume 46, pp. 1173-1193) defined so-called chaotic set in plane $\mathbb{R}^2$ which existence leads to an existence of Devaney, Li-Yorke and distributional chaos. In this paper, we follow up on Stockman, Raines and we show that chaos in plane $\mathbb{R}^2$ with two "classical" (with non-zero determinant of Jacobi's matrix) hyperbolic singular points of both branches not lying in the same point in $\mathbb{R}^2$ is always admitted. But the chaos existence is caused also by set of solutions of Euler equation branching which have to fulfil conditions following from the definition of so-called chaotic set. So, we research this set of solutions. In the second part we create new overall macroeconomic equilibrium model called IS-LM/QY-ML. The construction of this model follows from the fundamental macroeconomic equilibrium model called IS-LM but we include every important economic phenomena like inflation effect, endogenous money supply, economic cycle etc. in contrast with the original IS-LM model. We research the dynamical behaviour of this new IS-LM/QY-ML model and show when a chaos exists with relevant economic interpretation.