Dynamics of a Continuous Piecewise Affine Map of the Square

TL;DR

This paper studies a family of area-preserving, piecewise affine maps of the square, exploring invariant structures and complex dynamics inspired by game theory, including explicit constructions and numerical experiments.

Contribution

It introduces a new one-parameter family of maps with explicit invariant circles and annuli, analyzing their coexistence of stochastic and periodic behaviors.

Findings

Existence of invariant circles with rational and irrational rotation numbers.

Numerical evidence of complex dynamics within invariant annuli.

Explicit constructions of invariant structures for specific parameters.

Abstract

We present a one-parameter family of continuous, piecewise affine, area preserving maps of the square, which are inspired by a dynamical system in game theory. Interested in the coexistence of stochastic and (quasi-)periodic behaviour, we investigate invariant annuli separated by invariant circles. For certain parameter values, we explicitly construct invariant circles both of rational and irrational rotation numbers, and present numerical experiments of the dynamics on the annuli bounded by these circles.