Bifurcations and invariant sets for a family of replicator maps from evolutionary games

TL;DR

This paper investigates the complex dynamics of a family of replicator maps from evolutionary games, proving hyperbolic chaos and addressing an open problem about periodic orbits with equal mean values.

Contribution

It demonstrates the existence of hyperbolic chaos in these maps and provides a partial solution to characterizing maps with uniform mean periodic orbits.

Findings

Proves hyperbolic chaos for the family of replicator maps.

Addresses an open problem regarding maps with all periodic orbits having the same mean.

Provides insights into the bifurcation structure of evolutionary game dynamics.

Abstract

We study the dynamics of a family of replicator maps, depending on two parameters. Such studies are motivated by the analysis of the dynamics of evolutionary games under selections. From the dynamics viewpoint, we prove the existence of hyperbolic chaos for the considered map. Moreover, we also give a partial solution of an open problem formulated in \cite{Misiurewicz1}: to describe all the one-dimensional maps with all the periodic orbits having the same mean value.