Asymptotic Poincar\'e Maps along the edges of Polytopes

TL;DR

This paper introduces a piecewise linear model for analyzing asymptotic dynamics on polytopes, enabling efficient computation of invariant structures and demonstrating chaos in certain Hamiltonian replicator systems.

Contribution

It presents a novel, computationally accessible model for asymptotic flow analysis on polytopes, applicable to high-dimensional systems and evolutionary game theory.

Findings

Model captures asymptotic dynamics along polytope edges

Enables numerical detection of invariant structures

Proves chaos existence in 5D Hamiltonian replicator systems

Abstract

For a class of flows on polytopes, including many examples from Evolutionary Game Theory, we describe a piecewise linear model which encapsulates the asymptotic dynamics along the heteroclinic network formed out of the polytope's vertexes and edges. This piecewise linear flow is easy to compute even in higher dimensions, which allows the usage of numeric algorithms to find invariant dynamical structures such as periodic, homoclinic or heteroclinic orbits, which if robust persist as invariant dynamical structures of the original flow. We apply this method to prove the existence of chaotic behavior in some Hamiltonian replicator systems on the five dimensional simplex.