Asymptotic Poincaré maps along the edges of polytopes

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 vertices and edges. This piecewise linear flow is easy to compute, even in higher dimensions, allowing the usage of numeric algorithms to detect robust invariant dynamical structures such as periodic, homoclinic and heteroclinic orbits. We apply this method to prove the existence of chaotic behaviour in some Hamiltonian replicator systems on the five dimensional simplex.