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.
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Recommended by Dr Hinke M Osinga
Footnotes
- 1
A d-dimensional sector is any closed convex cone bounded by d transversal facets.
- 2
The precise definition of fan requires that the intersection of any two family members is either empty or else a common face and that faces of family members are also in the family.
- 3
The line graph of a directed graph G, denoted by
, is the graph whose vertices are the edges of G, and where is an edge of if the end-point of coincides with the start-point of . - 4
These are not KAM islands because the piecewise linearity of
does not allow any twist.
