On the non-transverse homoclinic channel of a center manifold

TL;DR

This paper analyzes the dynamics near a center manifold where stable and unstable manifolds coincide, deriving normal forms and exploring complex behaviors like Henon-like maps and Lorenz attractors, with applications to game theory.

Contribution

It introduces a normal form analysis for systems with coinciding stable and unstable manifolds on a center manifold, revealing complex limit dynamics including Lorenz-like attractors.

Findings

Normal form of the governing ODE derived

Return map can be Henon-like or satisfy cone-field condition

Complex attractors such as Lorenz-like are possible

Abstract

We consider a scenario when a stable and unstable manifolds of compact center manifold of a saddle-center coincide. The normal form of the ODE governing the system near the center manifold is derived and so is the normal form of the return map to the neighbourhood of the center manifold. The limit dynamics of the return map is investigated by showing that it might take the form of a Henon-like map possessing a Lorenz-like attractor or satisfy 'cone-field condition' resulting in partial hyperbolicity. We consider also motivating example from game theory.