Chaotic phenomena in generic unfoldings of the Hamilton Hopf bifurcation with emphasis on the restricted planar circular 3-body problem beyond the Gascheau-Routh mass ratio

TL;DR

This paper proves that generic unfoldings of Hamiltonian Hopf singularities exhibit chaotic dynamics with transverse homoclinic orbits, and applies this to the restricted planar circular three-body problem near L4, contingent on a subtle condition involving the Stokes constant.

Contribution

It establishes the existence of chaotic dynamics in Hamiltonian systems near Hopf bifurcations and applies this to the three-body problem, highlighting the role of the Stokes constant.

Findings

Presence of transverse homoclinic orbits near the bifurcation

Chaotic dynamics with arbitrarily large topological entropy

Conditional results for the three-body problem near L4

Abstract

In this work, we prove that a generic unfolding of an analytic Hamiltonian Hopf singularity (in an open set with codimension 1 boundary) possesses transverse homoclinic orbits for subcritical values of the parameter close to the bifurcation parameter. As a consequence, these systems display chaotic dynamics with arbitrarily large topological entropy. We verify that the Hamiltonian of the restricted planar circular three-body problem (RPC3BP) close to the Lagrangian point $L_4$ falls within this open set. The generic condition ensuring the presence of transversal homoclinic intersections is subtle and involves the so-called Stokes constant. Thus, in the case of the RPC3BP close to $L_4$, our result holds conditionally on the value of this constant.