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 work analyzes generic unfoldings of analytic Hamiltonian Hopf bifurcations with codimension-1 boundaries and proves that subcritical unfoldings possess transverse homoclinic orbits, implying chaotic dynamics and arbitrarily large topological entropy. The authors develop a two-scale (outer/inner) complex-analytic framework, derive a versal normal form near the Hopf point, and show that invariant manifolds split at exponentially small scale via inner/outer matching and Hamilton–Jacobi equations. They apply the theory to the restricted planar circular three-body problem near the Lagrangian point L4, showing transversal intersections conditional on a Stokes-constant-type parameter, Theta, which is expected generically to be nonzero. The results provide a constructive route to detect transversal homoclinic intersections in Hamiltonian systems with Hopf bifurcations and illuminate why Melnikov-type predictions fail in exponentially small splitting regimes. Overall, the paper establishes that chaotic dynamics are generic in a natural open set of subcritical Hopf unfoldings and offers precise asymptotics for homoclinic distance in RPC3BP 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.

Figures (7)

• Figure 1: The primaries, as black dots, and the equilibria, as red diamonds, of the RPC3BP in rotating coordinates. The black arrows represent the forces acting on the massless particle at $q$.
• Figure 2: Evolution in the complex plane of the eigenvalues associated to the origin of $\mathbf{H}_{\mu}$ with respect to the bifurcation parameter $\mu$
• Figure 3: The representation of the homoclinic surface $W^{\mathrm{u},\mathrm{s}}_0(0)$. The axis are $(y_1,y_2, \sqrt{x^2_1+ x^2_2})$. On the left, it is represented by a piece of the homoclinic surface, just to clarify its geometry. On the right, the homoclinic surface is shown.
• Figure 4: The domains $D^{\mathrm{out}, *}_{\kappa}$, $D^{\mathrm{out}, *}_{\kappa, \varsigma}$, and $D^{\mathrm{out}, *}_{\kappa, \infty}$ with $*=\mathrm{u}, \mathrm{s}$ defined by \ref{['def:domains_outer_s']} and \ref{['def:domains_outer_u']}.
• Figure 5: The dashed red-colored domain is $D^{\mathrm{out}, \mathrm{u}}_{\kappa}$, while the dashed green-colored one is $D^{\mathrm{out}, \mathrm{s}}_{\kappa}$. The domain in blue represents $D_{\kappa}^{\mathrm{ext}}$.

Theorems & Definitions (81)

• Theorem 1.1
• Theorem 1.2
• Corollary 1.3
• Remark 2.1
• Theorem 2.2
• Remark 2.3
• Proposition 2.4
• proof
• Lemma 2.5
• proof