Separatrix splitting at a Hamiltonian $0^2 i\omega$ bifurcation

TL;DR

This paper analyzes the exponentially small splitting of a separatrix in a Hamiltonian system near a degenerate equilibrium with a zero eigenvalue and purely imaginary eigenvalues, revealing implications for homoclinic orbits and normal form divergence.

Contribution

It provides an asymptotic expression for the separatrix splitting in a Hamiltonian bifurcation with a zero eigenvalue, linking normal form theory and complex analysis.

Findings

Splitting is exponentially small relative to the unfolding parameter.

Normal form series diverge due to separatrix splitting.

Results connect asymptotic analysis with complex continuation behavior.

Abstract

We discuss the splitting of a separatrix in a generic unfolding of a degenerate equilibrium in a Hamiltonian system with two degrees of freedom. We assume that the unperturbed fixed point has two purely imaginary eigenvalues and a double zero one. It is well known that an one-parametric unfolding of the corresponding Hamiltonian can be described by an integrable normal form. The normal form has a normally elliptic invariant manifold of dimension two. On this manifold, the truncated normal form has a separatrix loop. This loop shrinks to a point when the unfolding parameter vanishes. Unlike the normal form, in the original system the stable and unstable trajectories of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies non-existence of single-round homoclinic orbits and divergence of series in the normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with behaviour of analytic continuation of the system in a complex neighbourhood of the equilibrium.