Splitting of separatrices for the Hamiltonian-Hopf bifurcation with the Swift-Hohenberg equation as an example

TL;DR

This paper investigates the exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation in the Swift-Hohenberg equation, combining asymptotic analysis with high-precision numerics to quantify transversality.

Contribution

It introduces an asymptotic expansion for the homoclinic invariant and numerically evaluates the Stokes constant, advancing understanding of manifold splitting beyond normal form theory.

Findings

Asymptotic expansion accurately describes manifold transversality.

Numerical methods confirm the exponential smallness of splitting.

Stokes constant computed with high precision supports theoretical predictions.

Abstract

We study homoclinic orbits of the Swift-Hohenberg equation near a Hamiltonian-Hopf bifurcation. It is well known that in this case the normal form of the equation is integrable at all orders. Therefore the difference between the stable and unstable manifolds is exponentially small and the study requires a method capable to detect phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for an homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to support validity of the asymptotic expansion and evaluate a Stokes constant numerically using two independent methods.