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

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 of detecting phenomena beyond all algebraic orders provided by the normal form theory. We propose an asymptotic expansion for a homoclinic invariant which quantitatively describes the transversality of the invariant manifolds. We perform high-precision numerical experiments to support the validity of the asymptotic expansion and evaluate a Stokes constant numerically using two independent methods.