Separatrix Splitting at a Hamiltonian $0^2 i\omega$ Bifurcation

We study 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 non-semisimple double zero one. It is well known that a one-parameter 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 separatrices of the equilibrium do not coincide in general. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies nonexistence of single-round homoclinic orbits and divergence of series in normal form theory. We derive an asymptotic expression for the separatrix splitting. We also discuss relations with the behavior of analytic continuation of the system in a complex neighborhood of the equilibrium.