Separatrix splitting in a Hamiltonian bifurcation
speaker: Vassili Gelfreich (Warwick university, UK)abstract: In this talk 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 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. The splitting of this loop is exponentially small compared to the small parameter. This phenomenon implies the 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 degenerate equilibrium.
timetable:
Mon 5 May, 10:30 - 11:30, Aula Dini
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