Exponentially small splitting of invariant manifolds near a Hamiltonian-Hopf bifurcation

Abstract

Consider an analytic two-degrees of freedom Hamiltonian system with an equilibrium point that undergoes a Hamiltonian-Hopf bifurcation, i.e., the eigenvalues of the linearized system at the equilibrium change from complex ±β ±iα (α,β > 0) for ε > 0 to pure imaginary ±iω1 and ±iω2 (ω1 ≠ ω2 ≠ 0) for ε < 0. At ε = 0 the equilibrium has a pair of doubled pure imaginary eigenvalues. Depending on the sign of a certain coefficient of the normal form there are two main bifurcation scenarios. In one of these (the stable case), two dimensional stable and unstable manifolds of the equilibrium shrink and disappear as ε → 0+. At any order of the normal form the stable and unstable manifolds coincide and the invariant manifolds are indistinguishable using classical perturbation theory. In particular, Melnikov’s method is not capable to evaluate the splitting. In this thesis we have addressed the problem of measuring the splitting of these manifolds for small values of the bifurcation parameter ε. We have estimated the size of the splitting which depends on a singular way from the bifurcation parameter. In order to measure the splitting we have introduced an homoclinic invariant ωε which extends the Lazutkin’s homoclinic invariant defined for area-preserving maps. The main result of this thesis is an asymptotic formula for the homoclinic invariant. Assuming reversibility, we have proved that there is a symmetric homoclinic orbit such that its homoclinic invariant can be estimated as follows, ωε = ±2e−πα/2β (ω0 + O(ε1−μ)). where μ > 0 is arbitrarily small and ω0 is known as the Stokes constant. This asymptotic formula implies that the splitting is exponentially small (with respect to ε). When ω0 ≠ 0 then the invariant manifolds intersect transversely. The Stokes constant ω0 is defined for the Hamiltonian at the moment of bifurcation only. We also prove that it does not vanish identically. Finally, we apply our methods to study homoclinic solutions in the Swift-Hohenberg equation. Our results show the existence of multi-pulse homoclinic solutions and a small scale chaos