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

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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.

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