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

Abstract

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

Exponentially small splitting of separatrices beyond Melnikov analysis: Rigorous results

“…The next theorem gives the existence of the invariant manifolds in the domains D * ∞,ρ with * = u, s defined in (34). We state the results for the unstable invariant manifold.…”

“…We look for the parameterizations of the local invariant manifolds in the domains D u,s ∞,ρ defined in (34).…”

Analytic invariants for the 1:-1 resonance

“…the matrix DX H (0) is not diagonalizable and has a pair of double imaginary eigenvalues ±iα, α > 0. Our study is motivated by the problem of estimating the size of the chaotic zone near a Hamiltonian-Hopf bifurcation [9,19,24]. This is a codimension one bifurcation of an equilibrium point in a two degrees of freedom Hamiltonian system in R 4 .…”

“…In [9] a quantity ω known as homoclinic invariant was introduced to measure the size of the splitting of stable and unstable manifolds. Roughly speaking, it is defined to be the symplectic area formed by a pair of normalized tangent vectors at a homoclinic point.…”

“…Thus, ω measures the splitting of the stable and unstable manifolds along the homoclinic orbit γ h . Based on analytical and numerical evidence, in [9] it is conjectured that the homoclinic invariant has the following asymptotic expansion,…”