Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation

E. Fontich, A. Vieiro
Departament de Matem\`atiques i Inform\`atica, Universitat de Barcelona (UB), Gran Via, 585, 08007 Barcelona, Spain
Centre de Recerca Matem\`atica (CRM), Campus Bellaterra, 08193 Bellaterra, Spain

Abstract:

We consider a one parameter family of 2-DOF
Hamiltonian systems having an equilibrium point that undergoes a Hamiltonian-Hopf
bifurcation. We briefly review the well-established normal form theory in this
case. Then we focus on the invariant manifolds when there are homoclinic orbits
to the complex-saddle equilibrium point, and we study the behavior of the
splitting of the 2D invariant manifolds. The symmetries of the normal form are
used to reduce the dynamics around the invariant manifolds to the dynamics of a
family of area-preserving near-identity Poincar\'e maps that can be extended
analytically to a suitable neighborhood of the separatrices. This allows, in
particular, to use well-known results for area-preserving maps and derive an
explicit upper bound of the splitting of separatrices for the Poincar\'e map.
We illustrate the results in a concrete example.
Different Poincar\'e sections are used to visualize the dynamics near the 2D
invariant manifolds. Last section deals with the derivation of
a separatrix map to study the chaotic dynamics near the 2D invariant manifolds.