Dynamics near the invariant manifolds after a Hamiltonian-Hopf bifurcation

We consider a one parameter family of 2-DOF Hamiltonian systems having an equi-librium 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 Poincare 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 Poincare map. We illustrate the results in a concrete example. Different Poincare 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.(c) 2022 The Author(s). Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Automated summary

We study a one-parameter family of 2-DOF Hamiltonian systems with a Hamiltonian-Hopf bifurcation at an equilibrium point. The normal form theory is reviewed, and the behavior of the splitting of 2D invariant manifolds is investigated when there are homoclinic orbits to the complex-saddle equilibrium point. Symmetries of the normal form are utilized to reduce the dynamics to a family of area-preserving Poincare maps, allowing for the derivation of an explicit upper bound for the splitting of separatrices. The results are illustrated with a concrete example and a separatrix map is derived to analyze the chaotic dynamics near the 2D invariant manifolds.