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Invariant curves near Hamiltonian–Hopf bifurcations of four-dimensional symplectic maps

In this paper, we give a numerical description of an extended neighbourhood of a fixed point of a symplectic map undergoing an irrational transition from linear stability to complex instability, i.e. the so-called Hamiltonian–Hopf bifurcation. We have considered both the direct and inverse cases. This study is based on numerical computation of the Lyapunov families of invariant curves near the fixed point. We show how these families, jointly with their invariant manifolds and the invariant manifolds of the fixed point, organize the phase space around the bifurcation.