The Takens--Bogdanov Bifurcation with O(2)-Symmetry

The versal deformation of a vector field of co-dimension two that is equivariant under a representation of the symmetry group O(2) and has a nilpotent linearization at the origin is studied. An appropriate scaling allows us to formulate the problem in terms of a central-force problem with a small dissipative perturbation. We derive and analyse averaged equations for the angular momentum and the energy of the classical motion. The unfolded system possesses four different types of non-trivial solutions: a steady-state and three others, which are referred to in a wave context as travelling waves, standing waves and modulated waves. The plane of unfolding parameters is divided into a number of regions by (approximately) straight lines corresponding to primary and secondary bifurcations. Crossing one of these lines leads to the appearance or disappearance of a particular solution. We locate secondary saddle-node, Hopf and pitchfork bifurcations as well as three different global, i.e. homoclinic and heteroclinic, bifurcations.