The Takens-Bogdanov bifurcation with O(2)-symmetry.

*(English)*Zbl 0635.58032The authors study the bifurcations in a four-dimensional system which is equivariant under a representation of the group O(2) and which has a nilpotent linearization at the origin. Using normal forms and rescaling (including a time rescale) this codimension two problem is reduced to the study of a dissipative perturbation of a classical central force problem in the plane. The persistence of the closed orbits of the unperturbed (integrable) system under the perturbation is studied using the averaging method. Together with some (linear) stability calculations this leads to a series of detailed bifurcation diagrams.

There are four types of nontrivial solutions: steady-state solutions, bifurcating from the trivial solution; standing wave and travelling wave solutions, bifurcating from either the trivial solution or from the steady-state solutions by Hopf bifurcations; and modulated wave solutions, originating by secondary bifurcations from branches of standing or travelling waves. One also finds saddle-nodes of standing waves, and three types of global bifurcations: homoclinic, heteroclinic and saddle-loop bifurcations.

There are four types of nontrivial solutions: steady-state solutions, bifurcating from the trivial solution; standing wave and travelling wave solutions, bifurcating from either the trivial solution or from the steady-state solutions by Hopf bifurcations; and modulated wave solutions, originating by secondary bifurcations from branches of standing or travelling waves. One also finds saddle-nodes of standing waves, and three types of global bifurcations: homoclinic, heteroclinic and saddle-loop bifurcations.

Reviewer: A.Vanderbauwhede

##### MSC:

37G99 | Local and nonlocal bifurcation theory for dynamical systems |