A general reduction method for periodic solutions in conservative and reversible systems

We introduce a general reduction method for the study of periodic solutions near equilibria in autonomous systems which are either conservative or reversible. We impose no restrictions on the linearization at the equilibrium, allowing higher multiplicities and all kinds of resonances. It is shown that the problem reduces to a similar problem for a reduced system, which is itself conservative or reversible, but also has an additionalS ¹-symmetry. This symmetry allows to immediately write down the bifurcation equations. Moreover, the reduced system can be calculated up to any order by a normal form reduction on the original system. The method of proof combines normal forms with the Liapunov-Schmidt method. A similar approach was already introduced for Hamiltonian systems in 9], and for equivariant systems in 3]; this paper extends the results of these papers to the cases of conservative and reversible systems.The research in this paper was supported by the EEC Science Project on Bifurcation Theory and Its Applications.