The Constrained Liapunov - Schmidt Procedure and Periodic Orbits

This paper develops the Liapunov-Schmidt procedure for systems with additional constraints such as having a first integral, being Hamiltonian, or being a gradient system. Similar developments for systems with symmetry, including reversibility, are well known, and the method of this paper augments and is consistent with that approach. One of the results states that the bifurcation equation for Hamiltonian systems is actually a Hamiltonian vector field. In general, we use "implicit constraints" to encode the information constraining the system. The method is applied to the Liapunov center theorem for reversible systems and systems with 'in integral, as well as to the Hamiltonian Hopf bifurcation and resonance bifurcations for Hamiltonian and reversible systems. 1991 Mathematics Subject Classification. Primary 58F05; Secondary 58F14. 70K30. "Research partially supported by NSF Grant DMS-9101836, the Texas Advanced Research Program (003652037) and The Fields Institute . .. Research partially supported by NSF Grant DM8-9302992 and The Fields Institute. t Research partially supported by The Fields Institute. the Science and Engineering Research Council of the UK. and a European Commwlity Laboratory Twinning grant (European Bifurcation Theory Group). lResearch partially supported by The Fields Institute. and a European Community Laboratory Twinning grant (European Bifurcation Theory Group). @ 1995 American Matbematical Society