M. Golubitsky, J. Marsden, I. Stewart and M. Dellnitz
The Constrained Liapunov-Schmidt Procedure and Periodic Orbits
Fields Institute Proceedings 4 (1995) 81-127
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 an integral, as well as to
the Hamiltonian Hopf bifurcation and resonance bifurcations for
Hamiltonian and reversible systems.