Periodic Solutions of Perturbed Autonomous Systems, and Locking-In

Periodic solutions to perturbed autonomous systems are discussed, where the autonomous system has an isolated periodic solution and the period of the perturbation is considered as a parameter. With typical assumptions on the variational equation it is found that there exists a periodic k-parameter family of periods and corresponding periodic solutions for all sufficiently small values of the k perturbation parameters. The major result generalizes a theorem of Poincaré in which the perturbation is autonomous. An intrinsic transformation allows use of the standard implicit function theorem, and a degeneracy previously encountered in the application is avoided. The results are much improved. A sufficient condition for locking-in to occur is found which generalizes a known sufficient condition. Also stability results are given.