Homoclinic Bifurcations with Weakly Expanding Center Manifolds

Interaction of homoclinic bifurcation and bifurcation on the center manifold is studied. We show that the occurrence of different types of solutions near the homoclinic orbit is determined asymptotically by a reduced system on the center manifold. The method is applied to cases where the center manifold is one- or two-dimensional. When the center manifold is one-dimensional, we can obtain all the solutions near the homoclinic orbit. When a Hopf bifurcation occurs on a two-dimensional center manifold, the system can have infinitely many periodic and aperiodic solutions. These solutions disappear in a manner predicted by the reduced system when the perturbation term is increased. We prove that certain periodic and aperiodic solutions disappear through inverse period doubling or saddle-node bifurcation.

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