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Multiple homoclinic orbits in conservative and reversible systems. (English) Zbl 1162.34034

Summary: We study dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. We consider the existence of two homoclinic orbits in the bellows configuration, where the homoclinic orbits approach the equilibrium along the same direction for positive and negative times.

In conservative systems one finds one parameter families of suspended horseshoes, parameterized by the level of the first integral. A somewhat similar picture occurs in reversible systems, with two homoclinic orbits that are both symmetric. The lack of a first integral implies that complete horseshoes do not exist. We provide a description of orbits that necessarily do exist.

A second possible configuration in reversible systems occurs if a non-symmetric homoclinic orbit exists and forms a bellows together with its symmetric image. We describe the nonwandering set in an unfolding. The nonwandering set is shown to simultaneously contain one-parameter families of periodic orbits, hyperbolic periodic orbits of different index, and heteroclinic cycles between these periodic orbits.

MSC:
34C37Homoclinic and heteroclinic solutions of ODE
34C14Symmetries, invariants (ODE)
37C29Homoclinic and heteroclinic orbits