Multiple Homoclinic Orbits in Conservative and Reversible Systems

I will consider dynamics near multiple homoclinic orbits to saddles in conservative and reversible flows. Suppose the system has 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. I will 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. I will 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.