Cascades of Reversible Homoclinic Orbits to a Saddle-Focus Equilibrium

In a reversible system, we consider a homoclinic orbit being bi-asymptotic to a saddle-focus equilibrium. As was proved by Devaney, there exists a one-parameter family of periodic orbits accumulating onto this homoclinic orbit. In the present paper, we show that for any n 2 there exist infinitely many n- homoclinic orbits in a neighborhood of the primary homoclinic orbit . Each of them is accompanied by one or more families of periodic orbits. Moreover, we indicate how these families of periodic orbits correspond to branches of subharmonic periodic orbits. 1 Introduction Homoclinic orbits are well-known to have a great influence on the dynamics in a neighborhood. In systems with a parameter, homoclinic orbits are typically destroyed under small perturbations. However, in many cases one can find periodic orbits, n-homoclinic orbits or shift dynamics at parameter values that are close to the value where the primary homoclinic orbit exists. The situation is different in reversible or Ha..