Coalescence of reversible homoclinic orbits causes elliptic resonance

Reversible flows can possess a robust homoclinic orbit to a saddle equilibrium: the orbit is preserved under small perturbations that do not destroy the reversibility of the system. Such a homoclinic orbit is a limit of a unique one-parameter family of periodic orbits. All these orbits are saddles if the equilibrium state is a saddle. There are both saddle and elliptic periodic orbits in this family if the equilibrium state is a saddle-focus. In the present paper, we study the coalescence of two such homoclinic orbits in a one-parameter family of reversible flows. We show that, even in the case where all eigenvalues of the corresponding equilibrium are real, a family of elliptic periodic orbits arises at this bifurcation.