Following , we consider a class of reversible systems and study bifurcations of homoclinic orbits to hyperbolic saddle equilibria. Here we concentrate on the case in which homoclinic orbits are symmetric, so that only one control parameter is enough to treat their bifurcations, as in Hamiltonian systems. This result also provides a fundamental theory in a joint work with Dr. Stachowiak , which was described in the colloquium last year.
First, we apply an argument given in  to show that if such bifurcations occur in four-dimensional systems, then variational equations around the homoclinic orbits are integrable. We next extend the Melnikov method of  to reversible systems and obtain theorems on saddle-node, transcritical and pitchfork bifurcations of symmetric homoclinic orbits. We illustrate our theory for a four-dimensional system, and demonstrate the theoretical results by numerical ones.
 D. Blazquez-Sanz and K. Yagasaki, Analytic and algebraic conditions for bifurcations of homoclinic orbits I: Saddle equilibria, J. Differential Equations, 253 (2012), 2916--2950.
 K. Yagasaki and T. Stachowiak, Bifurcations in a coupled elliptic system with critical growth, in preparation.