Homoclinic bifurcations in reversible Hamiltonian systems

Abstract

We study the existence of homoclic solutions for reversible Hamiltonian systems taking the family of differential equations u iv + au″ − u + f( u, b) = 0 as a model, where f is an analytic function and a, b real parameters. These equations are important in several physical situations such as solitons and in the existence of “finite energy” stationary states of partial differential equations, but no assumptions of any kind of discrete symmetry is made and the analysis here developed can be extended to others Hamiltonian systems and successfully employed in situations where standard methods fail. We reduce the problem of computing these orbits to that of finding the intersection of the unstable manifold with a suitable set and then apply it to concrete situations. We also plot the homoclinic values configuration in parameters space, giving a picture of the structural distribution and a geometrical view of homoclinic bifurcations.