Variational construction of homoclinics and chaos in presence of a saddle-saddle equilibrium

We consider autonomous Lagrangian systems with two degrees of freedom, having a hyperbolic equilibrium of saddle-saddle type (that is, the eingenvalues of the linearized system about the equilibrium are real and distinct. We assume that the system possesses two homoclinic orbits. Under a nondegeneracy assumption on the homoclinics and under suitable conditions on the geometric behaviour of these homoclinics near the equilibrium we prove, by variational methods, that they give rise to an infinite family of multibump homoclinic solutions and that the topological entropy at the zero energy level is positive. A method to deal also with homoclinics satisfying a weaker nondegeneracy condition is developed. An application to a perturbation of an uncoupled system is also given.