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

Abstract

We consider autonomous Lagrangian systems with two degrees of freedom, having an hyperbolic equilibrium of saddle-saddle type (that is the eingenvalues of the linearized system about the equilibrium are $\pm \lambda_1, \pm \lambda_2 $, $\lambda_1, \lambda_2 > 0$). We assume that $\lambda_1 > \lambda_2$ and 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, then 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 and it is applied, for simplicity, when $\lambda_1 \approx \lambda_2$. An application to a perturbation of a uncoupled system is also given.