Homotopy Classes for Stable Periodic and Chaotic¶Patterns in Fourth-Order Hamiltonian Systems

Abstract:

We investigate periodic and chaotic solutions of Hamiltonian systems in ℝ⁴ which arise in the study of stationary solutions of a class of bistable evolution equations. Under very mild hypotheses, variational techniques are used to show that, in the presence of two saddle-focus equilibria, minimizing solutions respect the topology of the configuration plane punctured at these points. By considering curves in appropriate covering spaces of this doubly punctured plane, we prove that minimizers of every homotopy type exist and characterize their topological properties.