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

Abstract

We investigate periodic and chaotic solutions of Hamiltonian systems in R 4 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.

Chaotic Braided Solutions via Rigorous Numerics: Chaos in the Swift–Hohenberg Equation

“…[2,7,8,25]) and variational methods (e.g. [6,20,22,27]) have been used extensively to study (2) and related fourth order equations, they have not succeeded in revealing chaos for the Swift-Hohenberg ODE.…”

Travelling Waves for Fourth Order Parabolic Equations

“…For a thorough understanding of (1.1), the stationary problem is of great importance. An extensive literature on this subject exists (see, e.g., [3,29,7,16,17,18,25,22,23,24]). Typically, depending on the parameter γ, the stationary problem displays a multitude of periodic, homoclinic, and heteroclinic solutions.…”

Generalized monotonicity from global minimization in fourth-order ordinary differential equations

“…(6) By condition ( 6) we have v1 @ v2 E U. This operator extends in a natural way to three or more arguments, with the coordinate system of the result being that of the first argument.…”