Chaotic-like transfers of energy in Hamiltonian PDEs

TL;DR

This paper demonstrates the existence of solutions in certain Hamiltonian PDEs that exhibit chaotic-like energy exchanges between Fourier modes, using heteroclinic connections and symbolic dynamics in the Birkhoff Normal Form.

Contribution

It introduces a novel method to construct solutions with chaotic energy transfer in Hamiltonian PDEs via heteroclinic connections and symbolic dynamics.

Findings

Existence of solutions with energy exchange in cubic Wave, Hartree, and Beam equations.

Chaotic-like energy transfer can be controlled by mode choice and transfer timing.

Construction of symbolic dynamics for the Birkhoff Normal Form truncation.

Abstract

We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations on $T^2$ and we prove the existence of different types of solutions which exchange energy between Fourier modes in certain time scales. This exchange can be considered \emph{chaotic-like} since either the choice of activated modes or the time spent in each transfer can be chosen randomly. The key point of the construction of those orbits is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form truncation of those equations.