The EMS Publishing House is now EMS Press and has its new home at ems.press.

Please find all EMS Press journals and articles on the new platform.

Rendiconti Lincei - Matematica e Applicazioni


Full-Text PDF (420 KB) | Metadata | Table of Contents | RLM summary
Volume 32, Issue 1, 2021, pp. 149–166
DOI: 10.4171/RLM/931

Published online: 2021-04-22

Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs

Filippo Giuliani[1], Marcel Guardia[2], Pau Martin[3] and Stefano Pasquali[4]

(1) Universitat Politècnica de Catalunya and Centro de Recerca Matemàtica, Barcelona, Spain
(2) Universitat Politècnica de Catalunya and Centro de Recerca Matemàtica, Barcelona, Spain
(3) Universitat Politècnica de Catalunya and Centro de Recerca Matemàtica, Barcelona, Spain
(4) Lund University, Sweden

The aim of this note is to present the recent results in [16] where we provide the existence of solutions of some nonlinear resonant PDEs on T2 exchanging energy among Fourier modes in a "chaotic-like" way. We say that a transition of energy is "chaotic-like" if either the choice of activated modes or the time spent in each transfer can be chosen randomly. We consider the nonlinear cubic Wave, the Hartree and the nonlinear cubic Beam equations. The key point of the construction of the special solutions is the existence of heteroclinic connections between invariant objects and the construction of symbolic dynamics (a Smale horseshoe) for the Birkhoff Normal Form of those equations.

Keywords: Transfer of energy, Birkhoff normal form, Hamiltonian PDEs

Giuliani Filippo, Guardia Marcel, Martin Pau, Pasquali Stefano: Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. 32 (2021), 149-166. doi: 10.4171/RLM/931