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Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs

Giuliani, Filippo ; Guardia, Marcel ; Martin, Pau and Pasquali, Stefano LU (2021) In Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni 32(1). p.149-166
Abstract

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 Birkho. Normal Form of those equations.

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author
; ; and
organization
publishing date
type
Contribution to journal
publication status
published
subject
keywords
Birkho. normal form, Hamiltonian PDEs, Transfer of energy
in
Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni
volume
32
issue
1
pages
18 pages
publisher
European Mathematical Society Publishing House
external identifiers
  • scopus:85106298226
ISSN
1120-6330
DOI
10.4171/RLM/931
language
English
LU publication?
yes
id
e979e65d-f692-44ca-ac5e-cc740f8ae9c6
date added to LUP
2022-01-04 14:52:38
date last changed
2022-04-27 07:00:50
@article{e979e65d-f692-44ca-ac5e-cc740f8ae9c6,
  abstract     = {{<p>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 Birkho. Normal Form of those equations.</p>}},
  author       = {{Giuliani, Filippo and Guardia, Marcel and Martin, Pau and Pasquali, Stefano}},
  issn         = {{1120-6330}},
  keywords     = {{Birkho. normal form; Hamiltonian PDEs; Transfer of energy}},
  language     = {{eng}},
  number       = {{1}},
  pages        = {{149--166}},
  publisher    = {{European Mathematical Society Publishing House}},
  series       = {{Atti della Accademia Nazionale dei Lincei, Classe di Scienze Fisiche, Matematiche e Naturali, Rendiconti Lincei Matematica e Applicazioni}},
  title        = {{Chaotic resonant dynamics and exchanges of energy in Hamiltonian PDEs}},
  url          = {{http://dx.doi.org/10.4171/RLM/931}},
  doi          = {{10.4171/RLM/931}},
  volume       = {{32}},
  year         = {{2021}},
}