Chaotic behavior of rapidly oscillating Lagrangian systems

Francesca Alessio; Vittorio Coti Zelati; Piero Montecchiari

doi:10.3934/dcds.2004.10.687

Article Contents

2004, Volume 10, Issue 3: 687-707. Doi: 10.3934/dcds.2004.10.687

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Chaotic behavior of rapidly oscillating Lagrangian systems

1.
Dipartimento di Matematica, Università di Torino
2.
Dipartimento di Matematica e Applicazioni "R. Caccioppoli", Università di Napoli
3.
Dipartimento di Scienze Matematiche, Università Politecnica delle Marche

Received: December 2002

Revised: May 2003

Published: July 2004

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Abstract

In the paper we prove that the Lagrangian system

$ \ddot{q} = \alpha(\omega t) V'(q), \quad t \in \mathbb R, q \in \mathbb R^N,$ $\qquad\qquad (L_\omega)$

has, for some classes of functions $\alpha$, a chaotic behavior---more precisely the system has multi-bump solutions---for all $\omega$ large. These classes of functions include some quasi-periodic and some limit-periodic ones, but not any periodic function.
We prove the result using global variational methods.

Keywords:

Homoclinic solutions,
chaotic behavior,
almost-periodic and quasiperiodic forcing.

Mathematics Subject Classification: 37J45, 37C29, 34C37, 70K44.

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