Eur. Phys. J. Special Topics 223, 665-676 (2014)
https://doi.org/10.1140/epjst/e2014-02132-0

Regular Article

Anti-phase synchronization and ergodicity in arrays of oscillators coupled by an elastic force

Instituto Superior Técnico, University of Lisbon, NonLinear Dynamics Group, Av. Rovisco Pais, 1049-001 Lisboa, Portugal

^a e-mail: ruidilao@tecnico.ulisboa.pt

Received: 3 March 2014
Revised: 18 March 2014
Published online: 28 April 2014

Abstract

We have proposed a mechanism of interaction between two non-linear dissipative oscillators, leading to exact and robust anti-phase and in-phase synchronization. The system we have analyzed is a model for the Huygens's two pendulum clocks system, as well as a model for synchronization mediated by an elastic media. Here, we extend these results to arrays, finite or infinite, of conservative pendula coupled by linear elastic forces. We show that, for two interacting pendula, this mechanism leads always to synchronized anti-phase small amplitude oscillations, and it is robust upon variation of the parameters. For three or more interacting pendula, this mechanism leads always to ergodic non-synchronized oscillations. In the continuum limit, the pattern of synchronization is described by a quasi-periodic longitudinal wave.