In chaotic systems, synchronization phenomena, i.e., resonances, can occur, the simplest being motions asymptotically developing over a low dimension attractor that is visited in synchrony with a weak time-periodic perturbing force. This is analogous to the synchronization (“phase locking”) occurring when an integrable system, perturbed by a periodic forcing and in the presence of friction, develops attracting periodic orbits with period being a simple fraction (1, 3/2, 2,. . .) of the forcing period. Formation of an attractor and existence of dissipation, i.e., of phase volume contraction, are concomitant. A simple illustration of the resonance phenomenon in a chaotic system is exhibited here and exactly solved by deriving the shape of the attractor. This turns out to be a continuous surface with weak smoothness properties, being a strange attractor with an exponent of Hoelder-continuity with a very small lower bound. The peculiarity of the class of systems treated is the possibility of determining analytically an attractor, whose existence can be rather easily seen in simulations.

Gallavotti, G., Gentile, G., Giuliani, A. (2012). Resonances within chaos. CHAOS, 22(2), 026108,1-6 [10.1063/1.3695370].

Resonances within chaos

GENTILE, Guido;GIULIANI, ALESSANDRO
2012-01-01

Abstract

In chaotic systems, synchronization phenomena, i.e., resonances, can occur, the simplest being motions asymptotically developing over a low dimension attractor that is visited in synchrony with a weak time-periodic perturbing force. This is analogous to the synchronization (“phase locking”) occurring when an integrable system, perturbed by a periodic forcing and in the presence of friction, develops attracting periodic orbits with period being a simple fraction (1, 3/2, 2,. . .) of the forcing period. Formation of an attractor and existence of dissipation, i.e., of phase volume contraction, are concomitant. A simple illustration of the resonance phenomenon in a chaotic system is exhibited here and exactly solved by deriving the shape of the attractor. This turns out to be a continuous surface with weak smoothness properties, being a strange attractor with an exponent of Hoelder-continuity with a very small lower bound. The peculiarity of the class of systems treated is the possibility of determining analytically an attractor, whose existence can be rather easily seen in simulations.
Gallavotti, G., Gentile, G., Giuliani, A. (2012). Resonances within chaos. CHAOS, 22(2), 026108,1-6 [10.1063/1.3695370].
File in questo prodotto:
Non ci sono file associati a questo prodotto.

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11590/278685
Citazioni
  • ???jsp.display-item.citation.pmc??? ND
  • Scopus 4
  • ???jsp.display-item.citation.isi??? 4
social impact