Lyapunov Analysis Captures the Collective Dynamics of Large Chaotic Systems

Abstract

We show, using generic globally coupled systems, that the collective dynamics of large chaotic systems is encoded in their Lyapunov spectra: most modes are typically localized on a few degrees of freedom, but some are delocalized, acting collectively on the trajectory. For globally coupled maps, we show, moreover, a quantitative correspondence between the collective modes and some of the so-called Perron-Frobenius dynamics. Our results imply that the conventional definition of extensivity must be changed as soon as collective dynamics sets in.

Collective Lyapunov modes

“…Lyapunov exponents measured in the PF dynamics have therefore been related to the collective dynamics [4], without, however, direct evidence being ever shown. In the next section, following our preceding letter [26], we will show that there is indeed a quantitative correspondence between the collective Lyapunov modes and some of such PF Lyapunov modes.…”

“…Using this property of the CLVs, we, in our preceding work [26], reported numerical evidence that collective behavior of large dynamical systems is indeed encoded in their Lyapunov spectrum: while most modes are localized on a few degrees of freedom, thus corresponding to microscopic fluctuations of the system, there exist some collective modes characterized by the delocalized CLVs, acting therefore collectively on the dynamical units. The delocalization of the CLVs is quantified by the mean value of the inverse participation ratio (IPR) [27] …”

“…The parameter values are set to be c 1 = −2.0, c 2 = 3.0, K = 0.47, which correspond to a regime of collective chaos [6,7,26]. We numerically integrate Equation (2) by the fourth-order Runge-Kutta method with time step 0.1 and compute LEs and CLVs using Ginelli et al's algorithm [24,25].…”

“…Following Nakagawa and Kuramoto's work [6,7] and our preceding letter [26], we consider here N (a) (b) globally-coupled limit-cycle oscillatorṡ…”

Covariant Lyapunov vectors

“…The situation has drastically changed a few years ago, when efficient algorithms have been developed [9,10,11], so that many scientists are now aware that CLVs can offer information on the local geometric structure of chaotic attractors, as opposed to LEs, which are powerful but global quantities. Several papers appeared, where CLVs have been successfully employed to better understand many aspects of chaotic dynamics [12,13,14,15,16,17,18,19,20,21,22]. Some of the relevant questions are touched in this Special Issue.…”