Synchronization in a population of globally coupled chaotic oscillators (original) (raw)

Abstract

PACS. 05.45+b -Theory and models of chaotic systems. PACS. 64.60Cn -Order-disorder and statistical mechanics of model systems.

Figures (5)

Fig. 1. — Variance of the mean field X vs. coupling parameter ¢ for different topologies of the Roéssler system (a = 0.15,wo = 1 for curves a) and 6); a = 0.25, wo = 0.97 for curves c) and d)) and different distributions of natural frequencies (Aw = 0 for curves a) and c); Aw = 0.02 for curves b) and d)). The number of oscillators is N = 5000 for curves a)-d). Curve e) differs from curve d) only in the size of the ensemble (N = 20000); it demonstrates the finite-size effect on the order parameter.

Fig. 1. — Variance of the mean field X vs. coupling parameter ¢ for different topologies of the Roéssler system (a = 0.15,wo = 1 for curves a) and 6); a = 0.25, wo = 0.97 for curves c) and d)) and different distributions of natural frequencies (Aw = 0 for curves a) and c); Aw = 0.02 for curves b) and d)). The number of oscillators is N = 5000 for curves a)-d). Curve e) differs from curve d) only in the size of the ensemble (N = 20000); it demonstrates the finite-size effect on the order parameter.

Fig. 2. — Projections of the phase portraits of the Réssler oscillators (left column) and of the mean fields X = (xi), Y = (y) in an ensemble of N = 5000 oscillators. a) Phase-coherent Réssler attractor, w) = 1, a= 0.15. b) Mean field in the ensemble of oscillators a) with Gaussian distribution of frequencies Aw = 0.02 and coupling ¢ = 0.1. c) Funnel attractor wo = 0.97, a = 0.25. d) Mean field in the ensemble of oscillators c) with Gaussian distribution of frequencies Aw = 0.02 and coupling e=0.15.

Fig. 2. — Projections of the phase portraits of the Réssler oscillators (left column) and of the mean fields X = (xi), Y = (y) in an ensemble of N = 5000 oscillators. a) Phase-coherent Réssler attractor, w) = 1, a= 0.15. b) Mean field in the ensemble of oscillators a) with Gaussian distribution of frequencies Aw = 0.02 and coupling ¢ = 0.1. c) Funnel attractor wo = 0.97, a = 0.25. d) Mean field in the ensemble of oscillators c) with Gaussian distribution of frequencies Aw = 0.02 and coupling e=0.15.

A. S. PIKOVSKY et al.: SYNCHRONIZATION IN A POPULATION OF GLOBALLY ETC.

A. S. PIKOVSKY et al.: SYNCHRONIZATION IN A POPULATION OF GLOBALLY ETC.

Fig. 4. — Successive maxima (upper panel) and observed frequencies, eq. (5) (bottom panel) vs. natural frequencies in the ensemble of coupled phase-coherent Réssler systems of fig. 2a). a) The coupling € = 0.05 is slightly below the transition threshold, the observed frequencies 92 are proportional to the natural frequencies w. b) Above threshold (€ = 0.1) most of the oscillators form a coherent cluster (plateau in the bottom panel), while the amplitudes remain chaotic (with the exception of the period-3 window for w = 0.97).

Fig. 4. — Successive maxima (upper panel) and observed frequencies, eq. (5) (bottom panel) vs. natural frequencies in the ensemble of coupled phase-coherent Réssler systems of fig. 2a). a) The coupling € = 0.05 is slightly below the transition threshold, the observed frequencies 92 are proportional to the natural frequencies w. b) Above threshold (€ = 0.1) most of the oscillators form a coherent cluster (plateau in the bottom panel), while the amplitudes remain chaotic (with the exception of the period-3 window for w = 0.97).

A. S. PIKOVSKY et al.: SYNCHRONIZATION IN A POPULATION OF GLOBALLY ETC. Fig. 5. — The observed frequency {2 in the ensemble of coupled funnel attractors with parameters of fig. 2d). The upper panel shows the deviation © from the linear fit: the tendency to synchronization is clearly seen in this panel, although it is rather small.

A. S. PIKOVSKY et al.: SYNCHRONIZATION IN A POPULATION OF GLOBALLY ETC. Fig. 5. — The observed frequency {2 in the ensemble of coupled funnel attractors with parameters of fig. 2d). The upper panel shows the deviation © from the linear fit: the tendency to synchronization is clearly seen in this panel, although it is rather small.

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