Coherence properties of collective modes in ensembles of oscillators

Abstract

Synchronization transition in oscillatory networks manifests itself as the appearance of a periodic global mode. While perfect in the thermodynamic limit, this mode fluctuates for finite ensembles. We characterize the coherence of this mode in terms of the phase diffusion constant. In several examples, we always observed normal diffusion, but the dependence of the diffusion constant on the system size $D\sim N^{-μ}$ depends on the nature of coupled units: for coupled chaotic systems $μ=1$, while for coupled periodic oscillators we observe, depending on the particular model, $μ=2$ and $μ=2.5$. These large values of the power index are attributed to the size-dependence of collective chaos in the finite ensemble, which disappears in the thermodynamic limit. We also show that in the standard Kuramoto model for a symmetric set of frequencies, there is an additional transition to a symmetric chaotic state with vanishing diffusion of the global phase.

Figures (5)

• Figure 1: Panel (a): Examples of phase variations $\Theta(t)-\Omega t$ for $N=100,\;200$. Panel (b): examples of dependencies of the phase variance $\langle (\Theta(T)-\Theta(0)-\Omega T)^2 \rangle$ vs time lag $T$; these dependencies are linear with high accuracy, indicating normal diffusion of the phase. Panel (c): log-log plot of the diffusion constants $D$ vs $N$ (in the range $100\leq N\leq 1000$). The best fit (green line) has slope $1.98$.
• Figure 2: Cumulative distributions of the diffusion constant $D$ for different system sizes, plotted for rescaled quantity $D (N/100)^2$. Parameters are the same as in the case of regular sampling: $\varepsilon=2.5$, $\alpha=\pi/4$.
• Figure 3: System size dependencies (log-log plots) of the global phase diffusion constant for three models discussed in the text and formulated in the Appendix. Panel (a): nonlinearly coupled SL oscillators \ref{['eq:sl1']}, for four values of the frequency distribution half-width $q$. The best fit slopes are (from top to bottom) $1.92,\;1.9,\;1.98,\;2.03$. Panel (b): coupled ML system of neural spiking \ref{['eq:mldet2']}. The slope of the best fit is $2.48$. Panel (c): coupled chaotic Roessler systems \ref{['eq:roes']}. The best fit slope is $1.02$.
• Figure 4: Panel (a): The transversal LE vs coupling parameter $\varepsilon$ for several ensemble sizes $N$. Panel (b): critical value of coupling $\varepsilon$ at which the transversal LE changes sign and the symmetric manifold becomes stable (blue circles). Red squares: critical value of coupling at which a well-defined global mode with $R_{min}>0.1$ appears.
• Figure 5: Panel (a): The variance of the global frequency vs. phase shift $\alpha$ in the KS system; for two values of coupling strength below and above the symmetry synchronization transition. Panel (b): The variance of the global frequency vs asymmetry parameter $\delta$ of the Beta-distribution of frequencies in the Kuramoto model.