Origin and scaling of chaos in weakly coupled phase oscillators

We discuss the behavior of the largest Lyapunov exponent $\lambda$ in the incoherent phase of large ensembles of heterogeneous, globally coupled, phase oscillators. We show that the scaling with the system size $N$ depends on the details of the spacing distribution of the oscillator frequencies. For sufficiently regular distributions $\lambda \sim 1/N$, while for strong fluctuations of the frequency spacing $\lambda \sim lnN/N$ (the standard setup of independent identically distributed variables belongs to the latter class). In spite of the coupling being small for large $N$, the development of a rigorous perturbative theory is not obvious. In fact, our analysis relies on a combination of various types of numerical simulations together with approximate analytical arguments, based on a suitable stochastic approximation for the tangent space evolution. In fact, the very reason for $\lambda$ being strictly larger than zero is the presence of finite-size fluctuations. We trace back the origin of the logarithmic correction to a weak synchronization between tangent and phase-space dynamics.

Figure 1

Finite-size scaling of the largest Lyapunov exponents in the incoherent phase (see text). (a) LLE $\lambda$ vs. system size for disordered Gaussian (DG, full red circles) and regular Gaussian (RG, empty circles) frequencies. (b) Same as (a) but for disordered uniform (DU, full red squares) and regular equispaced (RE, empty squares) frequencies. Axes are in double logarithmic scale, and the black dashed lines mark a decay $\sim 1/N$. [(c) and (d)] The LLEs of panels (a) and (b) are multiplied by system size $N$ to better show the leading logarithmic correction to the disordered frequencies choices. Axes are now in log-lin scale, and the red dashed lines represent the best logarithmic fit of the disordered frequencies data.

Figure 2

Marginal self-averaging properties. (a) Probability distributions for the rescaled LLE ${\mathrm{\Lambda }}_{\alpha }$ (see main text) for different system sizes (size is increasing along the blue arrow). Each probability distribution is estimated from ${10}^3$ realizations of the disordered Gaussian frequencies. (b) Inverse ratio $\lambda /\mathrm{\Delta }\overline{\lambda }$ vs. system size in a log-lin scale. The dashed red line marks a logarithmic fit.

Figure 3

(a) Frequency shift $\mathrm{\Delta }{\omega }_j$ vs. the natural frequency ${\omega }_j$ for different system sizes $N$ for one typical realization of DG frequencies and an average time of $T\approx {10}^6$ time units. (b) Same as (a) but for RG frequencies. [(c) and (d)] Frequency shift multiplied by the system size $N$ to show the finite-size scaling $1/N$.

Figure 4

Quasiperiodic approximation (a) LLEs vs. system size for the quasiperiodic (squares) and full dynamics (circles) in the disordered (full symbols, data averaged over 100 different frequency realizations) and regular (open symbols) Gaussian frequencies setups. Axes are in a doubly logarithmic scale. (b) The LLE of panel (a) are multiplied by system size $N$ to highlight the leading logarithmic correction to the disordered frequencies setups. Axes are now in log-lin scale, and the red dashed line represents the best logarithmic fit of the disordered frequencies data.

Figure 5

Rescaled largest Lyapunov exponent for the discrete Kuramoto model vs. system size. Disordered uniformly distributed (DU) frequencies are marked by full red circles, while empty black squares refer to perturbed ($p=1$) regular equispaced frequencies. Axes are in lin-log scale to highlight the logarithmic correction to the disordered ensemble (black dashed line fit). Data have been averaged over 100 different realizations.

Figure 6

Quasiperiodic Kuramoto model with disordered frequencies ensembles generated via the frequency spacing distribution method (see text). (a) Graphical depiction of the different frequency spacing distribution tested here. (b) Rescaled largest Lyapunov exponent. The dashed lines have been obtained as the best fit of the numerical data (see text). Data have been averaged over 100 different realizations.

Figure 7

Discrete-time stochastic approximation. Numerical stationary probability distribution $S\left(v\right)$ (symbols) compared with the theory (dashed full lines). Simulations are performed by iterating a system of $4\times {10}^7$ elements with ${\sigma }_{\chi }^2={10}^{2}f$ and ${\sigma }_{\eta }^2={10}^2(1-f)$ for $f$ such as to have $a=3$ (green diamonds), $a=1$ (red circles), and $a=0.3$ (black squares).

Figure 8

The two average observables $Q_Z^2/\lambda$ and $Q_{\beta }^2/\lambda$ for different values of $N$ and 100 different realizations of the frequencies in the discrete Kuramoto model (23). (a) Disordered uniform frequency setup. (b) Perturbed regular equispaced frequencies ($p=0.8$). Single sample time averages are performed over about $4\times {10}^5$ time units.

Figure 9

(a) Finite-size scaling of the reshuffled average ${\tilde{Z}}^2$ for DU frequencies (see text) in the discrete Kuramoto model. Ensemble averages are taken over 100 different realizations of the frequencies. (b) Time-averaged (over $T=4\times {10}^5$ time units) square amplitude of the Lyapunov vector components as a function of the corresponding natural frequency for a single frequency realization.

Figure 10

Finite-size scaling of the tangent vector inverse participation ratio for the full Kuramoto model (1) and different frequency setups: disordered Gaussian (DG, full red circles), disordered uniform (DU, full blue squares), regular Gaussian (RG, empty red circles) and regular equispaced (RE, empty blue squares). Axes are in a double-logarithmic scale. Simulation parameters as in Sec. 2b. The dashed black line marks a power-law decay $N^{-0.4}$.