Statistics of Lyapunov exponent spectrum in randomly coupled Kuramoto oscillators

Characterization of spatiotemporal dynamics of coupled oscillatory systems can be done by computing the Lyapunov exponents. We study the spatiotemporal dynamics of randomly coupled network of Kuramoto oscillators and find that the spectral statistics obtained from the Lyapunov exponent spectrum show interesting sensitivity to the coupling matrix. Our results indicate that in the weak coupling limit the gap distribution of the Lyapunov spectrum is Poissonian, while in the limit of strong coupling the gap distribution shows level repulsion. Moreover, the oscillators settle to an inhomogeneous oscillatory state, and it is also possible to infer the random network properties from the Lyapunov exponent spectrum.

Figure 1

Eigenvalue statistics of random uniform matrix. (a) Eigenvalue spectrum of banded coupling matrix, ${\lambda }_{i}vsi/N$, shown for two system sizes, $N=32,64$ and $W=2,N$. The eigenvalue spectrum is symmetric and the rescaled values ${\overline{\lambda }}_i=2{\lambda }_i/({\lambda }_{\text{max}}-{\lambda }_{\text{min}})$ exhibit data collapse as shown in inset. (b) The distributions of rescaled eigenvalues show agreement with Wigner semicircle law. (c) The eigenvalue gap distribution $P\left(S\right)$, where $S$ is the gap between successive eigenvalues are shown for band-widths $W=2,N$, where $N=32$. The continuous curves are theoretical predictions. (d) Same as (c) for $N=64$. An ensemble average is taken over for 1000 realizations.

Figure 2

Spatiotemporal dynamics of $\text{sin}(\theta ({\omega }_i,t))$ for randomly coupled Kuramoto oscillators $\left(N=64\right)$. The oscillator index is ordered according to the magnitude of the frequency. (a) $K=2,W=2$, (b) $K=2,W=N$, (c) $K=100,W=2$, (d) $K=100,W=N$.

Figure 3

The Lyapunov spectrum for the Kuramoto oscillators with normally distributed frequencies. The theoretical curve [28] and the numerical simulation results for two stability parameter values $\left(\beta =2,20\right)$ are shown for a single realization. The numerical integration has been done for $N=64$ oscillators with a step size of ${10}^{-2}$ and ${10}^5$ time steps. Inset shows the convergence of the maximum and the minimum LE for a single realization.

Figure 4

Lyapunov exponent statistics of randomly coupled Kuramoto oscillators for strong coupling, $K=100$. (a) Lyapunov spectrum of banded coupling matrix, ${\mathrm{\Lambda }}_{i}vsi/N$, shown for two system sizes, $N=32,64$ and $W=2,N$. (b) The distributions of rescaled Lyapunov exponents ${\overline{\mathrm{\Lambda }}}_i$, where the rescaled values ${\overline{\mathrm{\Lambda }}}_i=(2{\mathrm{\Lambda }}_i+|{\mathrm{\Lambda }}_{\text{min}}\left|\right)/|{\mathrm{\Lambda }}_{\text{min}}|$. (c) The Lyapunov exponent gap distribution $P\left(S\right)$, where $S$ is the gap between successive Lyapunov exponents are shown for bandwidths $W=2,N$, where $N=32$. The continuous curves are theoretical predictions. (d) Same as (c) for $N=64$. An ensemble average is taken over for $\sim 2\times {10}^3$ realizations.

Figure 5

The Brody parameter $\alpha$ is shown for varying matrix bandwidth $W$. The coupling strength $K=100$.

Figure 6

Lyapunov exponent statistics of randomly coupled Kuramoto oscillators for weak coupling, $K=5$. Lyapunov spectrum of banded coupling matrix, ${\mathrm{\Lambda }}_{i}vsi/N$, is shown for two system sizes, $N=32,64$ for bandwidth (a) $W=2$, (b) $W=N$. The LS is an ensemble average taken over for $\sim {10}^3$ realizations. The arrows indicate the range in which the LEs are $\sim 0$. (c) The spacing distribution for ${\mathrm{\Lambda }}_i<0$ is shown for $W=2$. In (d) the spacing distributions are shown separately for positive and negative LS for $W=N$.

Figure 7

Lyapunov exponent statistics of randomly all-to-all coupled Kuramoto oscillators for varying coupling strength $K$ for system size (a) $N=32$, (b) $N=64$. The ensemble averages are taken over $\sim {10}^3$ realizations. The insets show the estimated Brody parameter as a function of $K$.

Figure 8

Lyapunov exponent statistics of randomly coupled HMF model $\left(N=32\right)$. (a) The largest LE as a function of the energy per particle $U$. (b) LS shown for $U=1,10$. The LS spacing distributions, for the positive and the negative LEs, are shown for (c) $U=1.0$, (d) $U=10.0$.