Dynamics and chaotic properties of the fully disordered Kuramoto model

TL;DR

The paper addresses the dynamics of the fully disordered Kuramoto (Daido-Stiller-Radons) model with quenched random couplings, asking whether frequency entrainment can occur and how chaos manifests in large populations. By combining extensive numerical simulations with a dynamical cavity method, the authors derive closed-form weak-coupling expressions for long-term frequencies and dissipation, and map out how reciprocity, disorder, and coupling strength shape the chaotic dynamics. They show that frequency entrainment is absent in the thermodynamic limit, while chaotic dynamics are pervasive, with a universal $\lambda(J) \sim c J^2$ scaling at small $J$ and a nontrivial $J$-dependent crossover to linear growth at larger $J$, strongly influenced by the reciprocity parameter $\eta$. In the infinite-coupling limit, reciprocity yields a glassy phase, whereas nonreciprocity induces chaos, and the chaotic region expands with system size, highlighting the delicate balance between frustration, nonreciprocity, and collective behavior in disordered oscillator networks. These results provide a rigorous baseline for oscillator glasses and spin-glass-inspired dynamics in populations of coupled phase oscillators and raise questions about chaos and entrainment in the thermodynamic limit.

Abstract

Frustrated random interactions are a key ingredient of spin glasses. From this perspective, we study the dynamics of the Kuramoto model with quenched random couplings: the simplest oscillator ensemble with fully disordered interactions. We answer some open questions by means of extensive numerical simulations and a perturbative calculation (the cavity method). We show frequency entrainment is not realized in the thermodynamic limit and that chaotic dynamics are pervasive in parameter space. In the weak coupling regime, we find closed formulas for the frequency shift and the dissipativeness of the model. Interestingly, the largest Lyapunov exponent is found to exhibit the same asymptotic dependence on the coupling constant irrespective of the coupling asymmetry, within the numerical accuracy.

Figures (7)

• Figure 1: Difference between average and natural frequencies versus the natural frequency, for $N=10^4$ oscillators. In the upper (lower) row the coupling strength is fixed to $J=0.1$ ($J=1$). From left to right, the three columns correspond to $\eta=-1$, $0$ and $1$, respectively. The dashed line indicates the perturbative theoretical result in Eq. \ref{['eq.freq']}. The straight dotted line indicates the vanishing long-term average frequency ($\Omega_j=0$).
• Figure 2: (a) Log-log plot of the dissipation ($- \cal S$) versus $J$ for different asymmetry parameters. The symbols correspond to numerical simulations with $N=5000$ oscillators, and the solid lines are the theoretical result in Eq. \ref{['eq.diss']}. (b) Largest Lyapunov exponent versus $J$ (in log-log scale) with $N=2000$, for different asymmetry parameters. The black solid line is $\lambda=J^2$.
• Figure 3: Largest Lyapunov exponent in the homogeneous limit ($\sigma=0$), $\lambda_0$ versus $\eta$ for $N=200$, $400$, $800$, $1600$, and $3200$ oscillators. The Lyapunov exponent has been averaged over 20 independent realizations, save for $N=3200$ with only 10 realizations. Solid (dashed) lines correspond to increasing (decreasing) $\eta$.
• Figure 4: Long-term average frequencies of a population of $10^4$ oscillators with coupling constant $J=10$, and $\eta=1$, $0$ and $-1$ in panels (a), (b) and (c), respectively. The time averages were performed over $10^5$ t.u. The dashed straight line is $\Omega_j=\omega_j$.
• Figure 5: Lyapunov exponents for one instance of a symmetric coupling matrix $K$. (a) Three largest Lyapunov exponents $\lambda_1\ge\lambda_2\ge\lambda_3$ in green circles, blue squares and red triangles, respectively. The system size is $N=800$. Ten different initial oscillator phases were implemented for every value of $J$, and each symbol size corresponds to one of these runs. (b) Largest Lyapunov exponent $\lambda=\lambda_1$ for ten initial conditions, and five system sizes.