Finite-size induced random switching of chimeras in a deterministic two-population Kuramoto-Sakaguchi model

The two-population Kuramoto-Sakaguchi model for interacting populations of phase oscillators exhibits chimera states whereby one population is synchronised, and the other is desynchronised. Which of the two populations is synchronised depends on the initial conditions. We show that this deterministic model exhibits random switches of their chimera states, alternating between which of the two populations is synchronised and which is not. We show that these random switches are induced by the finite size of the network. We provide numerical evidence that the switches are governed by a Poisson process and that the time between switches grows exponentially with the system size, rendering switches unobservable for all practical purposes in sufficiently large networks. We develop a reduced stochastic model for the synchronised population, based on a central limit theorem controlling the collective effect of the desynchronised population on the synchronised one, and show that this stochastic model well reproduces the statistical behaviour of the full deterministic model. We further determine critical fluctuation sizes capable of inducing switches and provide estimates for the mean switching times from an associated Kramers problem.

💡 Research Summary

The paper investigates a deterministic two‑population Kuramoto‑Sakaguchi (KS) model consisting of N oscillators in each population, with intra‑population coupling strength K, inter‑population coupling strength κ (0 < κ < K), and a phase lag λ. Natural frequencies are drawn from the same Gaussian distribution for both populations, ensuring perfect symmetry. In the thermodynamic limit (N → ∞) the model admits stationary chimera states: one population is phase‑locked (order parameter r≈1) while the other remains partially desynchronised (r≈O(N⁻¹ᐟ²)).

The authors focus on finite‑size effects. By initializing one population fully synchronised and the other uniformly random, they integrate the deterministic equations using a high‑precision Runge‑Kutta scheme for various N (6 ≤ N ≤ 128). For small N the system exhibits spontaneous, irregular switches: the synchronised cluster loses coherence while the previously desynchronised cluster becomes coherent, and vice versa. These switches are not periodic; the waiting times τ between successive switches follow an exponential distribution, indicating a Poisson process. The mean waiting time ⟨τ⟩ grows exponentially with N, fitted as ⟨τ⟩≈0.0158 exp(0.889 N). For N≈21 the mean waiting time exceeds 10⁶ time units, making switches practically unobservable in larger networks.

Statistical analysis of the complex mean field Z generated by the desynchronised population shows that its real and imaginary parts are Gaussian with zero mean and variance scaling as 1/N. This scaling suggests that the fluctuations of Z are governed by a central limit theorem: the sum of many independent oscillator contributions yields a Gaussian process. Consequently, Z(t) can be modelled as a complex Ornstein‑Uhlenbeck (OU) process with covariance proportional to 1/N.

Using this insight, the authors construct a stochastic reduction for the synchronised population. The interaction term from the desynchronised group is replaced by a Gaussian white‑noise term of strength σ²∝1/N, leading to stochastic differential equations for the phases of the synchronised oscillators:

dθ_i =