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

TL;DR

The paper shows that finite-size effects in a deterministic two-population Kuramoto-Sakaguchi model induce random switching of chimera states between which population is synchronised and which is desynchronised, with switching times following a Poisson process and a mean that grows exponentially with system size $N$. A two-tier reduction is developed: (i) a deterministic mean-field Ott–Antonsen reduction yielding a closed equation for the desynchronised population variable $Z$, and (ii) a stochastic reduction where $Z$ is modeled as a complex Ornstein–Uhlenbeck process matched to the full system’s mean and covariance. This stochastic model reproduces the full system’s switching statistics, invariant densities, and phase fluctuations across moderate $N$ and enables a Kramers-type estimate of mean switching times. The work highlights finite-size fluctuations as a mechanism for chimera switching and provides a practical framework for predicting switching rates in finite networks, with potential extensions to other chimera regimes.

Abstract

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.

Figures (24)

• Figure 1: Order parameters $r_1$ (blue) and $r_2$ (orange) for the two-population KS model \ref{['eq:abrams_1']}--\ref{['eq:abrams_2']} exhibiting switching chimeras as a function of time for various numbers of oscillators $N$. Equation parameters are $K=100$, $\kappa=60$ and $\lambda=\frac{\pi}{2}-0.075$ and native frequencies $\omega$ are drawn equiprobably from a standard normal distribution $\mathcal{N}(0,1)$.
• Figure 2: Mean switching time $\bar{\tau}$ as a function of the number of oscillators $N$ (blue circles). The orange line shows a best-fit exponential regression model with $\bar{\tau}= \exp(0.889N - 4.15)$. Equation parameters are as in Figure \ref{['fig:r']}.
• Figure 3: Probability of switching times $P(\tau\geq\tau^\star)$ as a function of $\tau^\star$ for a network of size $N=12$, estimated from $968$ switching events. Blue circles denote samples of the switching times. The orange line shows a best-fit regression model corresponding to $P(\tau\geq \tau^\star)=\exp(-\tau^\star/\bar{\tau})$ with $\bar{\tau}\approx 486.8$. Equation parameters are as in Figure \ref{['fig:r']}.
• Figure 4: Empirical joint (top) and marginal (bottom) distributions of $Z$ obtained from simulating \ref{['eq:abrams_1']}--\ref{['eq:abrams_2']} with $N=16$. The filled orange circles denote the thermodynamic value of the real and imaginary part of $Z$ as calculated from the self-consistency relations \ref{['eq:selconst_r']}--\ref{['eq:selconst_Z']} of the mean-field theory. The red lines are best-fit Gaussian distributions. Equation parameters are as in Figure \ref{['fig:r']}.
• Figure 5: Variance of the real (left) and imaginary (right) component of $Z$ as a function of the number of oscillators $N$. The orange lines show a power law best fit indicating a scaling with $N^{-1.13}$ and $N^{-1.10}$ for the real and imaginary parts, respectively. Equation parameters are as in Figure \ref{['fig:r']}.