Multistability and Control in Ring Networks of Phase Oscillators with Frequency Heterogeneity and Phase Lag

Abstract

Many oscillator networks are multistable, meaning that different synchronization states are realized depending on the initial conditions. In this paper, we numerically analyze a ring network of phase oscillators, in which synchronous states with different wavenumbers are simultaneously stable. This model is an extension of the one studied in detail in previous studies by introducing inhomogeneities in the natural frequencies and the phase lag in the interaction, which are essential factors in the application. We investigate basin size distribution, which characterizes the size of the initial value set that converges to each synchronous state, showing that the basin size of synchronous states with higher wave-numbers broadens as the phase lag increases up to a certain extent. Weak inhomogeneities in the natural frequencies are also found to broaden the basin size of synchronous states with lower wave-numbers, i.e., more synchronous states. The latter result is seemingly counter-intuitive, but occurs because the higher wavenumber states are more vulnerable to inhomogeneity. Finally, we propose a control method that exploits inhomogeneity and phase lag to steer the system into a synchronized state with a specific wavenumber. This research furthers our understanding of the design principles and control of oscillator networks.

Figures (6)

• Figure 1: Examples of uniformly $q$-twisted states defined by Eq. \ref{['twist']}. In a $q$-twisted state, the phase increment between neighboring oscillators is constant, $\phi_{k+1}-\phi_k = 2\pi q/N$, so that the phase winds by $2\pi q$ over one circuit of the ring.
• Figure 2: Phase-locking probability on the $(\alpha,\sigma)$ parameter plane for Eq. \ref{['SKmodel']} ($N=80$). For each parameter pair, we perform $M=10^4$ independent runs with random initial phases and random frequency realizations; the color indicates the fraction of runs that converge to a phase-locked state (i.e., the number of phase-locked outcomes divided by $M$). A run is judged to be phase-locked when the Kuramoto order parameter becomes stationary, $|r(t+1)-r(t)|<10^{-7}$ (see Sec. III A). The plot is shown as a rectangular tiling to emphasize the shape of the phase-locking region.
• Figure 3: Basin-size statistics of phase-locked $q$-twisted states, obtained from random initial phases and random frequency realizations (Sec. III A, $N=80$). (a) Empirical distribution of the winding number $q$ for $\sigma=0$ and several values of $\alpha$. (b) Distribution of $q$ for $\alpha=0$ and several values of $\sigma$. (c) Distribution of $q$ for $\alpha=0.3\pi$ and several values of $\sigma$. In (a)--(c), frequencies are shown as relative frequencies conditioned on phase-locking outcomes. (d) Standard deviation of the $q$ distribution as a function of $(\alpha,\sigma)$, summarizing the broadening/narrowing trends in (a)--(c).
• Figure 4: Critical heterogeneity $\sigma_{\rm c}(q)$ for each $q$-twisted state for three representative frequency sets (Sec. III B, $N=80$). For each $q$, the simulation starts from the homogeneous $q$-twisted state and $\sigma$ is increased in steps of $\Delta\sigma=0.01$ until the trajectory no longer converges to a phase-locked state with the same winding number $q$. (a) $\alpha=0$. (b) $\alpha=0.1\pi$.
• Figure 5: Ensemble statistics of the critical heterogeneity $\sigma_{\rm c}(q)$ over 100 independent frequency realizations $\{\hat{\omega}_k\}$ (Sec. III B). Symbols show the mean of $\sigma_{\rm c}(q)$ and error bars indicate the standard deviation across realizations, for each value of $\alpha$.