Basin Riddling in Coupled Phase Oscillators

Abstract

We investigate the global basin structure of twisted states in nearest-neighbor coupled phase oscillators with a common phase shift $α$. As $α$ increases, basin boundaries become progressively more complex, with their fractal dimension growing toward that of the full ambient phase space. We conjecture that the basins eventually become riddled as the system approaches the limit $α\to \fracπ{2}$, where the dynamics becomes volume-preserving. We characterize the transient dynamics via the stabilization time of the winding number and demonstrate that it grows with system size. The scaling accelerates at larger phase shifts, transitioning from logarithmic to power-law behavior. We further analyze the dynamical origin of these long transients. Our results demonstrate how a single phase-shift governs fractal basin complexity and provide new insights into the global geometry and transient dynamics of multistable, yet non-chaotic, coupled phase oscillators.

Figures (5)

• Figure 1: Basin metamorphosis of the $q$-twisted states on an $(\epsilon_1, \epsilon_2)$-slice centered at a random point $\boldsymbol{\theta}_b\in [-\pi, \pi)^N$: (a) $\alpha = 0$, (b) $\alpha=0.3\pi$, (c) $\alpha=0.36\pi$, (d) $\alpha=0.4\pi$, (e) $\alpha=0.42\pi$, and (f) $\alpha=0.44\pi$. Each point represents a phase-space state with $\boldsymbol{\theta}_b + (\epsilon_1, \epsilon_2, 0, ..., 0)$ and $N=80$. Resolution of each plot is $1000\times 1000$. The numerical scheme is specified in note1.
• Figure 2: Fractal dimension $D$ of the basin boundaries of the $q$-twisted states on the two-dimensional $(\epsilon_1, \epsilon_2)$-slice: the solid line shows the average over $9$ slices and small $|q| < \frac{N}{4}$, with gray region indicating variations across different $q$; $N=80$.
• Figure 3: Averaged winding-number stabilization time $t_s$ as a function of system size $N$ for various $\alpha$ in log-log scale: $t_s$ grows faster than $\log(N)$ for any $\alpha>0$; each point is averaged over $10^5$ random trajectories.
• Figure 4: (a) Typical time series of $\psi_1(t)$, corresponding winding number $q(t)$, order parameter $r_{\psi}(t)$ and (b) corresponding spatiotemporal evolution of $\boldsymbol{\psi}(t)$; here $\alpha=0.4\pi$, a random initial state and $N=80$.
• Figure 5: (a) Winding-number stabilization time $t_s$ for initial points on a $(\epsilon_1, \epsilon_2)$-slice at $\alpha=0.4\pi$ (cf. Fig. \ref{['fig:basins-delpi']}(d)); (b) distribution of $t_s$ separating basins boundary and non-boundary points, in semi-log scale.