Basin Riddling in Coupled Phase Oscillators

TL;DR

This paper explores how the basin boundaries of twisted states in coupled phase oscillators become increasingly complex and riddled as the phase shift approaches $rac{ ext{pi}}{2}$, revealing insights into transient dynamics and basin geometry.

Contribution

It demonstrates how a single phase shift parameter influences fractal basin complexity and transient behavior in coupled phase oscillators, providing new understanding of their global dynamics.

Findings

Basin boundaries become more fractal with increasing phase shift.

Transient stabilization times grow with system size and shift magnitude.

Long transients are linked to the dynamical origin of basin structure.

Abstract

We investigate the global basin structure of twisted states in nearest-neighbor coupled phase oscillators with a common phase shift $\alpha$. As $\alpha$ 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 $\alpha\to \frac{\pi}{2}$, 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.