Rotating Wave Dynamics in Rings of Coupled Oscillators: A Comprehensive Review

Abstract

This comprehensive review paper provides a thorough survey of the extensive research conducted on rotating waves observed in rings of coupled oscillators. These waves manifest as stable periodic, quasiperiodic, or chaotic orbits, arising from the phase difference between neighboring oscillators. The research encompasses a wide range of nonlinear systems, including electrical circuits, neural models, Duffing, Lorenz, and R¨ossler oscillators, among others. While exploring the behavior of rotating waves, particular emphasis is placed on neural oscillators, as neural rings in the brain play a crucial role in working memory. The intricate dynamics of rotating waves are elucidated through the application of various techniques, including time series analysis, phase-space analysis, bifurcation diagrams, spectral analysis, and basins of attraction. These methods carefully uncover the complex routes from coexisting stable equilibria to hyperchaos. The study highlights a sequence of bifurcations occurring with increasing coupling strength, such as the Andronov–Hopf, torus, and crisis bifurcations. Moreover, the coexistence of multiple rotating waves under the same system parameters is examined. The vast body of research on rotating waves provides insights that are essential for a wide range of scientific disciplines and real-world applications, including lasers, chemical reactions, cardiorespiratory systems, and even beyond, with particular relevance to neural networks and brain functions.