arXiv subjects

I. Belykh

Publications and source records attributed to I. Belykh.

3 recordsLinked to original sources

arXiv2026

Birth and breakdown of breathing and rotobreathing cyclops states in Kuramoto networks with higher-mode coupling

Cluster synchronization states often serve as organizing centers for collective dynamics, but their destabilization can open unexpected routes to both coherent and incoherent behavior. We study such transitions for cyclops states in globally coupled networks of identical Kuramoto-Sakaguchi rotators with inertia and two-harmonic coupling. Stationary cyclops states consist of two coherent clusters and a solitary oscillator that maintains fixed phase differences with the clusters. Using Floquet analysis and numerical continuation of periodic orbits, we trace bifurcation routes for the birth and breakdown of breathing and rotobreathing cyclops states with nonstationary intercluster phase differences. Breathing cyclops states, born from their stationary counterparts, correspond to bounded oscillations of the intercluster phases. They lose stability via period-doubling bifurcations, which produce phase-split cyclops states whose intercluster motion repeats only after two cycles of the parent breather, or via cluster-destruction bifurcations. Rotobreathing cyclops states, in which the intercluster phase differences undergo full rotations, are not merely the large-amplitude continuation of breathing cyclops states; instead, they form a separate family of nonstationary cyclops dynamics born through global bifurcations involving heteroclinic-contour-like structures of saddle cluster states. We further show that these states have wide, often global, basins of attraction, persist in large odd-sized networks, and contrast sharply with even-sized networks, where stationary multi-cluster states dominate. These results identify higher-harmonic coupling and solitary-oscillator-mediated rotations as generic mechanisms for organizing complex cluster motion in phase oscillator networks.

nlin.PS↗

arXiv2025

From Delay to Inertia and Triadic Interactions: A Predictive Model for Time-Delayed Oscillator Networks

Time-delayed oscillator networks underlie diverse biological and physical systems, yet standard first-order phase reductions fail to capture their high-dimensional collective dynamics. In this Letter, we develop a universal second-order predictive reduction for time-delayed Kuramoto-Daido networks that maps delayed one-dimensional phase dynamics to a delay-free network of two-dimensional rotators. Delay induces effective inertia and triadic interactions, yielding accurate predictions of nontrivial attractors and their collective-state statistics, including splay, cyclops, and chimera states. The reduction reveals a division of roles: inertia organizes higher-dimensional dynamics, whereas triadic terms are crucial for lower-dimensional patterns such as chimeras. Applicable to arbitrary topology, higher harmonics, and intrinsic-frequency heterogeneity, it provides a compact, parameter-explicit reduced model. The same framework also extends to time-delayed amplitude-phase oscillator networks, including swarmalators, yielding analogous reduced equations with emergent inertia and triadic higher-order couplings. This unified and readily deployable description enables systematic prediction and analysis of delay-controlled collective dynamics across oscillator networks.

nlin.PS↗

arXiv2022

Cyclops states in repulsive Kuramoto networks: the role of higher-order coupling

Repulsive oscillator networks can exhibit multiple cooperative rhythms, including chimera and cluster splay states. Yet, understanding which rhythm prevails remains challenging. Here, we address this fundamental question in the context of Kuramoto-Sakaguchi networks of identical rotators with higher-order coupling. Through analysis and numerics, we show that three-cluster splay states with two distinct coherent clusters and a solitary oscillator are the prevalent rhythms in networks with an odd number of units. We denote such tripod patterns cyclops states with the solitary oscillator reminiscent of the Cyclops's eye. As their mythological counterparts, the cyclops states are giants that dominate the system's phase space in weakly repulsive networks with first-order coupling. Astonishingly, the addition of the second or third harmonics to the Kuramoto coupling function makes the cyclops states global attractors practically across the full range of coupling's repulsion. At a more general level, our results suggest clues for finding dominant rhythms in repulsive physical and biological networks.

nlin.AO↗