Twisted states in low-dimensional hypercubic lattices

Abstract

Twisted states with non-zero winding numbers composed of sinusoidally coupled identical oscillators have been observed in a ring. The phase of each oscillator in these states constantly shifts, following its preceding neighbor in a clockwise direction, and the summation of such phase shifts around the ring over $2\pi$ characterizes the winding number of each state. In this work, we consider finite-sized $d$-dimensional hypercubic lattices, namely square ($d=2$) and cubic ($d=3$) lattices with periodic boundary conditions. For identical oscillators, we observe new states in which the oscillators belonging to each line (plane) for $d=2$ ($d=3$) are phase synchronized with non-zero winding numbers along the perpendicular direction. These states can be reduced into twisted states in a ring with the same winding number if we regard each subset of phase-synchronized oscillators as one single oscillator. For nonidentical oscillators with heterogeneous natural frequencies, we observe similar patterns with slightly heterogeneous phases in each line $(d=2)$ and plane $(d=3)$. We show that these states generally appear for random configurations when the global coupling strength is larger than the critical values for the states.

References in corpus (16)

• Synchronization in complex networks
• Spontaneous synchrony in power-grid networks
• Symmetries, Cluster Synchronization, and Isolated Desynchronization in Complex Networks
• Analysis of a power grid using the Kuramoto-like model
• The sound of many hands clapping
• Remote synchronization reveals network symmetries and functional modules
• Complete Characterization of Stability of Cluster Synchronization in Complex Dynamical Networks
• Amplitude Expansions for Instabilities in Populations of Globally-Coupled Oscillators
• Scaling and singularities in the entrainment of globally-coupled oscillators
• Collective synchronization in spatially extended systems of coupled oscillators with random frequencies
• Stable Chimeras and Independently Synchronizable Clusters
• Cycle flows and multistabilty in oscillatory networks: an overview
• The Size of the Sync Basin Revisited
• Finite-size scaling, dynamic fluctuations, and hyperscaling relation in the Kuramoto model
• Multistability of Phase-Locking and Topological Winding Numbers in Locally Coupled Kuramoto Models on Single-Loop Networks
• Multistability of Phase-Locking in Equal-Frequency Kuramoto Models on Planar Graphs

Cited by in corpus (5)

• Attracting Poisson Chimeras in Two-population Networks
• Decoupled synchronized states in networks of linearly coupled limit cycle oscillators
• Heteroclinic Switching between Chimeras in a Ring of Six Oscillator Populations
• Stability of twisted states on lattices of Kuramoto oscillators
• Concurrent formation of nearly synchronous clusters in each intertwined cluster set with parameter mismatches