Time-delayed Kuramoto model in the Watts-Strogatz small-world networks

Area of Science:

• Complex systems
• Network science
• Nonlinear dynamics

Background:

• Phase oscillators are fundamental in modeling coupled systems.
• Small-world networks exhibit unique topological properties.
• Time delays significantly impact network dynamics and synchronization.

Purpose of the Study:

• To investigate the effect of uniform time delay on the synchronization of identical coupled phase oscillators in small-world networks.
• To characterize the transitions between different dynamical states, including partially synchronized and glassy phases.
• To identify indicators of discontinuous transitions and explore the emergence of Chimera states.

Main Methods:

• Simulations of Kuramoto phase oscillators on small-world networks.
• Systematic variation of intrinsic frequency, coupling constant, and time delay.
• Analysis of frequency distributions, phase-locking patterns, and order parameters.
• Identification of hysteresis loops and Chimera states.

Main Results:

• Synchronization enhancement observed within specific time delay ranges.
• Discontinuous transitions identified between partially synchronized states and a glassy phase.
• Bimodal frequency distributions and hysteresis loops confirm discontinuous transitions.
• Chimera states observed at the onset of these transitions.

Conclusions:

• Time delay is a critical parameter controlling synchronization in small-world oscillator networks.
• Discontinuous transitions and complex phenomena like Chimera states emerge due to time delays.
• The findings provide insights into the rich dynamics of delayed complex networks.