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Phase-lag controllers are widely used in control systems to improve stability and reduce steady-state errors. A dimmer switch controlling the brightness of a light bulb serves as a practical example of phase-lag control, gradually adjusting the bulb's brightness. Mathematically, phase-lag control or low-pass filtering is represented when the factor 'a' is less than 1.
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Synchronization of phase oscillators with frequency-weighted coupling.

Can Xu1,2, Yuting Sun2, Jian Gao2

  • 1College of Information Science and Engineering, Huaqiao University, Xiamen 361021, China.

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We reveal how coupled phase oscillators synchronize, finding their order parameter decays exponentially below the synchronization threshold, similar to Landau damping. This work clarifies synchronization transitions in complex networks.

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Area of Science:

  • Complex Systems
  • Nonlinear Dynamics
  • Statistical Physics

Background:

  • The first-order synchronization transition in coupled phase oscillators is a key area of research.
  • Understanding synchronization dynamics in networks with heterogeneous couplings is crucial.

Purpose of the Study:

  • To propose a framework for investigating synchronization in the frequency-weighted Kuramoto model with all-to-all couplings.
  • To analyze the stability of steady states and understand bifurcation mechanisms near the synchronization threshold.

Main Methods:

  • Rigorous mean-field analysis to predict steady states.
  • Detailed linear stability analysis to examine the incoherent state.
  • Amplitude expansion theory to reveal bifurcation mechanisms.

Main Results:

  • The incoherent state is neutrally stable below the synchronization threshold.
  • Order parameter amplitude exhibits exponential decay, analogous to Landau damping.
  • Explicit expressions for critical coupling strength were determined; oscillating standing wave states identified.

Conclusions:

  • Theoretical analysis and numerical results are consistent, validating the proposed framework.
  • The study provides insights into synchronization transitions in general networks with heterogeneous couplings.