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Area of Science:

  • Dynamical Systems and Control Theory
  • Network Science
  • Nonlinear Dynamics

Background:

  • Amplitude death (AD) is a phenomenon in coupled oscillators where oscillations cease.
  • Analyzing AD in systems with heterogeneous connection delays is complex.
  • Directed graphs offer a framework for studying coupled systems.

Purpose of the Study:

  • To simplify the linear stability analysis of amplitude death in delay-coupled oscillators on directed graphs.
  • To investigate the role of heterogeneous connection delays in amplitude death.
  • To identify key network structures influencing stability.

Main Methods:

  • Linear stability analysis applied to delay-coupled oscillators.
  • Factorization of the characteristic function of steady states.
  • Focus on directed cycles within the graph topology.
  • Numerical simulations for verification.

Main Results:

  • The characteristic function for stability analysis can be factorized, simplifying independent analysis.
  • Stability analysis complexity is reduced by focusing on directed cycles.
  • System stability depends on the sum of delays along directed cycles.
  • Delays on edges not part of directed cycles do not affect stability.

Conclusions:

  • A simplified method for analyzing amplitude death stability in heterogeneous delay-coupled systems is presented.
  • Directed cycles are crucial structural elements determining stability in such networks.
  • The findings offer a more efficient approach to understanding complex oscillatory networks.