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Will a large complex system with time delays be stable?

Viktor K Jirsa1, Mingzhou Ding

  • 1Center for Complex Systems and Brain Sciences, Florida Atlantic University, Boca Raton, Florida 33431, USA.

Physical Review Letters
|August 25, 2004
PubMed
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System stability requires a specific inequality relating system size and coupling strength. This finding, previously shown for random coupling, now extends to systems with time delays.

Area of Science:

  • Dynamical Systems
  • Network Theory
  • Control Theory

Background:

  • In 1972, Robert May established a critical inequality for the stability of large linear systems with random coupling.
  • This inequality relates system size and average coupling strength to ensure equilibrium point stability.
  • Previous analyses primarily focused on systems without time delays.

Purpose of the Study:

  • To extend May's stability analysis to systems with time-delayed coupling.
  • To investigate the impact of discrete and distributed delays on system stability.
  • To determine if the original stability inequality holds for delay-coupled systems.

Main Methods:

  • Mathematical analysis of delay differential equations.
  • Derivation of stability conditions for randomly coupled systems with delays.

Related Experiment Videos

  • Comparison of stability criteria for delayed versus non-delayed systems.

    • Main Results:

    • The same inequality derived by May for non-delayed systems also constrains the stability of delay-coupled systems.
    • This generalized inequality applies to systems with both discrete and distributed delays.
    • The analysis confirms the robustness of the stability condition across different coupling types.

    Conclusions:

    • The fundamental inequality for system size and coupling strength is a universal condition for stability in randomly coupled systems, even with time delays.
    • The findings provide a unified framework for understanding stability in complex networks with delayed interactions.
    • This research offers critical insights for designing and controlling large-scale dynamical systems with inherent delays.