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Turing-like instabilities from a limit cycle.

Joseph D Challenger1,2, Raffaella Burioni3, Duccio Fanelli2

  • 1Department of Infectious Disease Epidemiology, Imperial College London, London, W2 1PG, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
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Summary
This summary is machine-generated.

Oscillation death in coupled oscillators is a Turing instability for their synchronous periodic state. This finding generalizes Turing

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Area of Science:

  • Mathematical and Computational Biology
  • Nonlinear Dynamics
  • Complex Systems

Background:

  • Turing instability drives pattern formation in reaction-diffusion systems.
  • Oscillation death, or quenching, yields stable patterns from synchronous oscillators.
  • Existing theories primarily address homogeneous fixed points, not periodic states.

Purpose of the Study:

  • To demonstrate that oscillation death is a Turing instability.
  • To generalize Turing instability conditions to systems with periodic homogeneous solutions.
  • To provide a unified framework for pattern formation in reaction-diffusion systems and coupled oscillators.

Main Methods:

  • Analysis of the first return map for synchronous periodic states.
  • Derivation of approximated closed conditions for Turing instability.
  • Generalization of original Turing relations for time-periodic solutions.
  • Numerical testing across diverse reaction schemes.

Main Results:

  • Oscillation death is identified as a Turing instability.
  • A generalized Turing condition is derived for periodic states.
  • The framework applies to both continuum and network systems.
  • Numerical simulations validate the theoretical predictions.

Conclusions:

  • Oscillation death is a specific manifestation of Turing instability.
  • The generalized Turing relations offer predictive power for pattern formation.
  • This work unifies pattern formation mechanisms in different dynamical systems.