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Pattern formation in systems with multiple delayed feedbacks.

Serhiy Yanchuk1, Giovanni Giacomelli2

  • 1Institute of Mathematics, Humboldt University of Berlin, Unter den Linden 6, 10099 Berlin, Germany.

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Summary
This summary is machine-generated.

Complex dynamical systems with multiple time delays can exhibit intricate instabilities. This study reveals how hierarchical delays encode spiral defects and turbulence, offering a new understanding of these phenomena.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Complex Systems

Background:

  • Dynamical systems with significant propagation times often display complex oscillatory instabilities.
  • Understanding systems with multiple, hierarchical time delays is crucial for analyzing phenomena like turbulence.

Purpose of the Study:

  • To introduce and analyze a class of dynamical systems with multiple, hierarchically long time delays.
  • To uncover hidden features in temporal dynamics using a space-time representation.
  • To establish an equivalence between these systems and the complex Ginzburg-Landau equation.

Main Methods:

  • Development of a suitable space-time representation to analyze temporal dynamics.
  • Application of multiple scale analysis to establish mathematical equivalences.
  • Introduction of a novel criterion for identifying the long-delay regime.
  • Experimental demonstration using a semiconductor laser with optical feedbacks.

Main Results:

  • The behavior of systems with two delays was shown to encode two-dimensional spiral defects and defect turbulence.
  • An equivalence was established between the studied systems and a complex Ginzburg-Landau equation.
  • A new criterion for the long-delay regime was introduced and demonstrated.

Conclusions:

  • Multiple, hierarchically long time delays in dynamical systems can lead to complex phenomena like spiral defects and turbulence.
  • The findings provide a new framework for understanding and analyzing such systems, with implications for fields like laser physics.