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Dispersive Time-Delay Dynamical Systems.

Alexander Pimenov1, Svetlana Slepneva2,3, Guillaume Huyet2,3,4,5

  • 1Weierstrass Institute, Mohrenstraße 39, Berlin 10117, Germany.

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Summary
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We developed a new theoretical method to study dispersion effects in time-delay dynamical systems. This approach reveals modulation instability in anomalous dispersion regimes, as seen in long cavity lasers.

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Area of Science:

  • Physics
  • Nonlinear Dynamics
  • Optical Engineering

Background:

  • Dynamical systems with time delays are often modeled without considering dispersion.
  • Dispersion effects are crucial for understanding wave propagation in extended systems.
  • Previous methods for incorporating dispersion can introduce artificial instabilities.

Purpose of the Study:

  • To develop a theoretical framework for investigating dispersion in time-delay dynamical systems.
  • To analyze the impact of dispersion on system stability, particularly in optical systems.
  • To identify and characterize instabilities arising from dispersion.

Main Methods:

  • Introduced a polarization equation to model dispersion as a distributed delay term.
  • Expanded the distributed delay term using power series expansion.
  • Applied the theoretical approach to a long cavity laser model.

Main Results:

  • The theoretical model accurately predicts stable operation in the normal dispersion regime for a long cavity laser.
  • A modulation instability, identified as the Benjamin-Feir instability, was observed in the anomalous dispersion regime.
  • The power series expansion can lead to spurious instabilities if not handled carefully.

Conclusions:

  • The proposed theoretical approach effectively incorporates dispersion into time-delay models.
  • Dispersion plays a critical role in the stability of dynamical systems, leading to phenomena like Benjamin-Feir instability.
  • Careful mathematical treatment is necessary when expanding delay terms to avoid artifacts.