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Lyapunov exponent in the Vicsek model.

L H Miranda-Filho1, T A Sobral2, A J F de Souza1

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This study quantifies chaotic behavior in self-propelled particle (SPP) systems using the largest Lyapunov exponent (LLE). A modified Vicsek model reveals a chaotic regime and noise-dependent LLE near phase transitions.

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Area of Science:

  • Physics
  • Complex Systems
  • Statistical Mechanics

Background:

  • The Vicsek model is a standard for flocking dynamics in self-propelled particles (SPPs).
  • Quantifying chaotic behavior in these systems has been a significant challenge.
  • Existing models lack direct measures for chaos.

Purpose of the Study:

  • To introduce a method for measuring chaotic behavior in Vicsek systems.
  • To investigate the dynamical phase transition using the largest Lyapunov exponent (LLE).
  • To propose a smooth Vicsek model for improved computational analysis.

Main Methods:

  • Numerical computation of the largest Lyapunov exponent (LLE).
  • Analysis of dynamical evolution in tangent space for up to two million SPPs.
  • Development and application of a smooth Vicsek model to overcome computational discontinuities.

Main Results:

  • Identification of a chaotic regime in the collective behavior of SPPs.
  • The LLE was successfully computed, providing a direct measure of chaos.
  • Observed significant changes in LLE dependence on noise near known Vicsek model transition points.

Conclusions:

  • The largest Lyapunov exponent (LLE) provides a robust measure for chaotic dynamics in Vicsek systems.
  • A smooth Vicsek model facilitates accurate computation of chaos.
  • Noise acts as a critical control parameter influencing chaotic behavior and phase transitions.