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Chaos meets stochasticity: A variance-based method for Lyapunov exponent estimation.

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This study introduces a new variance-based method using intrusive polynomial chaos (IPC) to calculate the Largest Lyapunov Exponent (LLE) for chaotic systems. This approach offers a robust alternative to traditional trajectory-tracking methods.

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Area of Science:

  • Nonlinear Dynamics
  • Chaos Theory
  • Computational Physics

Background:

  • Largest Lyapunov Exponent (LLE) quantifies chaos in dynamical systems.
  • Classical LLE computation methods (e.g., Wolf's algorithm) struggle with noise, efficiency, and scalability.
  • High-dimensional systems pose significant challenges for trajectory-based LLE estimation.

Purpose of the Study:

  • To develop a novel variance-based methodology for computing the LLE.
  • To utilize intrusive polynomial chaos (IPC) for uncertainty quantification in chaotic systems.
  • To establish a probabilistic approach for LLE estimation, connecting deterministic chaos with statistical descriptions.

Main Methods:

  • Employed intrusive polynomial chaos (IPC) to evolve the probability distribution of initial conditions.
  • Extracted LLE from the exponential growth rate of ensemble variance.
  • Validated the IPC-based method against classical trajectory-based algorithms on benchmark chaotic systems (Lorenz, Rössler, Al-Azzawi/Al-Obeidi).

Main Results:

  • The IPC approach demonstrated comparable accuracy and convergence rates to trajectory-based methods.
  • Achieved excellent agreement in convergence histories, probability density functions of instantaneous Lyapunov exponents, and statistical error measures.
  • The method directly computes the full statistical structure of ensemble dynamics, a key advantage.

Conclusions:

  • Variance-based LLE estimation via polynomial chaos is a robust and viable alternative to traditional methods.
  • IPC offers a powerful framework for analyzing chaotic dynamics and quantifying associated uncertainties.
  • The proposed methodology enhances the computational analysis of nonlinear dynamical systems.