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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Probabilistic density function method for nonlinear dynamical systems driven by colored noise.

David A Barajas-Solano1, Alexandre M Tartakovsky1

  • 1Pacific Northwest National Laboratory, Richland, Washington 99352, USA.

Physical Review. E
|June 15, 2016
PubMed
Summary
This summary is machine-generated.

We developed a new probability density function (PDF) method for nonlinear stochastic systems with colored noise. This approach accurately models system dynamics across various noise conditions.

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Area of Science:

  • Applied Mathematics
  • Computational Physics
  • Nonlinear Dynamics

Background:

  • Stochastic ordinary differential equations (SODEs) are crucial for modeling complex systems.
  • Colored noise, characterized by autocorrelation, presents unique challenges in SODE analysis.
  • Existing methods struggle to accurately capture the temporal evolution of system states under colored noise.

Purpose of the Study:

  • To introduce a novel probability density function (PDF) method for systems driven by colored noise.
  • To develop a closure approximation that accounts for advective transport in PDF evolution.
  • To enable accurate analysis of nonlinear systems and power grid dynamics.

Main Methods:

  • Developed a probability density function (PDF) method for nonlinear stochastic ordinary differential equations (SODEs).
  • Employed a modified large-eddy-diffusivity (LED) closure to approximate temporal deconvolution.
  • Introduced a generalized local linearization approximation for a computable second-order partial differential equation.

Main Results:

  • The proposed PDF method accurately describes phase space dynamics for arbitrary noise autocorrelation times.
  • Analysis of Kramers equations with colored noise revealed insights into nonlinear oscillators and power grid stability.
  • Method accuracy is high when noise autocorrelation time is significantly shorter or longer than system relaxation time.

Conclusions:

  • The novel PDF method offers a robust framework for analyzing nonlinear stochastic systems with colored noise.
  • The modified LED closure and local linearization approximation enhance predictive accuracy.
  • Accuracy is sensitive to the ratio of noise autocorrelation to system relaxation times and noise standard deviation.