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Density evolution in stochastic dynamical systems with memory: A universal algorithm.

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This study introduces a universal method for calculating probability density evolution in stochastic dynamical systems with memory. The computationally efficient approach uses a discrete model derived from the Euler scheme, enabling broader applications.

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Area of Science:

  • Dynamical Systems and Chaos Theory
  • Computational Mathematics
  • Climate Science

Background:

  • Stochastic dynamical systems with memory are often modeled using stochastic functional differential equations (SFDEs).
  • Quantifying probability density evolution in SFDEs is crucial for practical applications but remains challenging due to a lack of efficient computational methods.
  • The general form of SFDEs restricts their widespread application.

Purpose of the Study:

  • To present a universal and computationally efficient approach for calculating the probability density evolution in a broad class of stochastic dynamical systems with memory.
  • To overcome the limitations imposed by the lack of efficient methods for SFDEs.
  • To enable wider practical applications of these complex systems.

Main Methods:

  • Approximation of the stochastic functional equation using a discrete model derived from the Euler scheme.
  • Recursive estimation of probability density by computing the density of the discretized counterpart.
  • The proposed method is deterministic and computationally efficient.

Main Results:

  • Successfully computed transient and long-term probability density evolution for stochastic dynamical systems with memory.
  • Demonstrated the effectiveness and efficiency of the novel approach.
  • Validated the method on typical climate models.

Conclusions:

  • The developed universal approach provides an efficient and deterministic method for computing probability density evolution in stochastic dynamical systems with memory.
  • This method significantly broadens the applicability of SFDEs in various scientific fields, including climate modeling.
  • The approach offers a robust tool for analyzing complex systems where memory effects are significant.