A new solution method for high-dimensional stochastic dynamical systems via delay embedding

Area of Science:

• Dynamical Systems and Control
• Computational Physics
• Applied Mathematics

Background:

• Stochastic dynamical systems are prevalent but challenging to solve, especially in high dimensions.
• Analyzing only a subset of system variables is often sufficient for practical applications.
• Effective dimensionality reduction is crucial for studying complex systems.

Purpose of the Study:

• To develop a data-driven computational framework for high-dimensional stochastic dynamical systems.
• To reduce the dimensionality of complex systems using delay embedding.
• To provide an efficient and accurate analysis tool for stochastic systems.

Main Methods:

• A data-driven computational framework based on delay embedding is proposed.
• Multiple time series are used to construct delay-embedding mappings.
• These mappings transform the high-dimensional system into a low-dimensional time-delay system.

Main Results:

• The framework was evaluated on 4D and 10D systems under Gaussian white noise.
• Probability density function analysis showed excellent agreement with Monte Carlo simulations.
• Substantial reductions in computational cost were achieved.

Conclusions:

• The proposed framework offers an efficient and accurate method for analyzing high-dimensional stochastic systems.
• Delay embedding provides a powerful tool for dimensionality reduction in stochastic dynamics.
• This approach facilitates the study of complex systems with reduced computational resources.