Constructing low-dimensional ordinary differential equations from chaotic time series of high- or...

Area of Science:

• Dynamical Systems and Chaos Theory
• Nonlinear Time Series Analysis
• Computational Physics

Background:

• Previous work introduced the radial-function-based regression (RfR) method for constructing ordinary differential equations from deterministic time series.
• The RfR method utilizes Gaussian radial basis functions and polynomial terms for robust modeling of chaotic behavior.

Purpose of the Study:

• To apply and validate the RfR method on diverse high- or infinite-dimensional deterministic systems.
• To construct low-dimensional differential equation models from complex time series data, including noisy observations.

Main Methods:

• Application of the radial-function-based regression (RfR) method.
• Regression using Gaussian radial basis functions and polynomial terms.
• Analysis of time series from partial differential equations, delay differential equations, turbulence models, and intermittent dynamics.

Main Results:

• Successfully constructed low-dimensional ordinary differential equation systems for various complex deterministic dynamics.
• Demonstrated effective modeling and forecasting capabilities, including reconstruction of invariant sets and densities.
• Validated the method's robustness in the presence of observational noise.
• Identified chaotic saddle trajectories using the stagger-and-step method in specific models.

Conclusions:

• The RfR method is a powerful tool for deriving low-dimensional dynamical models from complex time series.
• The constructed models exhibit predictive accuracy and accurately represent the underlying chaotic dynamics.
• The method's applicability extends to systems with noise and high/infinite dimensions.