Optimal reconstruction of dynamical systems: a noise amplification approach

Area of Science:

• Dynamical systems theory
• Time series analysis
• Nonlinear dynamics

Background:

• State space reconstruction is crucial for understanding dynamical systems from time series data.
• Existing methods for reconstruction have limitations in parameter selection and comparison.
• Noise and attractor complexity can significantly impact reconstruction accuracy.

Purpose of the Study:

• To propose a novel objective function for guiding state space reconstruction.
• To enable direct comparison of different reconstruction approaches.
• To facilitate the selection of optimal parameters for reconstruction strategies.

Main Methods:

• Developed a cost function based on noise amplification, attractor complexity, and local stretch.
• Applied the objective function to evaluate various reconstruction methods like delay vectors, PCA, and Legendre coordinates.
• Demonstrated the method's utility in optimizing delay time and embedding dimension.

Main Results:

• The objective function effectively guides the search for optimal state space reconstructions.
• It allows for quantitative comparison of different reconstruction techniques.
• Parameter selection, including optimal delay time and embedding dimension, is improved.

Conclusions:

• The proposed objective function offers a robust framework for state space reconstruction.
• It enhances the reliability and interpretability of dynamical system analysis from time series.
• The method is validated on both synthetic and experimental data.