Reconstructing bifurcation diagrams from noisy time series using nonlinear autoregressive models

Area of Science:

• Dynamical Systems and Nonlinear Science
• Time Series Analysis
• Computational Physics

Background:

• Reconstructing bifurcation diagrams from experimental data is challenging due to noise.
• Existing methods often struggle with limited or noisy time series.
• Understanding system dynamics through bifurcation analysis is crucial in many scientific fields.

Purpose of the Study:

• To develop a robust formalism for reconstructing bifurcation diagrams from noisy time series.
• To introduce a two-stage algorithm for model selection and parameter identification.
• To validate the method's effectiveness with limited data.

Main Methods:

• A formalism for reconstructing bifurcation diagrams from noisy time series.
• A two-stage algorithm: 1. Model Selection (using nonlinear autoregressive models with polynomial terms) and 2. Bifurcation Parameter Identification.
• Employing a parametrized predictor function to match the bifurcation structure of the original system.

Main Results:

• Successfully reconstructed bifurcation diagrams from noisy time series.
• The nonlinear autoregressive model effectively represents the given time series.
• The bifurcation parameter identification stage accurately isolates key dynamic parameters.
• The algorithm demonstrates efficacy even with a limited number of time series.

Conclusions:

• The proposed formalism provides a reliable method for bifurcation diagram reconstruction.
• The two-stage algorithm is effective for analyzing complex dynamical systems from observational data.
• This approach enhances the ability to study nonlinear dynamics in the presence of experimental noise.