Incomplete phase-space method to reveal time delay from scalar time series

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Time Series Analysis

Background:

• Characterizing chaotic systems often requires identifying their time delay parameter.
• Traditional methods can be computationally intensive or require extensive data.
• Scalar time series present challenges for direct time delay estimation.

Purpose of the Study:

• To develop a computationally efficient and conceptually simple method for recovering the time delay of chaotic systems from scalar time series.
• To introduce a component reordering procedure and a segmented mean variance (SMV) metric for time delay identification.
• To demonstrate the robustness and effectiveness of the proposed method using numerical and experimental data.

Main Methods:

• Reconstruction of a 2D phase space from scalar time series.
• Component reordering to enhance local clustering of orbits.
• Calculation of segmented mean variance (SMV) from the reordered component.
• Validation using the Mackey-Glass equation in a chaotic regime.

Main Results:

• The component reordering reveals local clustering related to the system's time delay.
• The SMV metric exhibits a clear maximum when the embedding delay matches the system's time delay.
• The method successfully recovers the time delay from noisy data, small datasets, and weak feedback strengths.
• The proposed method demonstrates low time complexity.

Conclusions:

• The developed method provides an effective and efficient approach for time delay estimation in chaotic systems.
• The component reordering and SMV technique offer a robust solution for time series analysis in nonlinear dynamics.
• This method has practical implications for analyzing experimental data from chaotic systems.