From phase space to frequency domain: a time-frequency analysis for chaotic time series

Area of Science:

• Nonlinear Dynamics
• Chaos Theory
• Signal Processing

Background:

• Chaotic systems exhibit complex dynamics often analyzed using time-frequency methods.
• Traditional methods may struggle with the irregular recurrences characteristic of chaotic attractors.
• Phase-space reconstruction reveals state recurrences, but their temporal irregularity poses analytical challenges.

Purpose of the Study:

• To develop a novel power spectrum estimator for time-frequency analysis of chaotic signals.
• To leverage redundant information from phase-space neighborhood recurrences.
• To differentiate between chaotic flow and colored noise using time-frequency features.

Main Methods:

• Phase-space reconstruction using time delay embedding.
• Development of a neighborhood-based spectrum estimator utilizing state recurrences.
• Application of the estimator to Lorenz and Rössler time series, experimental laser data, and colored noise.

Main Results:

• Demonstrated the relationship between reference phase points and their nearest neighbors.
• Showcased the ability of the neighborhood-based estimator to utilize redundant information from recurrences.
• Generated spectrograms revealing distinct features for chaotic data versus colored noise.

Conclusions:

• The neighborhood-based spectrum estimator is effective for time-frequency analysis of chaotic dynamics.
• The method successfully distinguishes chaotic flow from colored noise based on spectrogram features.
• This approach offers a new tool for analyzing complex, irregular time series data.