Involution symmetry quantification using recurrences.

Area of Science:

• Complex Systems
• Nonlinear Dynamics
• Time Series Analysis

Background:

• Symmetries are fundamental in scientific theories, aiding pattern recognition in mathematical models and natural phenomena.
• Quantifying symmetry in time series data is crucial for understanding underlying dynamical systems.
• Existing methods may not fully capture the nuanced symmetries present in complex time series.

Purpose of the Study:

• To introduce a novel method for assessing the extent of symmetry within time series data.
• To explore both microscopic and macroscopic features from recurrence plots for symmetry detection.
• To validate the method's versatility across different dynamical models and phenomena.

Main Methods:

• Utilizing recurrence plots to analyze both microscale and macroscale features of time series.
• Analyzing statistics of small recurrence matrices to probe microscale dynamics.
• Identifying symmetric time series segments via diagonal macroscale structures on recurrence plots.

Main Results:

• Successfully quantified involution symmetries in 3D dynamical models, including rotational symmetry in the Lorenz '63 model and inversion symmetry in the Chua circuit.
• Demonstrated the method's ability to detect symmetry breaking in a modified Lorenz model relevant to the El Niño phenomenon.
• Validated the approach's applicability to both 3D trajectories and univariate time series.

Conclusions:

• The developed method provides a robust way to quantify symmetry in time series.
• This technique enhances the understanding and modeling of dynamical systems by profiling their symmetries.
• Symmetry quantification in time series offers promising applications in various scientific domains.