Involution symmetry quantification using recurrences

Symmetries are ubiquitous in science, aiding theoretical comprehension by discerning patterns in mathematical models and natural phenomena. This work introduces a method for assessing the extent of symmetry within a time series. We explore both microscopic and macroscopic features extracted from a recurrence plot. By analyzing the statistics of small recurrence matrices, our approach delves into microscale dynamics, facilitating the identification of symmetric time series segments through diagonal macroscale structures on a recurrence plot. We validate our approach by successfully quantifying involution symmetries for three-dimensional dynamical models, specifically, order-2 rotational symmetry in the Lorenz '63 model, and inversion symmetry in the Chua circuit. Our quantifier also detects symmetry breaking in the modified Lorenz model for El Niño phenomenon. The method can be applied in a versatile manner, not only to three-dimensional trajectories but also to univariate time series. Symmetry quantification in time series is promising for enhancing dynamical system modeling and profiling.