Enhancing recursive feature extraction in nonlinear time-series via fuzzy membership functions

Traditional recurrence analysis methods relying on preset thresholds inevitably introduce subjectivity into analytical results. To enhance the objectivity and robustness of recurrence analysis outcomes, this study proposes a novel recursive feature extraction approach based on fuzzy membership functions. The core mechanism of the method comprises three key steps: first, mapping the distance matrix between phase points to grayscale values within the interval [0, 1] (enabling unthresholded quantification); second, constructing texture recurrence plots (TRPs) for intuitive visual characterization; and third, calculating singular spectral entropy (Es) to quantitatively assess system complexity. The analyses of theoretical chaotic sequences (e.g., the Lorenz system) demonstrate that TRPs and Es derived from Z-type and Gaussian membership functions can effectively elucidate the intrinsic dynamical properties of diverse systems. Furthermore, the method was applied to the characterization of measured friction-induced signals. The results reveal that TRPs and Es generated by the Gaussian membership functions proficiently capture the self-organizing friction-induced attractors, providing reliable support for friction state identification. This research advances the refined extraction of nonlinear temporal recursive features and offers a novel perspective for characterizing the nonlinear attributes of complex dynamical systems.