Cardinality-based analysis of bifurcation structure in the stochastic Ueda oscillator

This study presents a systematic comparison of three quantitative measures for characterizing bifurcation structure in the pseudo-random Ueda oscillator: (i) the largest Lyapunov exponent (LLE), (ii) the Minkowski–Bouligand (MB), known also as the box-counting fractal dimension (iii) the cardinality number measure (CNM), and (iv) the spectral entropy (SE). The performance of each method is evaluated across a broad range of noise intensities in order to assess their computational efficiency and robustness under pseudo-random noise (PRN). It is revealed that LLE provide a useful indicator of dynamical complexity and stability; however, this measure degrades significantly in the presence of noise, particularly near bifurcation boundaries where pseudo-random fluctuations obscure deterministic structure. In contrast, the CNM emerges as a fast and noise-resilient diagnostic. By directly counting occupied grid cells in Poincaré projections, the CNM avoids explicit phase-space reconstruction and scaling analysis, resulting in substantially reduced computational cost. Moreover, it maintains reliable discrimination between periodic, quasiperiodic, and chaotic regimes even at elevated noise levels. These properties make the cardinality-based approach a practical and robust tool for the quantitative analysis of bifurcation diagrams in pseudo-random nonlinear oscillators.