Enhancing spectral analysis in nonlinear dynamics with pseudoeigenfunctions from continuous spectra

The analysis of complex behavior in empirical data poses significant challenges in various scientific and engineering disciplines. Dynamic Mode Decomposition (DMD) is a widely used method to reveal the spectral features of nonlinear dynamical systems without prior knowledge. However, because of its infinite dimensions, analyzing the continuous spectrum resulting from chaos and noise is problematic. We propose a clustering-based method to analyze dynamics represented by pseudoeigenfunctions associated with continuous spectra. This paper describes data-driven algorithms for comparing pseudoeigenfunctions using subspaces. We used the recently proposed Residual Dynamic Mode Decomposition (ResDMD) to approximate spectral properties from the data. To validate the effectiveness of our method, we analyzed 1D signal data affected by thermal noise and 2D-time series of coupled chaotic systems exhibiting generalized synchronization. The results reveal dynamic patterns previously obscured by conventional DMD analyses and provide insights into coupled chaos's complexities.