Enhancing Spectral Analysis in Nonlinear Dynamics: An Approach Based on Generalized Eigenfunctions 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 spectral features of nonlinear dynamical systems without any prior knowledge. However, analyzing the continuous spectrum resulting from chaos and noise is problematic due to its infinite dimensions. We propose a clustering-based method designed to analyze dynamics represented by generalized eigenfunctions associated with continuous spectra. This paper describes data-driven algorithms for comparing generalized eigenfunctions using subspaces. To compute approximate generalized eigenfunctions from data, we use the recently proposed Residual Dynamic Mode Decomposition (ResDMD). 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 some insights into the complexities of coupled chaos.