A novel embedding method for characterization of low-dimensional nonlinear dynamical systems
Characterization of dynamical systems remains a central challenge in real-world applications because, in most cases, governing equations of the systems cannot be obtained explicitly. Moreover, the behaviors of the systems are complex and possibly unpredictable, even when the system is deterministic. In this situation, nonlinear time series analysis provides an efficient tool for characterizing such dynamical systems. As one of the key techniques in nonlinear time series analysis, phase space reconstruction (PSR) can represent a univariate time series by a multidimensional phase space. Nevertheless, such representation relies on the conscientious selection of the time delay and embedding dimension. Although various algorithms have been developed to optimize these embedding parameters, there are still some limitations restricting their applicability, such as subjective effects, erratic results, and the consumption of time. Herein, we propose a novel Constant embedding parameters and Principal component analysis-based PSR (CPPSR) method, to characterize the low-dimensional (≤3) dynamical systems. The effectiveness and accuracy of the CPPSR method are numerically verified on various dynamical systems. The numerical results demonstrate that the CPPSR method can produce reconstructed attractors with precise correlation dimension and the largest Lyapunov exponent. The comparison with conventional and Hankel matrix-based PSR methods shows the superiority of the CPPSR method, demonstrating greater accuracy, higher efficiency, and stronger reliability. Moreover, the CPPSR method shows a high potential for the detection of singularity in applications such as structural health monitoring and abnormality diagnosis in ECG signals.
