A Preliminary Study On The Whole Analysis Of Time Series Of Nonlinear Dynamical System

In the fast-developing Internet era,systems in different fields will generate large amounts of data all the time.Most of these data are chaotic and disorganized.By modeling and analyzing the data,people further study the characteristics of the system that generates the data,and even use models to predict the future development trend of the system.Time series analysis has become an important research topic in data analysis.However,the analysis of nonlinear time series still faces severe challenges such as noise interference and nonlinear term processing.In response to the above challenges,the paper analyzes the time series of nonlinear dynamical systems.The main work and innovation results of the paper are as follows:1.The thesis adopts the theory of renormalization group to establish a fitting model for nonlinear time series,realizes the approximate characterization of the system dynamics expression,and predicts the short-term development trend of the system.The advantage of this model is that the influence of smaller noise on the fitting of the time series can be approximately ignored;for larger noise,the paper also realizes the separation of pure data and noise variance from the noise time series.2.The paper uses the Koopman operator to act on the observables of the nonlinear dynamical system.Through spectral analysis of the Koopman operator and calculation of its eigenvalues and eigenfunctions,the division of the invariant set in the phase space of the system is realized.With the help of Koopman operator,not only can the geometry of dynamics be realized,and the main dynamical modes of the system can be extracted,but also the effective dimensionality reduction or efficient control of high-dimensional systems or complex systems can be further processed.