Koopman Analysis Of Nonlinear Coupled Oscillator System

With the development of probing and computing technology,the study of complex systems becomes a necessity in many science and engineering problems.Except for linear systems,it is usually very difficult to resolve nonlinear systems,especially in high-dimensions.The study of dynamics focused on qualitive rather than quantitative features.In order to better.understand the dynamic characteristics of the system,it is necessary to extract the main mode.A good extraction scheme must involve dimension reduction or coordinate transformation.With the advancement of data-collecting technology,it becomes possible to obtain large amount of data for complex systems with relatively low cost.How to deduce dynamical or statistical properties of nonlinear systems from these data constitutes a large part of applied dynamics investigation in the modern era.The quest for efficient data-driven methods ever continues and appears more and more urgent in either theoretical or applied disciplines.common linear analysis method,such as POD and balanced POD,the linear characteristics of which hinder the effective representation of highly nonlinear characteristics in phase space.Some common nonlinear techniques,such as equidistance mapping and diffusion mapping,are mainly used to describe the geometric structure of phase space.However,in high-dimensional nonlinear systems,the geometric structure of phase space is very complex,and the description will become very difficult.It remains a great challenge to qualitatively describe the invariant structures at a global scale.The spectral properties of the operator and the Koopman modes of a typical orbit reveal interesting invariant structures near the orbit.In the evolution process of dynamical system,Koopman analysis focuses on how to extract the main modes of the system and what statistical characteristics can be used to track the bifurcation or chaos of the dynamical system.This method can control and predict the system better in the promotion of high dimension,which has great application value in various fields.In this paper,we mainly study the use of Koopman mode decomposition algorithm for system dynamics mode extraction,and use these modes to track the properties of system dynamics.Firstly,the Koopman analysis method is applied to a single orbit of a two degree of freedom coupled oscillator system,and the basic frequency information,growth rate and corresponding spatial structure of the mode are obtained,which proves the effectiveness of the method.Then,by tracking the amplitude change of the main mode of the corresponding orbit when the initial point changes along a straight line on the Poincare section,the change of dynamic properties of the phase space along the straight line is detected.Finally,the method is extended to the three degree of freedom coupled oscillator model,and similar conclusions are obtained.By this method of local tracing dynamics,the orbit strcture is divided into several invariant subsets to avoid the possible overall complexity of the nonlinear system orbits.Even in the case of high-dimensional,we can effectively identify the important orbital characteristics of part of the phase space,which is helpful for us to grasp the characteristics of the phase space as a whole,to coarsen the high-dimensional phase space,and to understand the qualitative changes in the process of system evolution.