Koopman Analysis Based On Renormalization Group Method

Synchronization is an important dynamics feature in coupled nonlinear systems,which has been studied extensively in recent years.However,analy-sis focused on individual orbits seems hard to extend to complex systems while a global statistical approach is overly cursory.Koopman operator technique seems to well balance the two points of view.In this paper,we extend Koop-man analysis to the study of synchronization of coupled oscillators by extract-ing important eigenvalues and eigenfunctions from the observed time series.The eigenfrequency can be computed from the eigenvalue,and the synchro-nization transition points could be located with quite high accuracy by check-ing the change of the correlation coefficient of neighbouring eigenfunctions at different coupling strengths,which may be readily applied to other nonlinear systems.In order to verify the reliability of Koopman's method,we apply it to Kuramoto model on a circle and a complex network.The results of analysis are in good agreement with the numerical results:the eigenfrequency corresponds to the average frequency,and the minimum of the correlation coefficient of the eigenfunction corresponds to the phase transition point.The importance of the average frequency and the correlation eigenfunction is further confirmed.In addition,a renormalization group approach is designed to derive of an analytic approximation of the eigen-system for weak coupling,which eliminates the res-onance term.We proposes renormalization frequency and frequency correction term,which can help to better understand the relationship between the average frequency and the critical coupling strength.