Application Of Koopman Analysis With Autoencoder To Nonlinear Vibration

The analysis of complex dynamical systems has always been a concern in various fields.Data of different dynamical systems,both from experiment and computer simulation,can be readily obtained and analyzed for research.Various approaches were proposed for dynamical systems data processing,Koopman operator is one of the outstanding methods and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics.However,it is difficult to identify and represent these eigenfunctions.In this paper,we used an autoencoder to achieve the reconstruction and evolution of complex dynamical systems by Koopman operators.According to the properties of the autoencoder,part of the encoder automatically becomes the Koopman eigenfunction,while the constrained evolutionary structure becomes the eigenvalue of the eigenfunction.Furthermore,since the autoencoder can minimize the loss function,the eigenfunctions corresponding to each encoder should be the most important ones of the system.The effectiveness of the method is verified on 1-d linear equation,Van der Pol equation and coupled oscillator system.