Theoretical and Computational Study of Nonlinear Series Expansion for Nonlinear System Identification

This article asks the question “Does a given nonlinear time series is generated by a unique nonlinear differential equation?” providing negative answer. We show that a given nonlinear time series is generated by a nonlinear differential equation with an infinite number of parameters, and each set of parameters has its own mathematical characteristics (a set of eigenvalues), corresponding to the identity (spectrogram) of the system. That is, we cannot identify the hidden nonlinear equation from the given time series. We show such situations in several ways. As a separate endeavor, we also investigate computational methods in nonlinear analysis, which allows us to obtain a single spectrogram from a given nonlinear time series. We here try both theoretical and computational approaches to a nonlinear system, to find which spectrogram is numerically chosen among an infinite number of theoretical spectrograms.