Wavelet analysis of time series for the Duffing oscillator: The detection of order within chaos.

Abstract

We consider a wavelet analysis of various time series for the Duffing oscillator, for which there is a potential maximum and a harmonic forcing term, and we focus on time series that return to the region of the potential maximum. When the dynamics is chaotic and the time series is highly nonstationary, there are many significant higher harmonics in a Fourier expansion and the usual Fourier analysis is problematic, especially for short total times. We show that the wavelet analysis is a robust tool that may be used to obtain qualitative information for highly nonstationary time series---specifically, that it may be used to detect a small-amplitude harmonic forcing term even when the dynamics is chaotic and even for short total times.