HIGHER-ORDER SPECTRAL ANALYSIS TO DETECT NONLINEAR INTERACTIONS IN MEASURED TIME SERIES AND AN APPLICATION TO CHUA’S CIRCUIT

Abstract

Higher-order spectra have been used to investigate nonlinear interactions between the Fourier components of measured time series in a remarkably wide range of random processes. The basic techniques of detecting and isolating nonlinear phase coupling in observed data using higher-order spectral analysis are reviewed here. These techniques are then used to investigate nonlinear interactions in time series of voltages measured from a realization of Chua’s circuit. For period-doubled limit cycles, quadratic and cubic nonlinear interactions result in phase coupling and energy exchange between increasing numbers of triads and quartets of Fourier components as the nonlinearity of the system is increased. For circuit parameters that result in a chaotic, Rössler-type attractor, bicoherence and tricoherence spectra indicate that both quadratic and cubic nonlinear interactions are important to the dynamics. For parameters that lead to the double-scroll chaotic attractor the bispectrum is zero, but the tricoherences are high, consistent with the importance of higher-than-second order nonlinear interactions during chaos associated with the double scroll.