Nonlinear characterization of a Rossler system under periodic closed-loop control via time-frequency and bispectral analysis Robert Bruce Alstrom a,⇑ , Stéphane Moreau a , Pier Marzocca b , Erik Bollt c a Aeroacoustics Group, Department of Mechanical Engineering, University of Sherbrooke, Canada b Aerospace Engineering and Aviation, School of Engineering, RMIT University, PO Box 71, Bundoora, VIC 3083, Australia c Dept of Mathematics, Clarkson University, Potsdam, NY 13699-5815, United States article info Article history: Received 7 October 2016 Received in revised form 2 May 2017 Accepted 4 June 2017 Keywords: Quadratic phase coupling Nonlinear system identification Bicoherence Synchronization Frequency entertainment abstract This study has two primary objectives; they are to investigate the nonlinear interactions (or quadratic phase-coupling) in a chaotic Rossler system under periodic closed-loop con- trol via wavelet bispectral analysis; and to further identify the component mechanisms of synchronization. It is observed that a fixed-gain, fixed-frequency controller produces quad- ratic phase-coupling and decoupling along lines of constant frequency and that are perpen- dicular to the diagonal of the bicoherence matrix. Further, it was also observed that for synchronization to occur, both frequency entrainment and quadratic phase-coupling must be present. It was found that forcing the Rossler system with a constant frequency did not reduce the amplitude of the resulting period-1 orbit at sufficiently high gains. For the con- troller with a fixed gain and time-varying error signal, it was found that the time varying forcing frequency (adjusted by an extremum seeking feedback loop) linearizes the Rossler system and in doing so, suppresses the phase coherence completely. The time-varying forc- ing frequency removes the conditions for frequency entrainment by providing broadband attenuation; the result is suppression without synchronization. Ó 2017 Published by Elsevier Ltd. 1. Introduction Power spectral analysis is sufficient for the analysis of linear systems, but they cannot provide information about the non- linear interaction between Fourier modes, nor can they resolve the changes in Fourier components in time. In general, the bicoherence, which is the normalized bispectrum a measure of the amount of phase coupling that occurs in a signal or between two signals. Phase coupling is said to occur when two component frequencies are simultaneously present in the signal (s) along with their sum (or difference) frequencies and the phase of these component frequencies remains constant. There are two types of bicoherence analysis, the first is Fourier based and the other is wavelet based; the wavelet based option will be used in this research. A formal definition of wavelet bicoherence is provided in Section 3 of this work. Bispec- tral analysis is applied to a wide variety of nonlinear systems. These systems include mathematical nonlinear systems with quadratic and cubic nonlinearities, mechanical systems, aeromechanical systems and fluid mechanics. The following brief literature review will highlight some of these examples. http://dx.doi.org/10.1016/j.ymssp.2017.06.001 0888-3270/Ó 2017 Published by Elsevier Ltd. ⇑ Corresponding author. E-mail addresses: Robert.Bruce.Alstrom@usherbrooke.ca (R.B. Alstrom), stephane.moreau@usherbrooke.ca (S. Moreau), pier.marzocca@rmit.edu.au (P. Marzocca), bolltem@clarkson.edu (E. Bollt). Mechanical Systems and Signal Processing 99 (2018) 567–585 Contents lists available at ScienceDirect Mechanical Systems and Signal Processing journal homepage: www.elsevier.com/locate/ymssp