account for sensitivities of the response to force and voltage magnitudes and to initial conditions, all of which can lead to large changes in response over short time. One cannot overestimate the importance of the need for an effective, straightforward, system identification and reduced-order modeling method for characterizing strongly nonlinear and nonstationary, complex, multi-component systems. However, the difficulty in developing a method of broad applicability is the well-recognized, highly individualistic nature of nonlinear systems which restricts the unifying dynamical features that are amenable to system identification. Reviews of nonlinear system identification (NSI) and reduced-order methods are presented in Kerschen et al. [4,5]. Moon [6] notes suitable methods for diagnosing chaotic motion which are also useful for determining when observed responses result from strongly nonlinear systems. These include analysis of time series of the response, interpretation of phase space trajectories and Poincaré maps, examination of power spectra of the response, and observation of system response when individual parameters are varied. Nichols and Virgin [7] presented a method for estimating damping in a system by chaotic interrogation, and Feeny et al. [8] performed NSI of a magneto-elastic two-well chaotic system. Typical nonparametric NSI methods include proper orthogonal decomposition, Volterra theory, and pattern recognition based on artificial neural networks. For example, Silva [9] performed NSI using Volterra theory on aeroelastic systems, developed computationally-efficient reduced-order models (ROMs) employing an Euler/Navier-Stokes fluid solver, and finally derived analytically Volterra kernels for nonlinear aeroelastic systems from data of flight flutter tests of an active aeroelastic wing aircraft. There are alternative well-established methods for nonlinear parameter estimation, such as the restoring force surface method [10], NARMAX (Nonlinear Auto-Regressive Moving Average models with eXogenous inputs) methods [11,12], methods based on Hilbert transforms [13,14], and others. Thothadri et al. [15] performed NSI of multi-degree-offreedom (MDOF) fluid-structure interaction systems using the principle of harmonic balance (HB). The main advantage of the HB technique is its usefulness to predict bifurcation behavior of a nonlinear system, for which nonparametric methods are not usually well suited. This is performed by exploiting the periodicity in the response of an experimental system, when parametric time-domain methods such as the NARMAX fail. A multi-staged approach for fitting the excitation of a nonlinear system in nonparametric form was developed in Masri et al. [16,17]. Also, a general data-based approach for developing ROMs for nonlinear MDOF systems was proposed assuming no information about the system mass [18]. reduced-order modeling based on nonlinear normal modes [19] has been discussed in Touzé et al. [20-22]. However, these techniques are only applicable to specific classes of dynamical systems; in addition, some type of functional form is assumed for modeling the system nonlinearity, and the main task becomes the determination of the corresponding coefficients. The key to a successful NSI method is the recognition that a single parametric model derived from data at a specific operating point in the frequency-energy space will not be globally descriptive. Rather, the method should non-parametrically provide a global picture of system behavior leading naturally to local parametric models capable of reproducing the strongly nonlinear phenomena at the operating points. With this idea in mind, we recently proposed a time-domain (nonparametric) NSI method based on equivalence or correspondence between analytical and empirical slow flows of a dynamical system [23,24], and studied its application to targeted energy transfers in 2-degree-of-freedom dynamical system, aeroelastic instability suppression [25], and a rod with an essentially nonlinear end attachment [26]. The proposed NSI method is feasible for broad classes of applications involving time-variant/time-invariant, linear/nonlinear, and smooth/non-smooth dynamical systems. Furthermore, it requires no a priori system information but only measured (or simulated) time series; i.e., it is purely an output-based approach. Empirical mode decomposition (EMD [27]) is employed to yield intrinsic mode functions (IMFs) of the measured time series as empirical slow flows, and its correspondence to the analytical slow-flow dynamics enables us to establish a nonlinear interaction model (NIM). The NIM consists of a set of intrinsic modal oscillators (IMO), an equivalent linear oscillator that can produce a given time series over different time scales and keeps any existing nonlinear modal interactions in its time-varying forcing term. An IMO, the solution of which reproduces the corresponding IMF, represents dynamical characteristics of the system under certain initial or parameter conditions (i.e., local aspects). By extracting energy-dependent frequency behavior from identified IMOs, a frequency-energy plot (in the modal space) can be constructed as global features of the dynamical system. 2. REDUCED SLOW-FLOW DYNAMICS AND EMD Slow-flow reduction of the dynamics is a useful tool for understanding the major features of a dynamical system. The reduced slow-flow model of a dynamical process is derived by introducing a slow/fast partition of the dynamics whereby the (non-essential) fast dynamics is averaged out to reveal the (essential) slow-flow modulations of appropriately defined amplitudes and phases. Perturbation tools have been developed to perform this task; e.g., the Linstedt-Poincaré method 514
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