We study the correspondence between analytical and empirical slow-flow analyses, which will form a basis for a time-domain nonparametric nonlinear system identification method. Given a sufficiently dense set of sensors, measured time series recorded throughout a mechanical or structural system contains all information regarding the dynamics of that system. Empirical mode decomposition (EMD) is a useful tool for decomposing the measured time series in terms of intrinsic mode functions (IMFs), which are oscillatory modes embedded in the data that fully reproduce the time series. The equivalence of responses of the analytical slow-flow models and the dominant IMFs derived from EMD provides a physics-based theoretical foundation for EMD, which currently is performed formally, in an ad hoc fashion. First deriving appropriate mathematical expressions governing the empirical slow flows and based on analyticity conditions, we demonstrate only close correspondence between analytical and empirical slow flows in a physical system that can be modeled as a two-degree-of-freedom strongly nonlinear coupled oscillators.