On the structure of time-delay embedding in linear models of non-linear dynamical systems

This work addresses fundamental issues related to the structure and conditioning of linear time-delayed models of non-linear dynamics on an attractor. While this approach has been well-studied in the asymptotic sense (e.g., for an infinite number of delays), the non-asymptotic setting is not well-understood. First, we show that the minimal time-delays required for perfect signal recovery are solely determined by the sparsity in the Fourier spectrum for scalar systems. For the vector case, we provide a rank test and a geometric interpretation for the necessary and sufficient conditions for the existence of an accurate linear time delayed model. Furthermore, we prove that the output controllability index of a linear system induced by the Fourier spectrum serves as a tight upper bound on the minimal number of time delays required. An explicit expression for the exact linear model in the spectral domain is also provided. From a numerical perspective, the effect of the sampling rate and the number of time delays on numerical conditioning is examined. An upper bound on the condition number is derived, with the implication that conditioning can be improved with additional time delays and/or decreasing sampling rates. Moreover, it is explicitly shown that the underlying dynamics can be accurately recovered using only a partial period of the attractor. Our analysis is first validated in simple periodic and quasiperiodic systems, and sensitivity to noise is also investigated. Finally, issues and practical strategies of choosing time delays in large-scale chaotic systems are discussed and demonstrated on 3D turbulent Rayleigh-Bénard convection.