Delay Time Window Research Articles

Nonlinear dynamics, delay times, and embedding windows

In order to construct an embedding of a nonlinear time series, one must choose an appropriate delay time τ d. Often, τ d is estimated using the autocorrelation function; however, this does not treat the nonlinearity appropriately, and it may yield an incorrect value for τ d. On the other hand, the correct value of τ d can be found from the mutual information, but this process is rather cumbersome computationally. Here, we suggest a simpler method for estimating τ d using the correlation integral. We call this the C–C method, and we test it on several nonlinear time series, obtaining estimates of τ d in agreement with those obtained using the mutual information. Furthermore, some researchers have suggested that one should not choose a fixed delay time τ d, independent of the embedding dimension m, but, rather, one should choose an appropriate value for the delay time window τ w=( m−1) τ, which is the total time spanned by the components of each embedded point. Unfortunately, τ w cannot be estimated using the autocorrelation function or the mutual information, and no standard procedure for estimating τ w has emerged. However, we show that the C–C method can also be used to estimate τ w. Basically τ w is the optimal time for independence of the data, while τ d is the first locally optimal time. As tests, we apply the C–C method to the Lorenz system, a three-dimensional irrational torus, the Rossler system, and the Rabinovich–Fabrikant system. We also demonstrate the robustness of this method to the presence of noise.

Delay time window and plateau onset of the correlation dimension for small data sets

The method of delays is widely used for reconstructing chaotic attractors from experimental observations. Many studies have used a fixed delay time ${\ensuremath{\tau}}_{d}$ as the embedding dimension m is increased, but this is not necessarily the best choice for obtaining good convergence of the correlation dimension. Recently, some researchers have suggested that it is better to fix the delay time window ${\ensuremath{\tau}}_{w}$ instead. Unfortunately, ${\ensuremath{\tau}}_{w}$ cannot be estimated using either the autocorrelation function or the mutual information, and no standard procedure for estimating ${\ensuremath{\tau}}_{w}$ has yet emerged. However, the recently introduced $C\ensuremath{-}C$ method can be used to estimate either ${\ensuremath{\tau}}_{d}$ or ${\ensuremath{\tau}}_{w}.$ Using this method, we show that, for small data sets, fixing ${\ensuremath{\tau}}_{w},$ rather than ${\ensuremath{\tau}}_{d},$ does indeed lead to a more rapid convergence of the correlation dimension as the embedding dimension m is increased.