Abstract
Throughout scientific research, the state space reconstruction that embeds a non-linear time series is the first and necessary step for characterizing and predicting the behavior of a complex system. This requires to choose appropriate values of time delay T and embedding dimension dE. Three methods are applied and discussed on nonlinear time series provided by the Rossler attractor equations set: Cao’s method, the C-C method developed by Kim et al. and the C-C-1 method developed by Cai et al. A way to fix a parameter necessary to implement the last method is given. Focus has been put on small size and/or noisy time series. The reconstruction quality is measured by using a criterion based on the transformation smoothness.
Highlights
Three methods are applied and discussed on nonlinear time series provided by the Rössler attractor equations set: Cao’s method, the C-C method developed by Kim et al and the C-C-1 method developed by Cai et al A way to fix a parameter necessary to implement the last method is given
In many fields of science and industry, complex systems are studied through temporal time series of scalar observations of a k dimensional dynamical system [1] [2] [3] [4] [5]
The original time series is subdivided by setting a parameter q which is independent of the time delay T
Summary
In many fields of science and industry, complex systems are studied through temporal time series of scalar observations of a k dimensional dynamical system [1] [2] [3] [4] [5]. When time series are limited or contaminated by noise, the theorem of Takens is silent and the delay time T is observed to vary with the embedding dimension dE In this case, as an irrelevant partnership between T and dE could affect the equivalence between the reconstructed space and the original one, another approach, based on the delay time window, = tw (dE −1)T selection, is used for the state space reconstruction [7] [8] [16] [17]. In the case of long free noise time series, as Takens embedding theorem ensures a topological equivalence between the original state space and the reconstructed one, the quality of the reconstruction is measured through the conservation of invariants such as the maximum Lyapunov exponent and the correlation dimension. Time Delay Reconstruction of the State Space by Sampling a Coordinate of the Rössler Attractor
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