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Introduction To Nonlinear Dynamics

Phase Space Reconstruction

Phase Space Reconstruction

In order to obtain the quantities and plots in the following sections from a (measured) scalar time series which is assumed to be generated by a (nonlinear) dynamical system with n degrees of freedom, one has to construct the appropriate series of state vectors of delay coordinates in the n-dimensional phase space according to the Takens theorem [8] as

where , i=1,...,N, and is the sampling time. n is called the embedding dimension.

For an infinite noise free data set the value of the delay time is in principle almost arbitrary. However, for a finite amount of data the choice of affects the quality of the reconstructed trajectory in phase space and thereby the values of the characteristic quantities under consideration [9]. Usually the correlation length, i.e. the first zero of the (vanishing) autocorrelation function, is a good value for to start with. Another possibility is to estimate by means of the peak-to-peak intervals (Peak-to-peak intervals).