## 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).