but these are not available, since the only data one has is the one-dimensional time series. Based on the conjecture that

(9) |

(10) |

In this way the one-dimensional time series is used to build a sequence of -dimensional vectors which form a trajectory in some -dimensional space. Numerical experiments done by Packard et al. show that this procedure gives reasonable results: the phase-space pictures one gets seem closely related to the corresponding pictures of the original dynamical system. In particular, topological properties and the geometrical form of the attractor seem to be conserved under this procedure. As an illustration of this method of phase space reconstruction, we present a result of Broomhead and King [7]: They considered the

(with the parameter values , , ). First they integrated this set of ordinary differential equations to get the ``complete solution''; the projection of this solution onto the -plane is shown in Fig. 1.a. They chose the -coordinate to be their ``observable'' and thus got the one-dimensional data stream as shown in Fig. 1.b. Then, after choosing some delay time , they obtained a discrete time series from the data stream and used this time series to construct a phase portrait, using the method of delays: See Fig. 1.c. Comparison of the original Lorenz attractor (Fig. 1.a) and the reconstructed one justifies Packard's approach: Although the pictures are not exactly the same and although Fig. 1.a is much coarser than Fig. 1.c, both of them seem to show an object with the same geometry. A shortcoming of Packard's method is that it does not give a general strategy how to choose . If one knows that the dynamical system is evolving in -dimensional phase space then one can use the idea mentioned above and take . But if one is analyzing a system about the dimensionality of which nothing is known a priori then how to proceed? This problem will be addressed in section 3. It is also not quite clear what should be when we try to reconstruct phase portrait of a system with infinitely many degrees of freedom, such as the Mackey-Glass equation ([5,12] and [14], chapter 5.3):

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- ... observable
^{2} -
One can compute an approximation to the derivatives from the time
series for example by using the formula

where the are determined by

This formula results from differentiating the interpolating polynomial. See [10], §§ 27 and 28, for details of this method of numerical differentiation. See [11], chapter 3.4, for a more sophisticated extrapolation method.