and it follows already from ``topological equivalence'' (for which we even only need to have a homeomorphism instead of a diffeomorphism) that [7]:

- having a singularity of in is equivalent to having a singularity of in ;
- closed orbits of correspond one-to-one to closed orbits of ;
- the forward and backward attractors of with respect to are the -pictures of the respective attractors of under action of ;
- in more general terms, the flow of the dynamical system
(1) and the flow of (14) are topologically
equivalent:

The dynamical systems of eqs. (1) and (14) are said to have the same qualitative dynamics. This situation is represented pictorially in Fig. 3.

(15) |

(16) |

(

(

Takens argues that these conditions are generically met by ; i.e. practically all functions meet these conditions

(At this stage, we are still restricted to , but below we will show that, in fact, nearly every can be chosen.) The difference between Packard's conjecture und Takens' approach is that Takens requires the embedding space to have a higher dimension than one would ad hoc expect (). This requirement

So both methods suggested by Packard rather intuitively are hereby justified, although slightly altered. With all these theorems one knows how to construct meaningful phase space vectors from one-dimensional data which has been measured with the sampling time . This result is interesting but not exhaustive, because one would like to have the possibility of adjusting to each individual situation rather than having to fix it at some given value; also there is obviously no physical reason for giving this very special role to the unit time interval. So we would like to generalize the above result to arbitrary time steps . Additionally, the important question if one can really reconstruct the

- ...
space
^{3} - For details about Whitney's theorem see e.g. [1].
- ...
^{4} - Refer to [2], for example, for a thorough treatment of embeddings.
- ... conditions
^{5} -
One can interpret the term ``generically'' in this case as follows:
Consider the function space of all -functions
which map from into ;
then every subset of consisting only of functions which
do not meet condition (
*i*) and (*ii*) has zero measure in . - ... well
^{6} - As hinted by Broomhead and King [7] it is not perfectly clear on generic grounds that one can exclude solutions with integer periods : In general one cannot argue that a perturbation of the generating function will automatically change the period of the flow. Broomhead and King circumvent this problem by not considering to be fixed; instead they make it small enough so that Takens' period-condition is met. See section 3.4.3.
- ... exists
^{7} - For attractors with a simple geometric structure a smaller embedding dimension may be sufficient, but the more complicated the structure of is (e.g. if the attractor is Cantor set-like or if there are many ``backfoldings'') the higher the embedding dimension must be [5]. The importance of Takens' result is that, no matter how complex the structure of , dimensions always suffice.
- ...
space
^{8} - There is one peculiarity for the method using time derivatives: Here, and must be at least -functions, and this stricter requirement may become a problem if the system or the observable are not that ``well-behaved''.