Dimensions and entropies

  Solutions of dissipative dynamical systems cannot fill a volume of the phase space, since dissipation is synonymous with a contraction of volume elements under the action of the equations of motion. Instead, trajectories are confined to lower dimensional subsets which have measure zero in the phase space. These subsets can be extremely complicated, and frequently they possess a fractal structure, which means that they are in a nontrivial way self-similar. Generalized dimensions are one class of quantities to characterize this fractality. The Hausdorff dimension is, from the mathematical point of view, the most natural concept to characterize fractal sets [67], whereas the information dimension takes into account the relative visitation frequencies and is therefore more attractive for physical systems. Finally, for the characterization of measured data, other similar concepts, like the correlation dimension, are more useful. One general remark is highly relevant in order to understand the limitations of any numerical approach: dimensions characterize a set or an invariant measure whose support is the set, whereas any data set contains only a finite number of points representing the set or the measure. By definition, the dimension of a finite set of points is zero. When we determine the dimension of an attractor numerically, we extrapolate from finite length scales, where the statistics we apply is insensitive to the finiteness of the number of data, to the infinitesimal scales, where the concept of dimensions is defined. This extrapolation can fail for many reasons which will be partly discussed below. Dimensions are invariant under smooth transformations and thus again computable in time delay embedding spaces.

Entropies are an information theoretical concept to characterize the amount of information needed to predict the next measurement with a certain precision. The most popular one is the Kolmogorov-Sinai entropy. We will discuss here only the correlation entropy, which can be computed in a much more robust way. The occurrence of entropies in a section on dimensions has to do with the fact that they can be determined both by the same statistical tool.

 
• Correlation dimension
  • Takens-Theiler estimator
  • Gaussian kernel correlation integral
• Information dimension
  • Entropy estimates