Practical Remarks on the Estimation of Dimension and Entropy from Experimental Data

Abstract

For non-linear dissipative dynamical system, invariant sets have volume zero in phase-space, and trajectories possess the property of sensitivity to initial conditions. The geometry and dynamics can be characterised by quantities, dimensions and entropies which will be invariant under a large class of coordinate changes. To calculate these quantities from experimental data one needs to imbed the data in R ⁿ. These projections generically are diffeomorphisms for n sufficiently large [1,2]. In practice, they are done using time-delays see [3] for example or building vectors from measurements in different locations. However the geometrical structure of these projections sets will determine the values obtained in practice for the dimensions and entropies. For simple model systems these sets can be studied in a semi-quantitative way using the Grassberger-Procaccia correlation integral [4]. This yields some criterions on the choice of the distance in R ⁿ, the different parameters: the number of vectors N and averages m, the sampling period, the embedding dimension and the delay. Deterministic chaotic data are projected onto manifolds which are locally the product of cantor sets and smooth manifolds, it will be shown that the curvature of these sets caused by the divergence of trajectories becomes predominant for the determination.