A comparison of correlation and Lyapunov dimensions

This paper investigates the relation between the correlation (D-2) and the Kaplan-Yorke dimension (D-KY) of three-dimensional chaotic flows. Besides the Kaplan-Yorke dimension, a new Lyapunov dimension (D-Sigma), derived using a polynomial interpolation instead of a linear one, is compared with D-KY and D-2. Various systems from the literature are used in this analysis together with some special cases that span a range of dimension 2 < D-KY less than or equal to 3. A linear regression to the data produces a new fitted Lyapunov dimension of the form D-fit = alpha - betagimel(1)/gimel(3), where gimel(1) and gimel(3) are the largest and smallest Lyapunov exponents, respectively. This form correlates better with the correlation dimension D-2 than do either D-KY or D-Sigma. Additional forms of the fitted dimension are investigated to improve the fit to D-2, and the results are discussed and interpreted with respect to the Kaplan-Yorke conjecture. (C) 2004 Elsevier B.V. All rights reserved.