A Comparison of Correlation and Lyapunov Dimensions

Konstantinos E. Chlouverakis and J. C. Sprott
Department of Electronic Systems Engineering, University of Essex, Wivenhoe Park, Colchester CO4 3SQ, UK
Department of Physics, University of Wisconsin, Madison, WI 53706, USA

Received 14 February 2004; received in revised form 6 September 2004; accepted 22 October 2004


This paper investigates the relation between the correlation (D2) and the Kaplan-Yorke dimension (DKY) of three-dimensional chaotic flows. Besides the Kaplan-Yorke dimension, a new Lyapunov dimension (DS), derived using a polynomial interpolation instead of a linear one, is compared with DKY and D2. Various systems from the literature are used in this analysis together with some special cases that span a range of dimension 2 < DKY < 3. A linear regression to the data produces a new fitted Lyapunov dimension of the form Dfit = αβλ1/λ3, where λ1 and λ3 are the largest and smallest Lyapunov exponents respectively. This form correlates better with the correlation dimension D2 than do either DKY or DS. Additional forms of the fitted dimension are investigated to improve the fit to D2, and the results are discussed and interpreted with respect to the Kaplan-Yorke conjecture.

Ref: K. E. Chlouverakis and J. C. Sprott, Physica D 200, 156-164 (2004)

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