A comparison of correlation and Lyapunov dimensions

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 (DΣ), 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 = α − βλ1/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 DΣ. 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.