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Kaplan-Yorke Dimension

The Kaplan-Yorke dimension is a dimension associated with ordered Lyapunov characteristic exponents lambda_1>=...>=lambda_n. Put S_0=0 and S_k=lambda_1+...+lambda_k, and let j=max{k in {0,1,...,n}:S_k>=0}. Then

D_(KY)={0 if j=0; j+S_j/|lambda_(j+1)| if 0<j<n; n if j=n.
(1)

The convention S_k>=0 includes neutral directions when a partial sum vanishes (Kaplan and Yorke 1979).

If nu=sigma=D, where nu is the correlation exponent, sigma the information dimension, and D the Hausdorff dimension, then

D<=D_(KY)
(2)

(Grassberger and Procaccia 1983).

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References

Grassberger, P. and Procaccia, I. "Measuring the Strangeness of Strange Attractors." Physica D 9, 189-208, 1983.Kaplan, J. L. and Yorke, J. A. "Chaotic Behavior of Multidimensional Difference Equations." In Functional Differential Equations and Approximation of Fixed Points (Ed. H.-O. Peitgen and H.-O. Walther). Berlin, Germany: Springer, pp. 204-227, 1979. https://doi.org/10.1007/BFb0064319.

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Kaplan-Yorke Dimension

Cite this as:

Weisstein, Eric W. "Kaplan-Yorke Dimension." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Kaplan-YorkeDimension.html

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