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In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents.[1][2] By arranging the Lyapunov exponents in order from largest to smallest \( \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n} \), let j be the index for which

\( {\displaystyle \sum _{i=1}^{j}\lambda _{i}\geqslant 0} \)

and

\( \sum _{{i=1}}^{{j+1}}\lambda _{i}<0. \)

Then the conjecture is that the dimension of the attractor is

\( D=j+{\frac {\sum _{{i=1}}^{j}\lambda _{i}}{|\lambda _{{j+1}}|}}. \)

This idea is used for the definition of the Lyapunov dimension[3].
Examples

Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to estimate the fractal dimension and the Hausdorff dimension of the corresponding attractor.[4][3]

The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents \( \lambda _{1}=0.603 \) and \( \lambda _{2}=-2.34 \). In this case, we find j = 1 and the dimension formula reduces to

\( {\displaystyle D=j+{\frac {\lambda _{1}}{|\lambda _{2}|}}=1+{\frac {0.603}{|{-2.34}|}}=1.26.} \)

The Lorenz system shows chaotic behavior at the parameter values \( \sigma =16 \), \( \rho =45.92 \) and\( \beta =4.0 \). The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find

\( {\displaystyle D=2+{\frac {2.16+0.00}{|-32.4|}}=2.07.} \)

References

Kaplan, J.; Yorke, J. (1979). "Chaotic behavior of multidimensional difference equations" (PDF). In Peitgen, H. O.; Walther, H. O. (eds.). Functional Differential Equations and the Approximation of Fixed Points. Lecture Notes in Mathematics. 730. Berlin: Springer. p. 204–227. ISBN 978-0-387-09518-9.
Frederickson, P.; Kaplan, J.; Yorke, E.; Yorke, J. (1983). "The Lyapunov Dimension of Strange Attractors". J. Diff. Eqs. 49 (2): 185–207. Bibcode:1983JDE....49..185F. doi:10.1016/0022-0396(83)90011-6.
Kuznetsov, Nikolay; Reitmann, Volker (2020). Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Cham: Springer.
Wolf, A.; Swift, A.; Jack, B.; Swinney, H. L.; Vastano, J. A. (1985). "Determining Lyapunov Exponents from a Time Series". Physica D. 16 (3): 285–317. Bibcode:1985PhyD...16..285W. CiteSeerX 10.1.1.152.3162. doi:10.1016/0167-2789(85)90011-9.

Mathematics Encyclopedia

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