Kaplan–Yorke conjecture

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 ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}}$, let j be the largest index for which

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

and

${\displaystyle \sum _{i=1}^{j+1}\lambda _{i}<0.}$

Then the conjecture is that the dimension of the attractor is

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

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 ${\displaystyle \lambda _{1}=0.603}$  and ${\displaystyle \lambda _{2}=-2.34}$ . 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 ${\displaystyle \sigma =16}$ , ${\displaystyle \rho =45.92}$  and ${\displaystyle \beta =4.0}$ . 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

1. 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. Vol. 730. Berlin: Springer. pp. 204–227. ISBN 978-0-387-09518-9. MR 0547989.
2. 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.
3. Kuznetsov, Nikolay; Reitmann, Volker (2020). Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Cham: Springer.
4. 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. S2CID 14411384.