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

\sum _{{i=1}}^{j}\lambda _{i}>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}}|}}.

Examples

Especially for chaotic systems, the Kaplan–Yorke conjecture is a useful tool in order to determine the fractal dimension of the corresponding attractor.^[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

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

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

References

↑ J. Kaplan and J. Yorke, "Chaotic behavior of multidimensional difference equations," in: Functional Differential Equations and the Approximation of Fixed Points, Lecture Notes in Mathematics, vol. 730, H.O. Peitgen and H.O. Walther, eds. (Springer, Berlin), p. 228.
↑ P. Frederickson, J. Kaplan, E. Yorke and J. Yorke, "The Lyapunov Dimension of Strange Attractors," J. Diff. Eqs. 49 (1983) 185.
↑ A. Wolf, A. Swift, B. Jack, H. L. Swinney and J.A. Vastano "Determining Lyapunov Exponents from a Time Series," Physica 16D, 1985, 16, pp. 285–317.

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