In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents. By arranging the Lyapunov exponents in order from largest to smallest λ 1 ≥ λ 2 ≥ ⋯ ≥ λ n {\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}} , let j be the largest index for which
∑ i = 1 j λ i ⩾ 0 {\displaystyle \sum _{i=1}^{j}\lambda _{i}\geqslant 0}
and
∑ i = 1 j + 1 λ i < 0. {\displaystyle \sum _{i=1}^{j+1}\lambda _{i}<0.}
Then the conjecture is that the dimension of the attractor is
D = j + ∑ i = 1 j λ i | λ j + 1 | . {\displaystyle D=j+{\frac {\sum _{i=1}^{j}\lambda _{i}}{|\lambda _{j+1}|}}.}
This idea is used for the definition of the Lyapunov dimension.
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.
The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents λ 1 = 0.603 {\displaystyle \lambda _{1}=0.603} and λ 2 = − 2.34 {\displaystyle \lambda _{2}=-2.34} . In this case, we find j = 1 and the dimension formula reduces to
D = j + λ 1 | λ 2 | = 1 + 0.603 | − 2.34 | = 1.26. {\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 σ = 16 {\displaystyle \sigma =16} , ρ = 45.92 {\displaystyle \rho =45.92} and β = 4.0 {\displaystyle \beta =4.0} . The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find
D = 2 + 2.16 + 0.00 | − 32.4 | = 2.07. {\displaystyle D=2+{\frac {2.16+0.00}{|-32.4|}}=2.07.}
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