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 , let j be the largest index for which
and
Then the conjecture is that the dimension of the attractor is
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 and . In this case, we find j = 1 and the dimension formula reduces to
The Lorenz system shows chaotic behavior at the parameter values , and . The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find
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