Kaplan–Yorke conjecture - Alchetron, the free social encyclopedia

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 , let j be the index for which

∑ i = 1 j λ i > 0

and

∑ i = 1 j + 1 λ i < 0.

Then the conjecture is that the dimension of the attractor is

D = j + ∑ i = 1 j λ i | λ 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.

The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents λ 1 = 0.603 and λ 2 = − 2.34 . In this case, we find j = 1 and the dimension formula reduces to

The Lorenz system shows chaotic behavior at the parameter values σ = 16 , ρ = 45.92 and β = 4.0 . The resulting Lyapunov exponents are {2.16, 0.00, -32.4}. Noting that j = 2, we find

References

Kaplan–Yorke conjecture Wikipedia

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