Kaplan–Yorke conjecture

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 [math]\displaystyle{ \lambda_1\geq\lambda_2\geq\dots\geq\lambda_n }[/math], let j be the largest index for which

[math]\displaystyle{ \sum_{i=1}^j \lambda_i \geqslant 0 }[/math]

and

[math]\displaystyle{ \sum_{i=1}^{j+1} \lambda_i \lt 0. }[/math]

Then the conjecture is that the dimension of the attractor is

[math]\displaystyle{ D=j+\frac{\sum_{i=1}^j\lambda_i}{|\lambda_{j+1}|}. }[/math]

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 [math]\displaystyle{ \lambda_1=0.603 }[/math] and [math]\displaystyle{ \lambda_2=-2.34 }[/math]. In this case, we find j = 1 and the dimension formula reduces to

[math]\displaystyle{ D=j+\frac{\lambda_1}{|\lambda_2|}=1+\frac{0.603}{|{-2.34}|}=1.26. }[/math]

• The Lorenz system shows chaotic behavior at the parameter values [math]\displaystyle{ \sigma=16 }[/math], [math]\displaystyle{ \rho=45.92 }[/math] and [math]\displaystyle{ \beta=4.0 }[/math]. The resulting Lyapunov exponents are {2.16, 0.00, −32.4}. Noting that j = 2, we find

[math]\displaystyle{ D=2+\frac{2.16 + 0.00}{|-32.4|}=2.07. }[/math]