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 $$\lambda_1\geq\lambda_2\geq\dots\geq\lambda_n$$, let j be the largest index for which


 * $$ \sum_{i=1}^j \lambda_i \geqslant 0 $$

and


 * $$ \sum_{i=1}^{j+1} \lambda_i < 0. $$

Then the conjecture is that the dimension of the attractor is


 * $$ D=j+\frac{\sum_{i=1}^j\lambda_i}{|\lambda_{j+1}|}. $$

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 $$\lambda_1=0.603$$ 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


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