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

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