Kaplan–Yorke conjecture

In applied mathematics, the Kaplan–Yorke conjecture concerns the dimension of an attractor, using Lyapunov exponents.^[1][2] By arranging the Lyapunov exponents in order from largest to smallest ${\displaystyle \lambda _{1}\geq \lambda _{2}\geq \dots \geq \lambda _{n}}$, let j be the largest index for which

${\displaystyle \sum _{i=1}^{j}\lambda _{i}\geqslant 0}$

and

${\displaystyle \sum _{i=1}^{j+1}\lambda _{i}<0.}$

Then the conjecture is that the dimension of the attractor is

${\displaystyle D=j+{\frac {\sum _{i=1}^{j}\lambda _{i}}{|\lambda _{j+1}|}}.}$

This idea is used for the definition of the Lyapunov dimension.^[3]

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.^[4][3]

• The Hénon map with parameters a = 1.4 and b = 0.3 has the ordered Lyapunov exponents ${\displaystyle \lambda _{1}=0.603}$ and ${\displaystyle \lambda _{2}=-2.34}$. In this case, we find j = 1 and the dimension formula reduces to

${\displaystyle D=j+{\frac {\lambda _{1}}{|\lambda _{2}|}}=1+{\frac {0.603}{|{-2.34}|}}=1.26.}$

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

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