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 ${\lambda }_1\ge {\lambda }_2\ge \dots \ge {\lambda }_n$, let j be the largest index for which

$\sum _{i=1}^j{\lambda }_i⩾0$

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

$D=j+\frac{{\lambda }_1}{|{\lambda }_2|}=1+\frac{0.603}{|-2.34|}=\mathrm{1.26.}$

• 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

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