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}=1+\frac{0.603}=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}=2.07. ==References==