Does Lyapunov exponent $\lim_{t\rightarrow\infty}\lim_{|\delta Z_0|\rightarrow 0}\frac{1}{t}\ln\frac{|\delta Z(t)|}{|\delta Z_0|}$ depends on initial state $Z_0$?
I'm asking this question since I think the exponent is trying to say that if we slightly perturb the initial state, how the deviation between perturbed solution and unperturbed solution grows over time. I think the different initial state we perturbed, then the different value for this exponent will be obtained.
p.s. I found in other document, it says the definition above is actually maximal lypunov exponent, I am wonder why it is? does maximal comes from $\lim_{t\rightarrow\infty}$?