Lyapunov Exponent and degree of chaos I am aware that having positive Lyapunov exponents in a system signifies that a system is chaotic. However, I would like to know if there is a means to know the degree of chaos in the system from the Lyapunov exponents. For example, does it signify anything if a system has 10 positive Lyapunov exponents out of 25, or all positive Lyapunov exponents. Thanks.
 A: Consider the map $T: \mathbb R^4 \to \mathbb R^4$ given by
$$
 T x = \begin{pmatrix} 1 & 1 & 0 & 0 \\\ 1 & 0  & 0 & 0 \\\
  0 & 0 & \cos(\theta) & \sin(\theta) \\\
  0 & 0 & -\sin(\theta) & \cos(\theta) \end{pmatrix}
$$
Then two Lyapunov exponents are $\neq 0$, and two are zero. Furthermore, one sees that the dynamic splits into a chaotic part given by $\begin{pmatrix} 1 & 1 \\\ 1 & 0 \end{pmatrix}$, the cat map, and by a completely regular one given by a rotation by $\theta$.
Hence, you want all Lyapunov exponent non-zero to get fully chaotic dynamics.
Of course, one might interpret all these things differently depending on what one means by chaos.
A: The (Kolmogorov--Sinai metric) entropy is a measure of chaos in a dynamical system w.r.t. an invariant measure. For a broad class of dynamical systems it is equal to the sum of all positive Lyapunov exponents. I would recommend reading  http://www.scholarpedia.org/article/Pesin_entropy_formula
Quoting this article, 
The content of this formula is that the entropy of a measure is given exactly by the total expansion in the system.
