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.

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 nonzero to get fully chaotic dynamics. Of course, one might interpret all these things differently depending on what one means by chaos. 


The (KolmogorovSinai 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. 

