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.



Lyapunov Exponent and degree of chaos
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.


