Does a greater Lyapunov exponent result in a more chaotic system?

Question

Let $\sigma_1$ be the Lyapunov exponent of a one-dimensional system $I_1$ and $\sigma_2$ be Lyapunov exponent of one-dimensional system $I_2$. Can I say that if $\sigma_1 > \sigma_2$, then $I_1$ is more chaotic than $I_2$? Is it possible to compare a one-dimensional system with a two-dimensional system?

Answer 1

What exactly do you mean by more chaotic?

If your answer is more sensitive to initial conditions, then the Lyapunov exponent quantifies exactly this (in a way that does not depend much on the system’s dimension). However, be aware that the value of the Lyapunov exponent also depends on the time scale of the system (further reading) and it usually makes sense to see it in relation to the typical length of oscillations of the system.

That being said, there are other characteristics of a dynamics that could be argued to quantify chaoticity, such as some fractal dimension (which via the Kaplan–Yorke conjecture is at least typically linked to the number of positive Lyapunov exponents in the system. For example, you might consider hyperchaos (at least two positive Lyapunov exponents) more chaotic than classic chaos (one positive Lyapunov exponent).

At the end of the day, the above characterisations of a system are much more specific and useful than talking about something being more or less chaotic. Hence chaos theorists usually avoid this notion.