Is it possible to quantify how chaotic a system is?

Question

In relation to this other question that I asked:

Is there anything more chaotic than fluid turbulence?

I had assumed that there are methods by which the level of 'chaotic-ness' of a system could be measured, for comparison with other non-linear systems. However, several comments called that into question, proposing that it's either/or: 'either a system is chaotic, or it's not'.

So, I am wondering if that is true? Or, are there parameters that can be used to determine and compare how chaotic one system is, compared to another?

One comment to the other question mentioned the Lyapunov exponent. I admit that I'm not very experienced in non-linear dynamical systems, but I was also thinking about other possible parameters, such as properties of the chaotic attractor; number or range of different distance scales that develop; or perhaps the speed or frequency of when bifurcations occur.

So, in general, is it possible to quantify the 'chaotic-ness' of a dynamical system? If so, what parameters are available?

Answer 1

There are a number of ways of quantifying chaos. For instance:

• Lyapunov exponents - Sandberg's answer covers the intensity of chaos in a chaotic system as measured by its Lyapunov exponents, which is certainly the main way of quantifying chaos. Summary: larger positive exponents and larger numbers of positive exponents correspond to stronger chaos.

• Relative size of the chaotic regions - An additional consideration is needed for systems which are not fully chaotic: these have regular regions mixed with chaotic ones in their phase space, and another relevant measure of chaoticity becomes the relative size of the chaotic regions. Such situation is very common and a standard example are Hamiltonian systems.

• Finite-time Lyapunov exponents - Still another situation is that of transient chaos (see e.g., Tamás Tél's paper, (e-print)), where the largest Lyapunov exponent might be negative, but the finite-time exponent, positive. One could say transient chaos is weaker than asymptotic chaos, though such comparisons won't always be straightforward or even meaningful.

• Hierarchy of ergodicity - Also worth mentioning is the concept of hierarchy of chaos. More than measuring the strength of chaos, it concerns itself with the nature of it. Detailed explained in its Stanford Encyclopedia of Philosophy's entry, I briefly summarized it in this answer:

  Bernoulli systems are the most chaotic, equivalent to shift maps. Kolmogorov systems (often simply K-systems) have positive Lyapunov exponents and correspond to what is most often considered a chaotic system. (Strongly) mixing systems intuitively have the behavior implied by their name and, while they don't necessarily have exponentially divergent trajectories, there is a degree of unpredictability which can justify calling them weakly chaotic. Ergodic systems, on the other hand, have time correlations that don't necessarily decay at all, so are clearly not chaotic.

Interesting, if tangentially related, is a bound on chaos conjectured to apply for a broad class of quantum systems, but I'm restricting this answer to classical systems and that bound on chaos diverges in the classical limit.

Answer 2

Lyapunov exponents are the standard method. If the dynamics is $\mathbf{x}'(t)=\mathbf{f}(\mathbf{x})$ and we follow a particular trajectory $\mathbf{x}_0(t)$ starting at $\mathbf{x}_0$, then a small ball of other starting points $\mathbf{x}_0+\mathbf{\epsilon}$ gives rise to a deformed ball (an ellipsoid) $\mathbf{x}_\epsilon(t)$ at later times. The axes of the ellipsoid grow as $\propto e^{\lambda_i t}$, where the $\lambda$s are the Lyapunov exponents. Normally they are averaged across trajectories as $t\rightarrow \infty$.

First, the sum of the exponents give a measure of how strongly trajectories are attracted to an attractor state (it indicates how much phase space volume decreases). If the sum is positive then trajectories will diverge to infinity, but this can be in a totally ordered way such as in exponential growth. If the sum is zero the system just mixes trajectories (this is where Hamiltonian systems show up). The usual case has a negative sum and converges to a finite attractor.

Second, the size of the positive exponents indicates how sensitive the system is to initial conditions. Big values indicate fast information loss and hence "more chaos".

Third, their signs indicate how many directions are chaotic in the attractor. In 3D the only possible chaotic case is $(-,0,+)$: convergence in one direction to the attractor, points along trajectories will keep their distance, and there is one direction pulling trajectories apart. In 4D one can have $(-,-,0,+)$ (vanilla attractor) or $(-,0,+,+)$ (hyperchaos) where trajectories are pulled apart in two directions. Higher dimensions allow further hyperchaos.

The distance scale spectrum is as far as I know rarely used to measure how chaotic a system is, but it does give useful information. Density of bifurcations is also not really used, since the question is usually asked about a given parameter setting rather than for the full parameter space. Still, being close to bifurcations typically produces intermittency dynamics that make the system look a bit less chaotic.