# Why would we want to calculate the Lyapunov exponent for experimental data?

Searching Google Scholar for "Lyapunov exponent from time series" turns up multiple papers (some of them highly cited) suggesting methods for estimating the largest Lyapunov exponent or sometimes even the whole Lyapunov spectrum from experimental time series.

None of these methods seem to be very precise, (as can be verified on synthetic data, which I have), and even if viewed as a binary classifier (i.e. used to determine if a system is chaotic/regular based on if the maximal Lyapunov exponent is positive/negative) yield both false negative and false positive results.

Do any of these methods have any practical (rather than theoretical) use?

In control theory, Lyapunov exponents correspond to the eigenvalues of the linearized nonlinear dynamical system which provide local information about the stability margin. This information can be used to design stabilizing controllers as in this paper.

As you already noted, a positive Lyapunov exponent (if properly ensured with surrogates, etc.) can tell you whether a system is chaotic. However, the differentiation to make is not only between chaos and regular dynamics¹, but between chaos and a stochastic dynamics². Somewhat simplified you want to distinguish between a finite positive and an infinite Lyapunov exponent.

Now, what is the practical value of this?

• If we know the system is chaotic and not stochastic, we may manipulate it to some extent, e.g., using chaos control.

• A chaotic system can be predicted to some extent, which in turn may allow for targeted interventions. The Lyapunov exponent tells us how far we can predict the system with a given knowledge about it. For example, the Lyapunov exponent of the weather gives us a natural limit of weather forecasts.

• It guides us on how to approach modelling a system, more precisely on whether to use a stochastic or deterministic model. This is admittedly not very practical, but the models in turn may help us understand a system and have a lot of practical implications.

• If we know that we are experimenting with a chaotic system, we can have some expectations on how replicates of the same setup will behave.

¹ If you really have a regular dynamics, it’s usually pretty obvious.
² More precisely: A dynamics dominated by inaccessible stochastic components. Most real systems have stochastic and deterministic components, where stochastic merely means that it is beyond the scope of our models and measurements, e.g., molecular billiard when investigating a chemical reaction in a tube.

• +1 Good point that the main differentiation here is rather chaotic/random than chaotic/regular. Commented Mar 3, 2020 at 9:25
• (1/2) Thank you for some good points. But surely, merely calculating a [maximal] Lyapunov exponent will not suffice to differentiate between determinism and stochasticity (And you are right of course, "stochastic" here only means that there are a larger number of degrees of freedom than we'd like to model). I'm pretty sure that all methods for calculating the LE will just return some random positive number when fed with a random time series, say a Gaussian random walk (I didn't test this though, so I may be wrong). Commented Mar 3, 2020 at 10:02
• (2/2 )It seems that when faced with an experimental time-series which is suspected to originate from a process which can be modeled by a low-dimensional dynamical system, you'd usually use something like the Grassberger-Procaccia algorithm for the first problem. Then perhaps if the result is some small number (a “deterministic” process), try to estimate the Lyapunov exponent. Is this correct? Commented Mar 3, 2020 at 10:03
• Come to think of it, wouldn't a symmetric random walk have LE of 0? Commented Mar 3, 2020 at 10:09
• @Student: But surely, merely calculating a [maximal] Lyapunov exponent will not suffice to differentiate between determinism and stochasticity – If you properly calculate a Lyapunov exponent with surrogates, the result you get is something like: “The Lyapunov exponent is higher than what can be expected for a linear stochastic dynamics (ARMA) with the same properties.” Commented Mar 3, 2020 at 11:25