Testing for chaos in data

Question

I have data for 3 variables ,each with respect to the discrete time values. How do I check for the existence of chaos for this 3D discrete system?(I don't have the analytic eqs.,just the data).

MY IDEAS ON CHECKING FOR CHAOS FROM DATA:(which of these are feasible an an algorithm?)

1.I have done a phase space reconstruction and the 3D plot doesn't look anything like a chaotic trajectory. It doesn't look like a attractor either. Can I check for chaos with the phase space reconstruction of these discrete data?

2.Since the data are discrete, does construction of a map($x_{t+1} vs x_{t}$, $y_{t+1}$ vs....so on) help in checking for chaos?Doyne Farmer used the same technique on a 1D system(see below).Can I use this for 3D systems also?

3.In Li and Yorke's paper, they describe "chaos" as the existence of orbits of all periods simultaneously(although they don't mention about the stability), I thought in this context that by using a Fourier transform,one can show visually the existence of periodic orbits and hence chaos.i.e a chaotic system would have a frequency(of oscillation) distributed over the entire range.

P.S: I just read in my text book that when the physicist Doyne Farmer gathered ultrasonic sound data from drops of water hitting the floor and used the time difference between 2 sound peaks as the variable x(i.e he plotted a 2D graph between $x_{t+1}$ and $x_t$), he observed a single hump in the $x_{t+1}$ vs $x_t$ graph,hence indicating the existence of a period∞ orbit a.k.a chaos.

[X(t+2)vsX(t+1)vsX(t)]plot

Answer 1

If you know that your data comes from a deterministic system, then finding broadband noise as opposed to a line spectrum would be indicative of chaos. However, there may be measurement noise (always assume there is), which would make this approach difficult.

The most popular method is the delay reconstruction of the phase space. This method assumes you have a time series of an observable $x_n$ that is usually scalar. Construct an $m$-dimensional vector using a delay time $\tau$: $$X_n = (x_{n-(m-1)\tau} \ldots, x_{n-\tau}, x_n)$$

Then find all pairs of points in this space that are very close together, and look at how much they diverge over one or a few time steps. The divergence rate will provide an estimate of the greatest Lyapunov exponent.

There are several difficulties with this method, such as estimating optimal values of $m$ and $\tau$. I'd recommend that you read Nonlinear Time Series Analysis by Kantz and Schreiber for getting an idea of how error prone this method can be if not done with care.

As an alternative, you could take a look at the 0-1 test for chaos.

Answer 2

What I would suggest to you would be two steps:

• test all information you have about your system beyond the data, whether chaos can be an option at all. You have 3 temporal modes but see whether you have reason to believe there is Chaos or not. For instance if you would have convincing argument via a model that the system is continuous and not discrete and not all 3 modes interdependent via a non-linearity rule.

• if you then are convinced that Chaos might be a possible case under certain parametric conditions, you need to Design a very diligent Test that uses the so called Lyapunov Exponent. For this, suggest you read diligently through literature accessible broadly and particularly the more simpller examples as done with for instance brain EEG data. See further for instance here. The Lyapunov Exponent can serve as a good measure to Chaos, if the test is properly and with diligence deisgned.

What you are asking for requires a strong theoretical understanding and I can just give you herewith the direction to obtain a strong solution.

Hope this answer helps.

Answer 3

Your graph is not clear enough. Try to plot again this graph without linking datas with lines. Then you could have a better view of the attractor looking on the cloud of points. If this cloud seems like hyperchaos (check this term, googling) try to compute numerically its fractal dimension.

René Lozi (Nice university)