Testing for chaos in data 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
 A: 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.
A: 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)
A: 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.
A: I do not know if this information is of any help at this point, still, I thought I would mention some quick ideas based on personal experience. Determining whether a time series is chaotic or not is quite tricky. 
A quick test for a deterministic time series (in the absence of noise) would be to check for the power spectrum of the signal to see if it exhibits a broadband structure or not. If it does, then your time series is chaotic. But as we all know, time series from any natural/engineered system is definitely noisy. 
I am assuming that you have picked an appropriate embedding dimension (checking for false nearest neighbors) and time-delay (the first minimum of mutual information) while reconstructing the attractor. 
Even if you have not, you should definitely have a look at the paper by A. Wolf, J. B. Swift, H. L. Swinney, and J. A. Vastano, Physica 16D (1985) 285-317 where they discuss an algorithm for calculation of Lyapunov exponents (LEs: give the rate of exponential convergence/divergence of nearby trajectories in different directions) from a time series and related details. 
Although they also talk about calculation of the higher order exponents but you do not need them for your purpose, you only needed the largest. 
If the largest LE > 0, then your time series is chaotic. You can find several papers in this direction on google scholar too. Good luck.
