Testing for chaos in data(method by Doyne Farmer for 3D discrete data)

[marellasunny]

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 discrete 3D system?(I don't have the analytic eqs.,just the data.)

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

1.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 using a Fourier transform can show me 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.

2.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?

3.Since the data are discrete, does construction of a Poincaré map help in checking for chaos?Doyne Farmer used the same technique on a 1D system(see below).Can I use for 3D systems also?

P.S:
I just read in my textbook 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.

 

[fresh_42]

You could perform some statistical tests to see if there is an underlying distribution or if all tests fail. If you have three trajectories, you could consider them one by one. It is in general difficult to find or reject a pattern on finitely many data, since they always can be explained by a polynomial. Therefore I guess that additional assumptions will have to be made.