Consider the time-series of one of the states (say x) in the Lorenz system. Also consider the time-series of one of the states (again say x) in the Rossler system.
If we only have the signal of x_Lorenz+x_Rossler (i.e. the mixture of those two signals), can we extract the original signals in the following conditions?
a) We know that the signal we have is the mixture of “x” state of Lorenz and “x” state of Rossler
b) We know that the signal we have is the mixture of one state of Lorenz and one state of Rossler
c) We know that the signal we have is the mixture two signals from two different chaotic flows
Dear Sajad, I think that it is difficult if not impossible. When we have a pure chaotic signal without noise of Lorenz type, we can probably distinguish it from pure random noise. But when it is mixed with noise, separation already becomes difficult. Only when we know the spectrum of this noise, we can try to subtract it.
Another question: what do you understand as a mixture of Lorenz and Rossler signals? Each of them is the solution to a particular system of differential equations. What equation governs the sum of processes? If you add R.H.S. of both equations, this will be a new DE, that might have neither Lorenz nor Rossler solution.
If you can give the practical application of your question (what problem in what science), it might be easier to give an advice.
a) and b) can be done by simple optimal fit. Should not be hard,
and even small noises can be part of he game.
c) is more tricky (for general flows which do not have a priori polynomial forms). Besides, mixing signal is more general than adding them. You did not make assumption on the smoothness of the flow (near the attractors). In fact, perhaps there is no attractor and two escapes to infinity compensate each other. c) must be highly specified before one can give a meaningful answer, I presume.
We used "multiplexing synchronization" to separate summed chaotic signals in Journal of Physics: Conference Series 23 (2005) 259–266. This approach is directly applicable to scenario (a).
Dear Prof. Jafari
I believe that reconstructing the phase space of the signal that you have might help in c). This can be done by studying the embedding dimension and the false nearest neighbors as described in the first chapters in Abarbanel's book:
Abarbanel, H. (2012). Analysis of observed chaotic data. Springer Science & Business Media.
Suppose there are two chaotic signals X(t) and Y(t). A new combination Z(t) is made by mixing the chaotic signals such that Z(t) = a X(t)+bY(t).
I think these signals can be separated with high fidelity at least in the following two situations.
1) Suppose if a>>b, then one should be able to separate X(t) and Y(t) just how an encrypted signal is decoded in Chaotic Cryptography.
2) You have at least two different mixtures. Z1(t)=a1 X(t) +b1 Y(t) and Z2(t)=a2 X(t) +b2 Y(t). Here the signals can be separated using Independent Component Analysis (see: http://research.ics.aalto.fi/ica/cocktail/cocktail_en.cgi) or may be some sort of machine learning :)
Maybe you can consider them as two different noisy data (this is not totally right but not so wrong), then there must be some condition for separability. For example, in Gaussian noise case, two noise can be detect if some of their properties, e.g. mean or variance be different. Here, maybe looking at scaling in frequency domain or Lyapunov exponents can help. Correlation analysis also could help but if you can compare it with typical correlation from models.
- Badges
- Science topic
- Similar topics
- Engineering
- Control Theory
- Control Systems
- Control Systems Engineering