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I am most interested in the difference of power spectrums between chaos and periodicity.

I read some paper on testing chaos against some more general alternatives (mostly stochastic systems) through means of either the embedding dimension methods or Lyapunov exponent.

However, I wonder if it is possible to tell chaos from periodics with lots noise just by looking at their power spectrum. Is it possible?

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My intuition is that if it is periodic, then the Fourier coefficients that constitute this periodicity will have different values than the rest of the spectrum (which, for example, would be kind of uniform for white noise). Of course, this probably requires many, many, many samples and might be not practical at all for a real world application. On the other hand, it is very hard to distinguish between random and non-random sequences with small number of samples. –  dtldarek Feb 12 '13 at 14:02

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I'm gonna guess that there is not. There is a paper by Kaplan and Glass that discuses this if not indirectly. They take a random time series with a nearly identical power spectrum of a chaotic one (I believe the lorenz system) and try to come up with a direct test for determinism. The fact is that while there may be some tell tale signs in extreme cases I don't think looking at the power spectrum is sufficient.

http://www.medicine.mcgill.ca/physio/glasslab/pub_pdf/directtest_1992.pdf

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