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Suppose I have a particle position history over an interval, i.e., $\mathbf{x}(t)$, where $t \in [t_1, t_2]$, that is the solution to a system that may be chaotic.

An anecdotal, less general, example would be that if $\mathbf{x}(t)$ is a solution to the Lorenz system, for which the parameters are unknown, to determine whether the system is chaotic.

More generally, is it possible to extract the system parameters from the position history alone?

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  • $\begingroup$ Just to add that there is a Dynamical Systems chat room where if any one interested in Dynamical systems and chaos theory can join,i admit that it is not so active though. $\endgroup$ – BAYMAX Sep 3 '17 at 17:23
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This is a typical task for the discipline of non-linear time-series analysis. However, with experimental data, it’s a quite complicated process with many pitfalls. Briefly, what you have to do is:

  • Reconstruct the phase space of your dynamics using Takens’ theorem. If all there is to your system is your particle’s position (and velocity, which can be estimated from this), you can skip this step. However, if you actually know this, you usually have more information about your system than just the position histories.

  • Estimate some characteristics from the time series that is capable of distinguishing between chaos and regular dynamics, e.g., the largest Lyapunov exponent.

  • Repeat the procedure for surrogate time series to ensure that your result is statistically robust and not a result of a flaw in your analysis process.

As already said, this is a discipline of its own and entire books have been written about (mostly) this, e.g., by Kantz and Schreiber.

Disclosure: One of the authors of said book is within my direct academic vicinity.

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