Chaoticity and randomness in a time series Suppose we have a time series : $X=\{X_t,t\in T\}$. How can we check if the data $X_t$ are random or they are the result of some chaotic behaviour of a nonlinear dynamical system? Is there some test useful to prove the chaoticity of the series? Thanks.

You can apply chaotic time series analysis to the data. There are some useful tools about this subject here. Such implementations try to find positive Lyapunov exponents from the data set based on the studies of Rosenstein et. al. ("A practical method for calculating largest Lyapunov exponents from small data sets") or Wolf et. al. ("Determining Lyapunov exponents from a time series"). You can also check the book of the Julien Sprott about this subject. 
If you can find positive Lyapunov exponents with the above methods you can say that the time series is chaotic. However, as Henning Makholm stated, randomness is a different case. You can apply statistical randomness tests to a sufficiently big data just to conclude that the hypothesis about randomness cannot be rejected for a significance level. 
