Fractional Fourier detection of L\'evy Flights: application to Hamiltonian chaotic trajectories

A signal processing method designed for the detection of linear (coherent) behaviors among random fluctuations is presented. It is dedicated to the study of data recorded from nonlinear physical systems. More precisely the method is suited for signals having chaotic variations and sporadically appearing regular linear patterns, possibly impaired by noise. We use time-frequency techniques and the Fractional Fourier transform in order to make it robust and easily implementable. The method is illustrated with an example of application: the analysis of chaotic trajectories of advected passive particles. The signal has a chaotic behavior and encounter L\'evy flights (straight lines). The method is able to detect and quantify these ballistic transport regions, even in noisy situations.