The Lévy Map: A two-dimensional nonlinear map characterized by tunable Lévy flights

Abstract

Once recognizing that point particles moving inside the extended version of the rippled billiard perform Lévy flights characterized by a Lévy-type distribution $P(\ell )\sim {\ell }^{-(1+\alpha )}$ with $\alpha =1$, we derive a generalized two-dimensional non-linear map $M_{\alpha }$ able to produce Lévy flights described by $P(\ell )$ with $0<\alpha <2$. Due to this property, we name $M_{\alpha }$ as the Lévy Map. Then, by applying Chirikov's overlapping resonance criteria we are able to identify the onset of global chaos as a function of the parameters of the map. With this, we state the conditions under which the Lévy Map could be used as a Lévy pseudo-random number generator and, furthermore, confirm its applicability by computing scattering properties of disordered wires.

6 pages, 5 figures

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