Two-dimensional nonlinear map characterized by tunable Lévy flights

Abstract

After recognizing that point particles moving inside the extended version of the rippled billiard perform Lévy flights characterized by a Lévy-type distribution P(l)∼l-(1+α) with α=1, we derive a generalized two-dimensional nonlinear map Mα able to produce Lévy flights described by P(l) with 0<α<2. Due to this property, we call Mα 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 pseudorandom number generator and furthermore confirm its applicability by computing scattering properties of disordered wires.