The L\'evy Map: A two-dimensional nonlinear map characterized by tunable L\'evy flights

Abstract

Once recognizing that point particles moving inside the extended version of the rippled billiard perform L\'evy flights characterized by a L\'evy-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 L\'evy flights described by $P(\ell)$ with $0<\alpha<2$. Due to this property, we name $M_\alpha$ as the L\'evy 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 L\'evy Map could be used as a L\'evy pseudo-random number generator and, furthermore, confirm its applicability by computing scattering properties of disordered wires.