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

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 $P(\mathrm{\ell })\sim {\mathrm{\ell }}^{-(1+\alpha )}$ with $\alpha =1$, we derive a generalized two-dimensional non-linear map $M_{\alpha }$ able to produce L'evy flights described by $P(\mathrm{\ell })$ 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.