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

TL;DR

The paper introduces the Lévvy Map, a two-dimensional nonlinear map capable of generating Lévvy flights with tunable parameters, and explores its chaotic behavior and potential as a pseudo-random number generator for disordered wire scattering simulations.

Contribution

A novel two-dimensional nonlinear map called the Lévvy Map is derived to produce Lévvy flights with adjustable parameters, expanding tools for modeling complex stochastic processes.

Findings

Identified conditions for the onset of global chaos in the Lévvy Map.

Demonstrated the map's capability as a Lévvy pseudo-random number generator.

Validated the map's applicability through scattering properties in disordered wires.

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.