Two-dimensional nonlinear map characterized by tunable Lévy flights

After recognizing that point particles moving inside the extended version of the rippled billiard perform Lévy flights characterized by a Lévy-type distribution $P\left(l\right)\sim l^{-(1+\alpha )}$ with $\alpha =1$, we derive a generalized two-dimensional nonlinear map $M_{\alpha }$ able to produce Lévy flights described by $P\left(l\right)$ with $0<\alpha <2$. Due to this property, we call $M_{\alpha }$ 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.

Figure 1

Geometry of the rippled billiard and definition of the variables of map $M$ [see Eq. (2)].

Figure 2

(a) Poincaré map $M$, (b) phase distribution $P\left(\theta \right)$, and (c) length distribution $P\left(l\right)$ for the ripple billiard with $\omega =2\pi /10$ and $d=2\pi$. A single initial condition $x_0={\theta }_0=0.1$ was iterated (a) ${10}^4$ and (b) and (c) ${10}^7$ times. The red solid curve in (c) is Eq. (3).

Figure 3

(a)–(d) Phase distributions $P\left(\theta \right)$ and (e)–(h) length distributions $\rho (lnl)$ for the Lévy map $M_{\alpha }$ with $\omega =\mathcal{C}=1$ and (a) and (e) $\alpha =1/4$, (b) and (f) $\alpha =1/2$, (c) and (g) $\alpha =1$, and (d) and (h) $\alpha =3/2$. To construct each histogram a single initial condition $x_0={\theta }_0=0.1$ was iterated ${10}^7$ times. The red solid curve in (e)–(h) is $\rho (lnl)=l^{-\alpha }$.

Figure 4

Typical value of $l,l_{\mathrm{typ}}=exp\langle lnl\rangle$, for the Lévy map $M_{\alpha }$ as a function of $\alpha$; $\omega =\mathcal{C}=1$ were used. The average was performed over ${10}^7$ values of $l$ obtained by iterating $M_{\alpha }$ from the single initial condition $x_0={\theta }_0=0.1$.

Figure 5

Average $\langle -lnG\rangle$ as a function of $L/l_{\mathrm{typ}}$ for the 1D Anderson model with off-diagonal Lévy-type disorder characterized by $\alpha$. We used an incoming wave with energy $E=0.1$. The dashed lines are fittings of the data with Eq. (15). Each symbol was calculated using ${10}^5$ wire realizations. Each wire realization was constructed from a single sequence of lengths $\left\{l\right\}$ generated by map $M_{\alpha }$ having random initial conditions uniformly distributed in the intervals $-\pi <x_0<\pi$ and $0<{\theta }_0<2\pi$.