Leaking from the phase space of the Riemann–Liouville fractional standard map

Abstract

In this work we characterize the escape of orbits from the phase space of the Riemann–Liouville (RL) fractional standard map (fSM). The RL-fSM, given in action–angle variables, is derived from the equation of motion of the kicked rotor when the second order derivative is substituted by a RL derivative of fractional order α. Thus, the RL-fSM is parameterized by K and α∈(1,2] which control the strength of nonlinearity and the fractional order of the RL derivative, respectively. Indeed, for α=2 and given initial conditions, the RL-fSM reproduces Chirikov’s standard map. By computing the survival probability PS(n) and the frequency of escape PE(n), for a hole of hight h placed in the action axis, we observe two scenarios: When the phase space is ergodic, both scattering functions are scale invariant with the typical escape time ntyp=exp〈lnn〉∝(h/K)2. In contrast, when the phase space is not ergodic, the scattering functions show a clear non-universal and parameter-dependent behavior.