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 $\alpha$. Thus, the RL-fSM is parameterized by $K$ and $\alpha \in (1,2]$ which control the strength of nonlinearity and the fractional order of the RL derivative, respectively. Indeed, for $\alpha =2$ and given initial conditions, the RL-fSM reproduces Chirikov's standard map. By computing the survival probability $P_{\text{S}}(n)$ and the frequency of escape $P_{\text{E}}(n)$, 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 $n_{\text{typ}}=\exp ⟨\ln n⟩\propto (h/K{)}^2$. In contrast, when the phase space is not ergodic, the scattering functions show a clear non-universal and parameter-dependent behavior.

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Cited by in corpus (2)

• Caputo Fractional Standard Map: Scaling Invariance Analyses
• Stochastic Web Map: Survival probability and escape frequency