Scaling properties of the action in the Riemann-Liouville fractional standard map

Abstract

The Riemann-Liouville fractional standard map (RL-fSM) is a two-dimensional nonlinear map with memory given in action-angle variables (I,θ). The RL-fSM is parameterized by K and αϵ(1,2], which control the strength of nonlinearity and the fractional order of the Riemann-Liouville derivative, respectively. In this work we present a scaling study of the average squared action (I2) of the RL-fSM along strongly chaotic orbits, i.e., for K≫1. We observe two scenarios depending on the initial action I0, I0≪K or I0≫K. However, we can show that (I2)/I02 is a universal function of the scaled discrete time nK2/I02 (n being the nth iteration of the RL-fSM). In addition, we note that (I2) is independent of α for K≫1. Analytical estimations support our numerical results.