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

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 α∈(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 K1. 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 K1. Analytical estimations support our numerical results.