2024/02/27 by J. A. Méndez‐Bermúdez, R. Aguilar-Sánchez, Mendez-Bermudez, J. A. +5
Mathematics · #Differential Equations and Boundary Problems #advanced mathematical theories #Nonlinear Differential Equations Analysis
paper · pdf · doi:10.48550/arxiv.2402.17673
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 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.