Scaling properties of discontinuous maps

Abstract

We study the scaling properties of discontinuous maps by analyzing the average value of the squared action variable I(2). We focus our study on two dynamical regimes separated by the critical value K(c) of the control parameter K: the slow diffusion (K < K(c)) and the quasilinear diffusion (K > K(c)) regimes. We found that the scaling of I(2) for discontinuous maps when K ≪ K(c) and K ≫ K(c) obeys the same scaling laws, in the appropriate limits, as Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to the absence of Kolmogorov-Arnold-Moser tori, we observed in both regimes that I(2) ∝ nK(β) for n ≫ 1 (n being the nth iteration of the map) with β ≈ 5/2 when K ≪ K(c) and β ≈ 2 for K ≫ K(c).