Scaling properties of discontinuous maps

TL;DR

This paper investigates the scaling behavior of discontinuous maps, revealing that their diffusion properties follow similar laws to the standard map but with distinct exponents due to the absence of KAM tori.

Contribution

It demonstrates that discontinuous maps exhibit universal scaling laws in different regimes, with specific power-law exponents, extending understanding of nonlinear dynamical systems without KAM tori.

Findings

Scaling laws for $I^2$ match those of the standard map in respective regimes.

Exponent $eta$ is approximately 5/2 for $K o 0$ and 2 for large $K$.

Absence of KAM tori leads to distinct diffusion scaling behavior.

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 $K\ll K_c$ and $K\gg K_c$ obeys the same scaling laws, in the appropriate limits, than Chirikov's standard map in the regimes of weak and strong nonlinearity, respectively. However, due to absence of KAM tori, we observed in both regimes that $I^2\propto nK^\beta$ for $n\gg 1$ (being $n$ the $n$-th iteration of the map) with $\beta\approx 5/2$ when $K\ll K_c$ and $\beta\approx 2$ for $K\gg K_c$.