Convergence Dynamics and Scaling Laws in the Dissipative Relativistic Kicked Rotator

TL;DR

This paper analyzes the convergence and scaling laws of the dissipative relativistic kicked rotator near bifurcations, revealing universal behaviors similar to one-dimensional unimodal maps through analytical and numerical methods.

Contribution

It provides the first detailed analytical and numerical characterization of the convergence dynamics and scaling laws in the dissipative relativistic kicked rotator near bifurcations.

Findings

Power-law decay at bifurcation threshold with exponent -1/2

Exponential relaxation below threshold with relaxation time diverging as (Kc - K)^{-1}

Shared universality class with one-dimensional unimodal maps

Abstract

We investigate the convergence dynamics of this system near period-doubling bifurcations by combining analytical derivations and large-scale numerical simulations. At the bifurcation threshold ($K = K_c$), the dynamics reduce to a normal form that produces a power-law decay $d(n) \propto n^{-1/2}$, from which the critical exponents $\alpha = 1$, $\beta = -1/2$, and $z = -2$ are derived. These analytical predictions are confirmed numerically and shown to satisfy the homogeneous scaling relation $z = \alpha / \beta$. Linearization of the map near the fixed point yields an exponential relaxation law $d_n = d_0 e^{-n/\tau}$ for $K < K_c$, with $\tau \propto (K_c - K)^{-1}$, leading to the relaxation exponent $\delta = -1$. The remarkable agreement between theory and simulation demonstrates that the dissipative relativistic kicked rotator shares the same universality class as one-dimensional unimodal maps, despite its higher dimensionality and relativistic corrections.