Eur. Phys. J. Spec. Top.
https://doi.org/10.1140/epjs/s11734-026-02325-8

Regular Article

Convergence dynamics and scaling laws in the dissipative relativistic kicked rotator

¹ School of Electrical Engineering and Computer Science, University of North Dakota, 58202, Grand Forks, ND, USA
² Institute of Geosciences and Exact Sciences, São Paulo State University (UNESP), Rio Claro, SP, Brazil

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Received: 23 May 2025
Accepted: 15 April 2026
Published online: 26 April 2026

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 ($K=K_{\text{c}}$), 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 $K<K_{\text{c}}$, with $\tau \propto {(K_{\text{c}}-K)}^{-1}$, 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.