Convergence Dynamics and Scaling Laws in the Dissipative Relativistic Kicked Rotator

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 $d(n)\propto n^{-1\mathrm{/}2}$, from which the critical exponents $\alpha =1$, $\beta =-1\mathrm{/}2$, and $z=-2$ are derived. These analytical predictions are confirmed numerically and shown to satisfy the homogeneous scaling relation $z=\alpha \mathrm{/}\beta$. Linearization of the map near the fixed point yields an exponential relaxation law $d_n=d_0{e}^{-n\mathrm{/}\tau }$ 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.