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 $d(n) \propto n^{-1/2}$, 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.