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 = Kc K = K c ), the dynamics reduce to a normal form that produces a power-law decay d(n) ∝ n-1/2 d ( n ) ∝ n - 1 / 2 , from which the critical exponents α = 1 α = 1 , β = -1/2 β = - 1 / 2 , and $$z = -2$$ z = - 2 are derived. These analytical predictions are confirmed numerically and shown to satisfy the homogeneous scaling relation z = α / β z = α / β . Linearization of the map near the fixed point yields an exponential relaxation law dₙ = d₀ e-n/τ d n = d 0 e - n / τ for K < Kc K < K c , with τ ∝ (Kc - K)⁻¹ τ ∝ ( K c - K ) - 1 , leading to the relaxation exponent δ = -1 δ = - 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.