Relativistic Dissipative Kicked Rotator

• The paper introduces the dissipative relativistic kicked rotator, extending the classical model with relativistic corrections and dissipation to reveal universal period-doubling bifurcation dynamics.
• It establishes a rigorous mathematical formulation through a discrete-time map and identifies key scaling laws with measured critical exponents: α≈1, β≈–1/2, z≈–2, and δ≈–1.
• The study employs extensive numerical simulations and normal form analysis to demonstrate that complex two-dimensional dynamics align with the universality of one-dimensional unimodal maps.

The dissipative relativistic kicked rotator is a discrete-time dynamical system exhibiting richness in both analytic tractability and nonlinear convergence phenomena. It generalizes the classical kicked rotator model to include relativistic corrections and dissipation, yielding a two-dimensional phase space with variables subject to periodic forcing and non-trivial nonlinearities. Recent investigations have established its scaling properties and convergence dynamics near period-doubling bifurcations, demonstrating universality with one-dimensional unimodal maps despite the model's higher dimensionality and physical generalizations (Borin et al., 4 Dec 2025).

1. Definition and Mathematical Formulation

The discrete-time map $$S\,:\,(\theta_n, I_n)\mapsto(\theta_{n+1}, I_{n+1})$$ is defined by: $\begin{cases} \displaystyle \theta_{n+1} =\Big[\theta_n + \frac{I_n}{\sqrt{1+(\rho\,I_n)^2} - \xi\,I_n}\Big] \bmod 2\pi, \[1em] \displaystyle I_{n+1} = (1-\psi)\,I_n + K\,\sin(\theta_{n+1})\,, \end{cases}$ where

• $K$ is the kick strength,
• $\rho$ the relativistic parameter,
• $\xi$ a velocity-coupling term,
• $\psi\in[0,1]$ the fractional dissipation per kick.

The Newtonian limit ($\rho\to 0$, $\xi\to 0$) recovers the standard dissipative map; setting $\psi=0$ yields the area-preserving relativistic standard map.

2. Bifurcation Structure and Normal Form Analysis

Period-doubling bifurcations occur at a critical threshold $K = K_c \approx 3.999999997$, identified via the largest Lyapunov exponent's zero crossing. At $K=K_c$, the fixed point $(\theta^*, I^*) = (\pi, 0)$ undergoes a pitchfork bifurcation. Defining the local state distance

$$d_n = \sqrt{(\theta_n-\theta^*)^2 + (I_n-I^*)^2}\,,$$

the system admits a normal form expansion for small $d_n$ and $|\mu|$: $$d_{n+1} = d_n - \mu\,d_n - d_n^3,\qquad \mu \equiv K_c - K\,.$$ At the bifurcation ($\mu=0$) one gets the cubic map $d_{n+1} - d_n = -d_n^3$, leading in the continuum limit to $\partial_n d = -d^3$ and the analytic solution

$d(n) = \frac{d_0}{\sqrt{1 + 2\,d_0^2\,n}}\,,$

with scaling properties:

• initial plateau $d(n)\approx d_0\propto d_0^\alpha$ ($\alpha=1$),
• late-time decay $d(n)\sim n^{-1/2}$ ($\beta=-\frac{1}{2}$),
• crossover $n_x\propto d_0^{-2}$ ($z=-2$),
• homogeneous scaling relation $z=\alpha/\beta=-2$.

3. Linear Relaxation and Critical Exponents Below Threshold

For $K<K_c$ $(\mu>0)$, near the fixed point ($d_n\to 0$), the dynamics linearize: $$\partial_n d = -\mu\,d \quad\implies\quad d(n) = d_0\,e^{-\mu n}\,.$$ With the relaxation time scale $\tau = \mu^{-1}$, this produces the relaxation exponent $\delta = -1$ as $\tau\propto (K_c - K)^{-1}$. Eigenvalues of the Jacobian linearization around $(\pi, 0)$, $\lambda_{1,2} \approx 1-\mu \pm O(\mu^2)$, confirm this exponential decay regime.

4. Numerical Simulation Protocols and Data

Simulations employ random initial conditions and long runs for bifurcation diagram and scaling law extraction:

• For each $K\in[0,11]$, $10$ random $(\theta_0, I_0)\in [0,2\pi]\times[-20,20]$; $10^4$ iterations; $9\times10^3$ transient steps discarded; $(\theta, I)$ plotted.
• Largest Lyapunov exponent via QR-decomposition used to locate $K_c$.
• Convergence curves at $K=K_c$: Initial distances $d_0$ sampled from $10^{-4}$ to $10^{-1}$, up to $n \sim 10^5$ iterations; log–log plots exhibit plateau and power-law decay $d\propto n^\beta$.
• Power-law fits yield $$\alpha = 0.996\pm0.002 \approx 1,\,\beta = -0.504\pm0.001,\,z = -1.978\pm0.003$$.
• Scaling collapse: rescaling $n \to n/d_0^z$, $d \to d/d_0^\alpha$ causes curves for different $d_0$ to fall on a master curve.
• Near-critical ($K<K_c$): $\mu$ varied down to $10^{-6}$, $10^6$ initial states within radius $r=10^{-1}$ of $(\pi, 0)$, $\tau$ defined as mean time to reach $d < 10^{-6}$; log–log fit $\tau$ vs.\ $\mu$ yields $\delta = -0.985\pm0.003\approx-1$.

5. Universality and Scaling Laws

Despite its two-dimensional phase space and explicit relativistic/dissipative terms, the system's local bifurcative normal form matches that of one-dimensional unimodal quadratic maps (e.g., logistic, Hénon). The measured critical exponents $\{\alpha, \beta, z, \delta\} = \{1,-1/2,-2,-1\}$ coincide with those for the dissipative Fermi–Ulam model and similar one-dimensional maps. This equivalence stems from the local quadratic normal form dominating the period-doubling critical behavior. The homogeneous-function formalism, manifest through scaling collapse and matching exponents, indicates identical universal properties and supports renormalization–group predictions that the Feigenbaum fixed point for period-doubling exhibits a single relevant eigenvalue. The scaling hypotheses — plateau, power law, crossover — arise from the self-similar structure near the bifurcation, independent of global model specifics.

6. Context and Implications in Nonlinear Dynamics

These findings demonstrate that universality of period-doubling bifurcations extends to higher-dimensional systems and those with physical modifications such as relativistic and dissipative effects, as long as the local nonlinearity remains quadratic. Analytic derivations and large-scale numerical simulations reveal that convergence dynamics, scaling exponents, and scaling functions in the dissipative relativistic kicked rotator mirror those of one-dimensional unimodal maps. This positions the model as a reference point for studies of universal scaling in complex nonlinear systems, as well as an exemplary testbed for renormalization-group theory and homogeneous-function self-similarity (Borin et al., 4 Dec 2025).