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 $\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}$0, the fixed point $\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}$1 undergoes a pitchfork bifurcation. Defining the local state distance

$\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}$2

the system admits a normal form expansion for small $\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}$3 and $\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}$4: $\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}$5 At the bifurcation ($\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}$6) one gets the cubic map $\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}$7, leading in the continuum limit to $\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}$8 and the analytic solution

$\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}$9

with scaling properties:

• initial plateau $K$0 ($K$1),
• late-time decay $K$2 ($K$3),
• crossover $K$4 ($K$5),
• homogeneous scaling relation $K$6.

3. Linear Relaxation and Critical Exponents Below Threshold

For $K$7 $K$8, near the fixed point ($K$9), the dynamics linearize: $\rho$0 With the relaxation time scale $\rho$1, this produces the relaxation exponent $\rho$2 as $\rho$3. Eigenvalues of the Jacobian linearization around $\rho$4, $\rho$5, 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 $\rho$6, $\rho$7 random $\rho$8; $\rho$9 iterations; $\xi$0 transient steps discarded; $\xi$1 plotted.
• Largest Lyapunov exponent via QR-decomposition used to locate $\xi$2.
• Convergence curves at $\xi$3: Initial distances $\xi$4 sampled from $\xi$5 to $\xi$6, up to $\xi$7 iterations; log–log plots exhibit plateau and power-law decay $\xi$8.
• Power-law fits yield $\xi$9.
• Scaling collapse: rescaling $\psi\in[0,1]$0, $\psi\in[0,1]$1 causes curves for different $\psi\in[0,1]$2 to fall on a master curve.
• Near-critical ($\psi\in[0,1]$3): $\psi\in[0,1]$4 varied down to $\psi\in[0,1]$5, $\psi\in[0,1]$6 initial states within radius $\psi\in[0,1]$7 of $\psi\in[0,1]$8, $\psi\in[0,1]$9 defined as mean time to reach $\rho\to 0$0; log–log fit $\rho\to 0$1 vs.\ $\rho\to 0$2 yields $\rho\to 0$3.

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 $\rho\to 0$4 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).