Convergence Dynamics and Scaling Laws in the Dissipative Relativistic Kicked Rotator

TL;DR

We address the convergence dynamics of a dissipative relativistic kicked rotator near period-doubling bifurcations using a local normal-form reduction and extensive simulations to extract universal scaling exponents. At the critical point $K=K_c$, the distance to the stationary state satisfies a plateau followed by a power-law decay $d(n) \sim n^{\beta}$ with $\alpha=1$ and $\beta=-1/2$, and the crossover scales as $n_x \sim d_0^{z}$ with $z=-2$, with a data-collapse consistent with the homogeneous-scaling form. For $K<K_c$ the relaxation is exponential, $d(n)=d_0 e^{-n/\tau}$, with $\tau \propto (K_c-K)^{\delta}$ and $\delta=-1$, indicating critical slowing down. Analytically, these exponents emerge from the local normal form, confirming universality with one-dimensional unimodal maps despite the model’s two-dimensionality and relativistic corrections, and highlighting the predictive power of the homogeneous-function framework for convergence near bifurcations.

Abstract

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 $α= 1$, $β= -1/2$, and $z = -2$ are derived. These analytical predictions are confirmed numerically and shown to satisfy the homogeneous scaling relation $z = α/ β$. Linearization of the map near the fixed point yields an exponential relaxation law $d_n = d_0 e^{-n/τ}$ for $K < K_c$, with $τ\propto (K_c - K)^{-1}$, leading to the relaxation exponent $δ= -1$. 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.

Figures (6)

• Figure 1: Phase space for $\psi = 0$, $\xi = 0.4$, $\rho = 0.15$, and $K = 1.55$. The conservative dynamics exhibits a mixed structure composed of KAM islands, chaotic regions, and invariant spanning curves.
• Figure 2: Phase space for the dissipative dynamics with $\psi = 0.01$, $\xi = 0.4$, $\rho = 0.15$, and $K = 1.55$. Dissipation induces phase-space contraction and the emergence of stable periodic attractors.
• Figure 3: Bifurcation cascade for (a) $I$ and (b) $\theta$ and (c) the lyapunov exponent $\lambda$ as functions of the control parameter $K$, showing the transition from periodic to chaotic dynamics. The blue dashed line indicates the critical parameter $K_c$, where the Lyapunov exponent becomes zero. Parameters: $\psi = 0.8$, $\xi = 0.4$, and $\rho = 0.15$.
• Figure 4: Behavior of $d(n)$ for different initial distances $d_{0}$ from the bifurcation point. A plateau regime ($n < n_{x}$) is followed by a power-law decay ($n > n_{x}$) with exponent $\beta \approx -1/2$, confirming the scaling relation $n_{x} \propto d_{0}^{\,z}$.
• Figure 5: (a) Method for determining the crossover time $n_x$. The plot shows $d(n)$ along with two additional curves: one representing the initial plateau $d_{ip}$ and the other describing the power-law decay. The crossover time $n_x$ is defined as the point where these two curves intersect. (b) Determination of the plateau exponent $\alpha$. The initial plateau $d_{ip}$ is plotted as a function of the initial distance $d_0$, and a power-law fit is performed, resulting in $\alpha = 0.996 \pm 0.002$. (c) Determination of the crossover exponent $z$. The values of $n_x$ are plotted against $d_0$, and a power-law fitting yields the exponent $z = -1.978 \pm 0.003$. (d) Scaling collapse of the convergence curves. After applying the transformations $n \rightarrow n / d_0^{z}$ and $d \rightarrow d / d_0^{\alpha}$, the curves describing the convergence dynamics collapse onto a single universal curve, confirming the scaling invariance of the observable.