Bifurcations and Intermittency in Coupled Dissipative Kicked Rotors

TL;DR

This work analyzes the emergence of complex spatiotemporal dynamics in a lattice of coupled dissipative kicked rotors by connecting collective behavior to microscopic bifurcations of a single rotor, described by the Zaslavsky map with dissipation $\gamma$ and nonlinearity $K_0$. Using pseudo arclength continuation, the authors trace cascades of bifurcations and basins of attraction for the single rotor and then study how these elementary dynamics seed spatial patterns in the coupled system, including alternating-patched and homogeneous-patched states. A combined order parameter $Q$, built from Kuramoto and Daido metrics, captures the spatial symmetry and phase transitions, revealing coexistence of regular and chaotic attractors near the onset of chaos and quantifying pattern formation across parameter space. Near the chaos threshold they observe spatiotemporal intermittency of type-I super-transient with percolation-like scaling of coherent domains, illustrating how simple-state bifurcations organize high-dimensional multistable dynamics with practical implications for predicting pattern formation in coupled map lattices.

Abstract

We investigate the emergence of complex dynamics in a system of coupled dissipative kicked rotors and show that critical transitions can be understood via bifurcations of simple states. We study multistability and bifurcations in the single rotor model, demonstrating how these give rise to a variety of coexisting spatial patterns in a coupled system. A combined order parameter is introduced to characterize different spatial patterns and to reveal the coexistence of chaotic and regular attractors. Finally, we illustrate an intermittent phenomenon near the onset of chaos.

Figures (9)

• Figure 1: Distinct typical snapshots in momenta $\{p_j\}$ ($j = 1, 2, ..., 100$) of the coupled dissipative kicked rotor system Eq.\ref{['eq-ckr']}, each drawn from a random initial condition with parameter values (a) $K_0=1.9, J=0.3$, (b) $K_0=1, J=0.8$, (c) $K_0=2, J=0.6$, (d) $K_0=4.8, J=0.2$, (e) $K_0=0.6, J=1.1$, (f) $K_0=5, J=0.5$.
• Figure 2: Bifurcations in the Zaslavsky map with $\gamma = 0.8$ and $K_0 \in [1, 7]$: (a) for momentum $p$ and (b) for angle $\theta$. The dashed curves connect a cascade of bifurcation points: fold (in red) and period-doubling (PD in short, in blue) bifurcations for fixed points $p^* = 2n\pi$, $n \in \mathbb{Z}$, and fold bifurcations for a period-$3$ state (in dark-green). Each plot also highlights the main $n$-resonances in violet, a subharmonic $(n, m) = (0, 4)$ resonance in cyan, and a subharmonic $(n, m) = (1, 3)$ resonance in light-green. Bifurcations for other $\gamma$ values are presented in Appendix \ref{['appA']}.
• Figure 3: Basins of attractions of the Zaslavsky map with $\gamma = 0.8$ and (a) $K_0 = 2$, (b) $K_0 = 2.7$ and (c) $K_0 = 6.6$.
• Figure 4: Order parameter $Q$ (Eq.\ref{['eq-Q']}) for a chain of $N=100$ coupled rotors at time $t = 2000$, with the two curves $C_1$ and $C_2$. The green stars with labels (a-f) correspond to the parameter values in Fig.\ref{['fig-intro']}. The heatmap is generated from a $100\times 100$ grid on $J\times K_0 \in [0, 1.43]\times [0, 6]$ and averaged over $30$ random initial conditions $(p_j(0), \theta_j(0)) \in \text{Uni}[-35, 35]\times \text{Uni}[-\pi, \pi]$.
• Figure 5: Fractions of alternating-patched (AP, solid) and homogeneous-patched (HP, dotted) states in varying the system parameters $(J, K_0)$ along the line $K_0(J) = K_0^*(J) + 1.0$, which is parallel to $C_1$ and inside the strip region in-between curves $C_1$ and $C_2$. Insets illustrate the two patched states: $(J, K_0) = (1.1, 0.2)$ for AP, and $(J, K_0) = (0.03, 4.48)$ for HP. Numerical settings: $N=30$, $t=10000$, and $30$ initial conditions are $\boldsymbol{p}(0) = \boldsymbol{0}$, $\boldsymbol{\theta}(0) + \boldsymbol{\epsilon}$, where $\boldsymbol{\theta}(0) = (1, -1, 1, -1, ..., 1, -1)$ and $\boldsymbol{\epsilon} = (\epsilon_1, ..., \epsilon_N)$, $\epsilon_j \in \text{Uni}[-0.01, 0.01]$, $j = 1, 2, ..., N$.