Coherence and Transients in Nonlocally Coupled Dissipative Kicked Rotors

TL;DR

This work investigates a network of nonlocally coupled dissipative kicked rotors to understand how coupling range $P_c$ and coupling strength $K$ shape spatiotemporal patterns. It combines linear stability analysis of the zero state with extensive simulations to map a phase diagram, revealing coherent period-2 states with distinct wavenumbers and a hierarchy of layers in parameter space. A key contribution is the analytic onset curve for period-doubling, $K=[2(1+\gamma)-K_0]\frac{P_c}{P_c-S_{\min}}$, together with a characterization of wavenumber selection and basin competition; it also uncovers a regime of super-long transient interfaces when the local map coexists with chaotic dynamics, where transient lifetimes scale exponentially with $P_c$ up to a threshold. The findings illuminate the role of nonlocal coupling in generating coherent patterns and long-lived mixed states, with potential relevance to neural networking, optical arrays, and information processing.

Abstract

The dynamics of nonlocally coupled dissipative kicked rotors is rich and complex. In this study, we consider a network of rotors where each interacts equally with a certain range of its neighbors. We focus on the influence of the coupling strength and the coupling range, and show both analytically and numerically the critical transitions in the phase diagram, which include bifurcations of simple spatiotemporal patterns and changes in basin sizes of coherent states with different wavenumbers. We highlight that this diagram is fundamentally different from those found in other coupled systems such as in coupled logistic maps or Lorenz systems. Finally, we show an interesting domain-wall phenomenon in the coupled chaotic rotors, where a super-long transient interface state (partially regular and partially chaotic) is observed and can persist exponentially long as the coupling range increases up to a critical threshold.

Figures (13)

• Figure 1: Snapshots of momenta for (a) $K=1.3$, (b) $K=1.4$ and (c) $K=3$. Other parameters: $\gamma=0.8$, $K_0=2$, $P_c=32$, $N=100$ and random initial conditions $(p_j(0), \theta_j(0)) \in \text{Uni}[-35, 35]\times \text{Uni}[0, 2\pi)$, $j = 1, 2, ..., N$. The corresponding angles behave similarly.
• Figure 2: Snapshots of momenta for (a) $P_c=23$, (b) $P_c=17$ and (c) $P_c=14$. Other parameters: $\gamma=0.8$, $K_0=2$, $K=3$, $N=100$ and random initial conditions $(p_j(0), \theta_j(0)) \in \text{Uni}[-35, 35]\times \text{Uni}[0, 2\pi)$, $j = 1, 2, ..., N$. The corresponding angles behave similarly.
• Figure 3: Regions of typical (a) wavenumbers $l \leq 8$ and (b) temporal periods $\tau \leq 8$ on the parameter $(P_c, K)$-plane, with $P_c \in \{ 1, 2, ..., \frac{N}{2}-1\}$ and $K \in [0.01, 5]$ for $100$ equidistant values. Each pixel color is decided by the mode over $20$ trajectories. The green curve is obtained from linear stability analysis of the zero synchronized state and the green dots are the corresponding wavenumber changes. The red dots mark the instability of the temporal period-$2$ states, obtained by numerical bifurcation analysis. Other parameters: $\gamma=0.8$, $K_0=2$, $N=100$, and random initial conditions $(p_j(0), \theta_j(0)) \in \text{Uni}[-35, 35]\times \text{Uni}[0, 2\pi)$, $j = 1, 2, ..., N$. Larger-size systems show similar phase transitions, cf. Appendix \ref{['app:large']}.
• Figure 4: Relative basin sizes for temporal period-$2$ states with wavenumbers $l$ as a function of the coupling strength $P_c$. The dashed lines denote $l > 8$. Other parameters: $\gamma=0.8$, $K_0=2$, $K=1.5$, and $N=100$.
• Figure 5: Transient behavior for an $N=1000$ system: (a) basin structure of the single rotor at $\gamma=0.8$ and $K_0=6.6$, the two sub-intervals are $I_0 := [\theta^* -0.05, \theta^* +0.05]$ with $p_j(0) = p^*$, where $(p^*, \theta^*) = (10\pi, \arcsin \frac{-(1-\gamma)p^*}{K_0})$ is the fixed point of the single-rotor model and $I_1 := [-\theta^* -0.05, -\theta^* +0.05]$; $I_0$ ($I_1$) is a subset of the basin the regular (chaotic) attractor; (b) averaged transient time on the parameter $(P_c, K)$-plane and (c) for $K=1$ in a semi-log scale: simulation data are shown in blue dots with fluctuations in blue ribbons; the gray dashed line marks the sharp transition.