Kink-Driven Chimera Motion with Quantized Velocity in a Chain of Interacting Particles

TL;DR

This work identifies a novel deterministic mechanism for directed motion of a chimera state in a ring of interacting particles: kink excitations in a damped dc-driven Frenkel–Kontorova chain drive a traveling chimera through a nonlocal phase network. The chimera drift velocity is shown to be linearly related to the number of kink pairs, implementing velocity quantization in units of a single kink pair and allowing transport control via initial perturbations. Robustness is demonstrated under random initial conditions, revealing regimes of decayed, multi-kink, or system-wide motion, with potential applicability to other interparticle potentials such as Lennard–Jones. Overall, the work links topological excitations to collective synchronization, advancing understanding of transport in periodic active-media systems and suggesting practical avenues for manipulating macroscopic motion through microscopic defects.

Abstract

We investigate chimera synchronization of internal oscillator states in a ring of interacting particles, using the damped dc-driven Frenkel--Kontorova chain model as an example. In a system with a spatially periodic potential, a dc external force, and dissipation, kinks spontaneously emerge and stabilize. We show that these kinks induce and govern a collective motion of the entire chimera pattern of internal states along the ring. In particular, the average velocity of this motion depends linearly on the number of kink pairs. This number is effectively determined by localized initial perturbations of particle positions, thereby opening a pathway for controlling macroscopic transport through microscopic excitations.

Figures (6)

• Figure 1: Results of simulations of Eqs. \ref{['eq:main_x']}--\ref{['eq:ker']} with initial conditions \ref{['eq:inits']}, \ref{['eq:x_pert']} corresponding to the excitation of a single kink pair driving chimera motion. Parameter values are $L = 1$, $N=256$, $\mu=1$, $\lambda=5$, $\gamma=1.1$, $\eta=25/512 \approx 0.049$, $\psi=3$, $\kappa=5.2$, $\alpha=1.457$, $A = 0.7$, $m = 3$, and $d = 128$. The calculated average velocities $\langle v_n \rangle$ of all the particles and the chimera coincide, $\langle v_{n} \rangle = \langle v_{\text{ch}} \rangle \approx 0.0013$. (a) Time dependence of the spatial coordinate $x_1(t)$ for an individual particle. The inset shows the time segment during which the particle transitions between adjacent potential wells of the substrate potential. (b--e) Snapshots of particle distribution in the coordinate--phase plane at sequential time instants: (b) $t=500$, (c) $t=650$, (d) $t=800$, and (e) $t=950$. The black crosses mark the particle with index $1$, whose spatial coordinate vs time is shown in panel (a). The red arrows indicate the direction of chimera motion (towards the average decrease of the substrate potential). (f) Absolute values of instantaneous particle velocities $|\dot{x}_n(t)|$ vs time for all particles. Two kinks counterpropagate and interact elastically when meeting. (g–-j) Snapshots of particle velocities $\dot{x}_n(t)$ at sequential time instants: (g) $t=992.4$, (h) $t=994.1$, (i) $t=995.8$, and (j) $t=997.5$. The green arrows indicate the directions of kink motion.
• Figure 2: Same as Fig. \ref{['fig:1']}, but for initial conditions \ref{['eq:inits']}, \ref{['eq:multipert']} corresponding to the excitation of $M=5$ kink pairs with $d_{j} =\lfloor (2j-1)N/2M \rfloor$. The calculated average velocities of all the particles and the chimera coincide, $\langle v_{n} \rangle = \langle v_{\text{ch}} \rangle \approx 0.0067$.
• Figure 3: Calculated dependences of the average particle velocity $\langle v_1 \rangle$ (circular markers) and the average chimera velocity $\langle v_{\text{ch}} \rangle$ (cross markers) on the number $M$ of kink pairs for several values of $\psi$: $\psi = 2$ (blue markers), $\psi = 3$ (purple markers), and $\psi = 4$ (cyan markers). All other parameters (except $\psi$ and $M$) are the same as in Figs. \ref{['fig:1']} and \ref{['fig:2']}. Different values of $\psi$ correspond to different kink velocities $v_{\mathrm{kink}}(M) = N\langle v_1\rangle/2$, which are indicated in the respective color for $M = 1$. The dashed straight lines represent the linear dependence $M v_k(1)$. On the right, each line ends at the critical value $M = M^*$---the maximum number of symmetric kink pairs which coexisted without merging for the initial conditions \ref{['eq:inits']}, \ref{['eq:multipert']} used.
• Figure 4: Results of simulations of Eq. \ref{['eq:main_x']} with initial conditions \ref{['eq:inits']}, \ref{['eq:multipert']} corresponding to the initial excitation of $M = 5$ kink pairs, which evolve into 3 asymmetric kink pairs through sequential kink merging. All the parameters are the same as in Fig. \ref{['fig:2']} except $\lambda=4$ and $\gamma=0.58$. (a, b) Time dependence of the spatial coordinate $x_1(t)$ for an individual particle. At $t = t_1$, the first instance of kink destruction occurs, leading to a noticeable decrease in time-average particle velocity (i.e., a reduction in the average slope). (c, d) Absolute values of instantaneous particle velocities $|\dot{x}_n(t)|$ vs time for all particles. Five initial symmetric kink pairs (c) finally transform into three asymmetric kink pairs (d).
• Figure 5: The average particle velocity $\langle v_1 \rangle$ dynamics, normalized by the propagation velocity of a single kink pair $v_k(1)$, is plotted against time for $10$ simulations. The normalized velocity $\langle v_1 \rangle / v_k(1)$ is proportional to the number of propagating kink pairs $M$. The observed decrease in $\langle v_1 \rangle / v_k(1)$ indicates a reduction in the number of active kinks within the medium. The system is simulated under initial conditions of the form \ref{['eq:multipert']} with (a) $M=16$ and (b) $M=32$, equidistantly located $d_j$, random uniformly distributed $m_j \in [1, 7]$, (a) $A_j \in [0.0, 0.005]$ and (b) $A_j \in [0.0, 0.008]$. Parameters: $\mu=1$, $\psi=3$, $\gamma = 0.58$, $\lambda = 4$, $\eta=25/512 \approx 0.049$, and different $L$ and $N$: (a) $L = 1$, $N=256$; (b) $L = 2$, $N=512$.