Atypical Chimera States in an Ensemble of Partially Mobile Particles

TL;DR

The study addresses how regular spatially inhomogeneous motion of oscillators on a ring with nonlocal coupling shapes chimera states. It introduces a moving subpopulation with coordinates $x_n(t)=x_n^{(0)}+A_n\sin{\Omega t}$ and a nonlocal coupling kernel $G(x)$, analyzing the mean field $H_n(t)$ and order parameter $R(t)$; the motion induces an effective kernel $G_{ ext{eff}}(x,\tilde{x},t)=G(\tilde{x}-x)+[\langle A\rangle(\tilde{x})-\langle A\rangle(x)]G'(\tilde{x}-x)\sin{\Omega t}$ that becomes asymmetric. Numerically, the authors observe a periodically traveling chimera and, for asymmetric amplitudes, an alternating chimera structure accompanied by two microscopic states: a nonuniformly twisted state when $\Delta=0$ and a coherent-incoherent-twisted state when $\Delta>0$, with cross-correlation analyses (via a Möbius transform) confirming the local phase-structure differences. The work highlights how mobility patterns can control and diversify chimera dynamics, with potential implications for active matter and engineered oscillator networks. Key contributions include the kernel-symmetry breaking mechanism, the characterization of nonuniformly twisted versus coherent-incoherent-twisted states, and the demonstration of amplitude-heterogeneity–driven alternations in chimera topology.

Abstract

We study the influence of nonuniform motion of oscillators in a ring chain with nonlocal coupling on their collective dynamics and reveal the mechanism behind the emergence of an atypical chimera state in such systems. The mechanism relies on regular spatially inhomogeneous motion of oscillators, which breaks the symmetry of the effective interaction kernel. This symmetry breaking induces spatial phase correlations in the asynchronous part of the system, giving rise to nonuniformly twisted and previously unobserved coherent-incoherent-twisted states.

Figures (7)

• Figure 1: Dynamics of the modulus of the global order parameter $|R(t)|$ for $30$ samples of random initial conditions $\varphi_n(0)$, uniformly distributed on $(-\pi,\pi]$, for a few samples of parameter sets $(\sigma, \mu, \Delta)$. Temporal evolutions with $\lvert R(t)\rvert = 1$ correspond to the fully synchronized state. Temporal evolutions with $\lvert R(t)\rvert < 1$ correspond to chimera-like regimes. Vertical red dashed lines indicate the time instants at which the particle coordinates $x_n(t)$ are equidistantly spaced. Parameters $\sigma = 0.03$ (left two columns), $\sigma = 0.1$ (right two columns), $\mu = 0$ (first and third columns), $\mu = 0.06$ (second and fourth columns), $\Delta = 0$ (first row), $\Delta = 0.06$ (second row), $\Delta = 0.18$ (third row). For parameter sets in which the fully synchronized state was observed, its probability of occurrence is shown.
• Figure 2: Periodically moving chimera state. (a--d) Phase snapshots at the time instants (a) $t = 1329$, (b) $t = 1407$, (c) $t = 1486$, (d) $t=1564$. Blue filled circular markers indicate stationary oscillators; red empty circular markers indicate moving oscillators. (e) Dynamics of $|R|$, black dotted horizontal lines indicate the time instants corresponding to panels (a--d). (f) Dynamics of $|H_n(t)|$. Parameters: $\sigma = 0.1, \mu = 0, \Delta = 0.06$.
• Figure 3: Alternating chimera state. Nonuniformly twisted state (from the perspective of microscopic dynamics). (a--f) The same as in Fig. \ref{['Fig2']}. Parameters: $\sigma = 0.1, \mu = 0.06, \Delta = 0$.
• Figure 4: Alternating chimera state. Coherent-incoherent-twisted state (from the perspective of microscopic dynamics). (a--f) The same as in Fig. \ref{['Fig2']}. Parameters: $\sigma = 0.1, \mu = 0.06, \Delta = 0.06$.
• Figure 5: Effective interaction kernels between particles with positions $\Tilde{x}$ and $x$ at the time instants $t_k = (\pi/2 + 2\pi k)/\Omega$ ($k = 0,1,2,\ldots$). The argument $y = \Tilde{x}-x$. $G$ denotes the effective kernel for particles on an equidistant grid (green continuous curve), $G_{\mathrm{ms}}$ denotes the interaction kernel between a moving and a stationary particle (blue dashed line). Parameters: $\sigma = 0.1, \mu = 0.06$.