Defect states as a precursor of the chimera states in a ring of non-locally coupled oscillators

TL;DR

The paper investigates how synchronized states transition to chimera states in a ring of non-locally coupled phase oscillators with phase delay $α$. It introduces defect states, traveling solitary waves in the phase gradient $Δ \hat{φ}_x$, as dynamical precursors to chimera nucleation and uses a large-ensemble statistical framework to classify states via the local order parameter $z_{loc}$, incoherence $ρ$, and $Δ \hat{φ}_x$ distributions. Key findings show that defect states appear prior to chimera formation, with defect fractions peaking near the crossover at $α ≈ 0.44$ and that defect dynamics can be largely independent of system size $N$ for certain $α$, while defect width $L_0$, number $n$, and total width $L_{total}$ depend on the coupling range $R$; satellite peaks in the phase-gradient distribution reveal structured defect dynamics. The results highlight a defect-mediated mechanism for chimera emergence, explain deviations from 1D directed percolation universality, and emphasize the role of the local coupling scale in defining the precursor's spatial extent, with implications for continuum limits and higher-dimensional systems.

Abstract

We investigate the transition from synchronized to chimera states in a ring of non-locally coupled phase oscillators. Our focus is on the intermediate defect states, where solitary waves in the phase gradient profile travel at a constant speed. These traveling defects serve as a dynamical precursor for the nucleation of chimera clusters. The fraction of samples exhibiting defect states increases with the phase delay $α$ and peaks at $α_{c}$, where the system crosses over to asynchronous states filled with chimera clusters. While the traveling speed, number, and width of these defects increase with $α$, the total spatial extent of the defects remains robust against the system size $N$. These results shed new light on the emergence of chimera states in frustrated coupled oscillators.

Figures (6)

• Figure 1: Spatiotemporal maps (left column) and spatial profiles of $\Delta\hat{\phi}_{x}$ at $t=9800$ (right column) for $N=1000$ and $R=5$. (a), (b) $\alpha=0.4$: defect states. (c), (d) $\alpha=0.437$: mixture of defect states and chimera states. (e), (f) $\alpha=0.45$: chimera states. The inset of (b) and (d) shows magnified plots with dotted lines in (b) corresponding to $\Delta \hat{\phi}_{x} =-0.004, -0.003$.
• Figure 2: Time evolution and time-dependent statistics of the incoherence parameter $\rho$. For $N=1000$ and $R=5$: (a) time evolution of $\rho$ for 13 different samples with $\alpha$ ranging from 0.36 to 0.48; (b), (c), (d), (e) time average and time standard deviation of the incoherence parameter, $\langle\rho\rangle_{t}$ and $\sigma_t(\rho)$, for 1000 samples with different random initial configurations. The statistics are taken within the time window $t=9000\text{--}10000$. Panels (b) and (c) show results for $\alpha = 0.40$ and $0.44$, respectively; the dotted lines indicate the theoretical values of $\rho$ for $q$-twisted states with $q=1\text{--}7$.
• Figure 3: Statistical characteristics of defects and state transitions for $R=5$ and $N=1000$. (a) Normalized distribution $\text{Prob}(\Delta \hat{\phi}_x)$ of the phase gradient $\Delta \hat{\phi}_{x}$, computed from 1000 samples for each $\alpha$ within the time window $t = 9000\text{--}10000$. (b) Fraction of samples in three different system states: coherent (red), defect (blue), and asynchronous (green). The results are shown for three consecutive time intervals: $t = 5000\text{--}10000$ (early stage, solid lines), $10000\text{--}15000$ (middle stage, dashed lines), and $15000\text{--}20000$ (late stage, dotted lines).
• Figure 4: Sample average traveling speed $\langle v \rangle_s$ of defects as a function of $\alpha$ for $R=5$. The statistics are calculated from all defect-state samples identified within an ensemble of 1000 samples for each $N$ and $\alpha$. The average is taken over the time window $t = 9000\text{--}10000$. Results for three different system sizes are shown: $N=500$ (red squares), $N=1000$ (blue circles), and $N=2000$ (green triangles).
• Figure 5: Statistical characteristics of defect width and number for $R=5$. (a) Normalized distribution $\text{Prob}(L_0)$ of the individual defect width $L_0$. (b) Sample average of the individual defect width $\langle L_0 \rangle_s$, (c) sample average number of defects $\langle n \rangle_s$, and (d) sample average of the total defect width $\langle L_{\rm total} \rangle_s$, plotted as functions of $\alpha$. Statistics are computed from all samples identified as defect states within an ensemble of 1000 samples for each $N$ and $\alpha$.