Chimeralike States in an Ensemble of Globally Coupled Oscillators

We demonstrate the emergence of a complex state in a homogeneous ensemble of globally coupled identical oscillators, reminiscent of chimera states in nonlocally coupled oscillator lattices. In this regime some part of the ensemble forms a regularly evolving cluster, while all other units irregularly oscillate and remain asynchronous. We argue that the chimera emerges because of effective bistability, which dynamically appears in the originally monostable system due to internal delayed feedback in individual units. Additionally, we present two examples of chimeras in bistable systems with frequency-dependent phase shift in the global coupling.

Figure 1

Chimera state in model (1). (a) Snapshot of the phases reveals that 64 oscillators (red circles, numbered with $k=1,\dots ,64$) are in the cluster and 36 oscillators (blue squares) belong to the cloud. For visibility, the radial coordinate is increased proportionally to the oscillator index $k$. (b) Temporal evolution ${\phi }_k(t)$, shown by color/grey coding. (c) Instantaneous frequencies of an oscillator from the cluster (upper red curve) and of an oscillator from the cloud (lower blue curve). The average values are $⟨{\dot{\phi }}_{syn}{⟩}_t=1.2897$ (cluster) and $⟨{\dot{\phi }}_{asyn}{⟩}_t=0.9033$ (cloud). (d) Amplitude of the mean field component contributed by the cluster, $R_1=|\sum _{k=1}^{64}{e}^{{i\phi }_k}|/100$ (red bold line), and by the cloud, $R_2=|\sum _{k=65}^{100}{e}^{{i\phi }_k}|/100$ (blue solid line). Black dotted line shows the amplitude $R$ of the total mean field.

Figure 2

Temporal evolution of the cluster and saturation of its size. (a) Growth of the relative cluster size $q=n/N$ for different initial conditions for $N=100$ oscillators. (b) Saturation of $q$ for different ensemble size: $N=250$ (red solid), $N=500$ (blue dashed), $N=750$ (green dash-dotted), and $N=1000$ (black dotted). (c) Standard deviation for the amplitude of the mean field component $R_2$ contributed by the cloud, for different ensemble size $N$.

Figure 3

(a) Approximate domain of chimera states (white region); $\omega$, $\alpha$, and $\beta$ are same as above, $N=256$. Symbol $\times$ marks the parameters used in Figs. 1, 2. In the black domain we observed multi-cluster states, while the gray domain corresponds to the states with zero mean field and equal rotation frequencies for all units. (b) Solution of Eq. (2): frequency of the one-cluster state $\mathrm{\Omega }$ as function of $\tau$, for uncoupled oscillators, $ϵ=0$, (blue dashed line) and for $ϵ=0.05$ (red bold line). Vertical black line marks $\tau =\pi -0.02$.

Figure 4

Chimera state in the system of identical Stuart-Landau-type oscillators in Eq. (6)). (a) Snapshot clearly demonstrates one cluster and a group of asynchronous units. Notice that for visibility, in the plot the amplitudes of all units are substituted as $r_k\to r_k+0.01k$. (b) Mean fields of two subgroups, $X_1=N^{-1}\sum _{j=1}^{N/2}{r}_k\cos {\phi }_j$ (bold black line) and $X_2=N^{-1}\sum _{j=N/2+1}^N{r}_k\cos {\phi }_j$ (solid red line).