Clustering as a Prerequisite for Chimera States in Globally Coupled Systems

The coexistence of coherently and incoherently oscillating parts in a system of identical oscillators with symmetrical coupling, i.e., a chimera state, is even observable with uniform global coupling. We address the question of the prerequisites for these states to occur in globally coupled systems. By analyzing two different types of chimera states found for nonlinear global coupling, we show that a clustering mechanism to split the ensemble into two groups is needed as a first step. In fact, the chimera states inherit properties from the cluster states in which they originate. Remarkably, they can exist in parameter space between cluster and chaotic states, as well as between cluster and synchronized states.

Figure 1

Evolutions in the complex plane and snapshots. Trajectories of the oscillators are shown as solid lines, whereas the symbols describe snapshots of the system. First row: type I dynamics. (a) Amplitude clusters ($\eta =0.9$). (b) Type I chimera ($\eta =1.02$), black lines and squares: incoherent group; cyan (gray) lines and circles: coherent group. (c) Complete synchronization ($\eta =1.2$). Other parameters: $c_2=0.58$ and $\nu =1.49$. Second row: type II dynamics. (d) Irregular dynamics ($\nu =-0.1$). (e) Type II chimera ($\nu =0.02$), black lines and squares: incoherent group; cyan (gray) lines and circles: coherent group. (f) Modulated amplitude clusters ($\nu =0.1$). Other parameters: $c_2=-0.6$, $\eta =0.7$.

Figure 2

(a),(c) Type I and II chimera states, modulus and real part versus time, respectively. The population splits into two groups, one being synchronized (cyan, gray) and one being desynchronized (black). (b),(d) Linear average, $\mathrm{Re}⟨W⟩$, (dashed lines) and nonlinear average, $\mathrm{Re}⟨{|W|}^{2}W⟩$, (solid lines) versus time for type I chimeras (b) and type II chimeras (d).

Figure 3

Chaos in the chimera states. (a),(c) Samples of time series of incoherent oscillators for type I and II chimeras, respectively. Identified peaks are marked with circles. (b),(d) Next-maximum maps for the peaks in (a),(c).

Figure 4

Heteroclinic connections between type I chimeras, 1-3 cluster states and amplitude clusters for $N=100$ oscillators. (a) Trajectory of $|\sigma |$ in time showing the transitions between the different states. (b) Exemplary dynamics of the 1-3 cluster state in the complex plane: lines depict time evolution and dots represent the configuration of the oscillators at one time step. (c) Histogram of phases in the 1-3 clusters state showing that it consists of 1 large cluster and 3 small clusters of approximately the same size.