Chimera States: The Existence Criteria Revisited

Chimera states, representing a spontaneous breakup of a population of identical oscillators that are identically coupled, into subpopulations displaying synchronized and desynchronized behavior, have traditionally been found to exist in weakly coupled systems and with some form of nonlocal coupling between the oscillators. Here we show that neither the weak-coupling approximation nor nonlocal coupling are essential conditions for their existence. We obtain, for the first time, amplitude-mediated chimera states in a system of globally coupled complex Ginzburg-Landau oscillators. We delineate the dynamical origins for the formation of such states from a bifurcation analysis of a reduced model equation and also discuss the practical implications of our discovery of this broader class of chimera states.

Figure 1

Phase diagram in the $C_1\text{−}K$ space for $C_2=2$. The entire region $S$ to the right of the thick solid curve supports stable synchronous (one-cluster) states. Region I below the dotted curve supports stable splay states while the dot-dashed curve shows the extended limit of the stability of the incoherent states which have nonuniformly distributed phases on the circle. CL-I and CL-II are regions where multicluster states exist with those in CL-II coexisting with the synchronous state. Region $C$ has chaotic states while the gray shaded area is where AMC states exist. Filled circles show typical AMC states coexistent with synchronous states while empty circles are for AMCs in the unstable region of synchronous states. The addition and cross symbols show positions of a typical three-cluster state and a chaotic state, respectively (see Fig. 3).

Figure 2

(a) Snapshots of the profiles of the phase ($\varphi$) and amplitude $|W|$ (multiplied by 10) of an AMC with $K=0.70$, $C_1=-1$, and $C_2=2.0$. The initial conditions (ICs) consist of a splay state. (b) Long-time average of the frequencies $\dot{\varphi }$ of the oscillators. (c) A histogram of the frequencies ($⟨\dot{\varphi }⟩$) in the incoherent segment with a corresponding Gaussian distribution curve.

Figure 3

Snapshots of the distribution of 201 oscillators in the complex plane for different values of $K$ with $C_1=-1.25$ and $C_2=2$ corresponding to the leftmost three points in Fig. 1. As $K$ decreases, two of the three clusters shown in (a) merge to form an AMC state in (b). (c) A further decrease in $K$ leads to merging of all the three one-cluster states leading to chaos.

Figure 4

(a) Time evolution of the real part of $\overline{W}$ along with the magnitude of $\overline{W}$. (b) Power spectrum of the real part of $\overline{W}$. The parameter values are as in Fig. 2.

Figure 5

Bifurcation diagram (adapted from Ref. [47]) of Eq. (2) in $F\text{−}\mathrm{\Omega }$ space, with the Hopf bifurcation line indicated by a solid curve and a dashed curve showing the saddle-node bifurcation line. The thin line between regions II and V is the homoclinic bifurcation line. The filled triangle and the square points are the Takens-Bogdanov bifurcation point and the codimension-two points, respectively. The filled and the empty circle markers correspond to the AMC points shown in Fig. 1. In the legend at the top left corner of the figure, the symbols circled plus, circled times, and circled dot denote a node, a saddle, and an attractive limit cycle, respectively. The filled and the empty diamonds denote stable and unstable spirals, respectively.