Chimera States: The Natural Link Between Coherence and Incoherence

Chimera states are remarkable spatiotemporal patterns in which coherence coexists with incoherence. As yet, chimera states have been considered as nongeneric, since they emerge only for particular initial conditions. In contrast, we show here that in a network of globally coupled oscillators delayed feedback stimulation with realistic (i.e., spatially decaying) stimulation profile generically induces chimera states. Intriguingly, a bifurcation analysis reveals that these chimera states are the natural link between the coherent and the incoherent states.

Figure 1

Chimera states induced by the delayed feedback with the exponential stimulation profile (3). Parameters: $a=1.0$, $C=0.1\pi$, $K=\pi$, $\omega =2\pi$, and $\tau =0.3$ (a) or $\tau =0.6$ (b). Equation (2) was approximated with an array of 101 oscillators uniformly distributed over the interval $[-1,1]$ and was integrated using a Runge-Kutta method starting from the coherent initial condition. Top: Snapshot of the oscillators’ phases (black dots). Middle: Stimulation profile. Bottom: Effective frequencies of the oscillators (black dots). Green thick lines in the top and bottom graphs represent the theoretical predictions obtained from Eqs. (6, 7, 8).

Figure 2

Order parameter $R$ from Eq. (7). Parameters: $C/\omega =0.05$, $K/\omega =0.5$, and exponential stimulation profile (3) with $a=1.0$. Solid lines represent stable branches of the solution: phase-locked state (arcs $A_1{B}_1$ and $A_2{B}_2$), chimera  I (arc $B_1{C}_1$) and chimera II (arc $B_2{C}_2$). Dashed lines $C_1{D}_1$ and $C_2{D}_2$ stand for unstable branches. The completely incoherent state is stable for delays taken from the interval $D_1{D}_2$.

Figure 3

(a),(b) Domains in the control parameters plane with one (yellow) or two (orange) stable chimera states. The incoherent state is stable inside the regions marked by blue lines. At least one stable coherent solution exists in the region outside of the green lines (see enlargement in Fig. 4). Parameters: exponential stimulation profile with $a=1.0$ and $C/\omega =0.05$ (a) or $C=0$ (b). Synchronization parameter of the chimera I (c) and chimera II (d) belonging to (a).

Figure 4

The enlarged left third of Fig. 3a shows the linkage between incoherent states (blue) and coherent states (violet). The incoherent state is stable in the blue region. At least one stable coherent solution exists in the violet region. There is a small white gap between the blue and violet region even at the bottom at $\tau /T\approx 0.5$: Incoherent and coherent states are not directly transformed into one another. Numbers in the plot indicate different types of stable solutions: incoherence (1), incoherence and chimera (2), incoherence and two chimeras (3), chimera (4), two chimeras (5), coherent solution (6), coherent solution and chimera (7), two coherent solutions (8), two coherent solutions and chimera (9), three coherent solutions (10), two coherent solutions and two chimeras (11).