Amplitude-mediated chimera states

We investigate the possibility of obtaining chimera state solutions of the nonlocal complex Ginzburg-Landau equation (NLCGLE) in the strong coupling limit when it is important to retain amplitude variations. Our numerical studies reveal the existence of a variety of amplitude-mediated chimera states (including stationary and nonstationary two-cluster chimera states) that display intermittent emergence and decay of amplitude dips in their phase incoherent regions. The existence regions of the single-cluster chimera state and both types of two-cluster chimera states are mapped numerically in the parameter space of $C_1$ and $C_2$, the linear and nonlinear dispersion coefficients, respectively, of the NLCGLE. They represent a new domain of dynamical behavior in the well-explored rich phase diagram of this system. The amplitude-mediated chimera states may find useful applications in understanding spatiotemporal patterns found in fluid flow experiments and other strongly coupled systems.

Figure 1

Snapshots of the stationary state spatial profiles of the phase ($\varphi$) and the amplitude $\left|W\right|$ (multiplied by 10 and shown in blue, the upper curves) for (a) a single cluster AMC with $K=0.40,C_1=-0.5$, $C_2=2.0$; (b) a two-cluster AMC with $K=0.40,C_1=-4.0$, $C_2=0.5$; and (c) a phase-only chimera with $K=0.05,C_1=-0.9$, $C_2=1.0$, respectively. $\kappa$ is fixed at 2 for all the cases. Panels (d)–(f) show the corresponding spatial profiles of the long-time average of the order parameter amplitudes ($R$) and the frequencies ($\omega =\langle \dot{\varphi }\rangle$). The computed value of $\Delta$ is marked with a horizontal dashed line (in black). The vertical dotted lines in the lower three panels (d)–(f) are drawn as a visual guidance to the coherent segments.

Figure 2

(a) The temporal patterns of the amplitude $\left|W\right|$ of an oscillator and amplitude $R$ (the lower curve in blue) of the order parameter. (b) The spatiotemporal pattern of the phases. The parameter values for this simulation are $\kappa =2,K=0.4,C_1=-8$, and $C_2=0.95$.

Figure 3

(a) Stability diagram of the synchronous state ($k=0$, solid curve) and the $k=1$ traveling wave (TW) state (dashed curve) in $C_1-K$ plane with $\kappa =2$ and $C_2=1$. The region below the solid curve and marked $U$ is unstable for both the states, region $S$ is stable for the synchronous state, and $B$ is a bistable region. (b), (c) Stability diagrams similar to (a) but in the phase plane of $C_1-C_2$. $K$ is chosen to be $0.05$ for (b) and $0.40$ for (c). The open circle symbols in (b) mark the phase-only chimeras that are found in the weak coupling limit. The filled circle marks the chimera state shown in Fig. 1c. In (c) the yellow colored region (upper shaded region) shows the existence domain of the single-cluster AMC and the green region (lower shaded region in bistable domain $B$) that of the two-cluster stationary and breather AMCs. The three filled circles mark the positions of the AMCs displayed in Figs. 1a, 1b, and 2.