Stable amplitude chimera in a network of coupled Stuart-Landau oscillators

In many natural systems, attractive coupling together with repulsive coupling plays a vital role in determining their evolutionary dynamics. We investigate the stabilization of amplitude chimera through repulsive coupling in the presence of attractive coupling in a system of nonlocally coupled oscillators. The nonlocal repulsive coupling can facilitate the emergence of stable amplitude chimera even for random initial conditions contrasting with the earlier investigations, where the amplitude chimera was observed just as a transient state and that too for a specific cluster initial conditions. The stability of the observed amplitude chimera is analyzed using Floquet theory. To elucidate the transition among the distinct dynamical states, we find the average number of inhomogeneous oscillators as a function of the coupling strength and show that the transition among the dynamical states exhibits hysteresis. Further, we deduce analytically the critical stability curve that separates the oscillatory (amplitude chimera and traveling wave) states from the death (multi-incoherent oscillation death, cluster chimera death, cluster oscillation death) states. We also analyze the influence of the nonisochronicity parameter and noise on the stable amplitude chimera. We report that the nonisochronicity parameter favors the traveling wave state from incoherent death through the stable amplitude chimera state.

Figure 1

The space-time plot and phase portrait of stable amplitude chimera at $\epsilon =2.02$ for (a) cluster initial conditions and (b) random initial conditions. In (c) and (d) the solid and dotted lines correspond to the homogeneous and inhomogeneous oscillations, respectively. Other parameters: $r_1=0.01, r_2=0.22, \omega =2.0, \lambda =1.0$, and $N=100.$

Figure 2

Floquet multipliers $|{\mu }_i|$ for the stable amplitude chimera state at $\epsilon =2.02, i=1,2,\cdots ,2N$ ($N=100$). The unfilled circles connected by a line represent $|{\mu }_i|$ for random initial conditions, and unfilled triangles with a line denote $|{\mu }_i|$ for cluster initial conditions.

Figure 3

The space-time plots of the variable $y_i$: (a) Traveling-wave state for $\epsilon =1.9$, (b) amplitude chimera for $\epsilon =2.02$, (c) multi-incoherent death state for $\epsilon =2.6$; (d) and (e) correspond to the center of mass of the left panel (a)–(c), respectively. Other parameter values are the same as in Fig. 1.

Figure 4

Two-parameter bifurcation diagram in ($r_2,\epsilon$) space for 100 realizations of distinct initial conditions. TW, AC, and MIOD represent the traveling wave, amplitude chimera, and multi-incoherent oscillation death state, respectively. $R_1, R_2$, and $R_3$ denote the multistability regions of TW-AC, TW-AC-MIOD, and AC-MIOD, respectively. Other parameters: $r_1=0.01, \lambda =1.0, \omega =2.0$, and $N=100$.

Figure 5

Probability of occurrence of TW, AC, and MIOD states with respect to the coupling strength $\epsilon$ for 100 realization of distinct cluster and random initial conditions. Other parameters: $r_1=0.01, r_2=0.22, \omega =2.0, \lambda =1.0$, and $N=100$.

Figure 6

The average number of oscillators in the inhomogeneous state ($K$) as a function of the coupling strength $\epsilon$ for the coupling range (a) $r_2=0.16$, (b) $r_2=0.28$, (c) $r_2=0.34$, and (d) $r_2=0.40$. Lines connected by open circles (blue) and stars (red) represent forward and reverse transitions, respectively. Other parameters: $r_1=0.01, \lambda =1.0, \omega =2.0$, and $N=100$.

Figure 7

Space-time plots of a dynamical transition along forward [panel (A)] and reverse [panel (B)] transition for $r_2=0.16$. Panel (A): (a) traveling wave state for $\epsilon =1.76$; (b)–(d) amplitude chimera state for $\epsilon =1.77, \epsilon =1.85$, and $\epsilon =1.9$; (e) multi-incoherent oscillation death for $\epsilon =2.0$. Panel (B): (a) multi-incoherent oscillation death for $\epsilon =2.0$; (b)–(d) amplitude chimera state for $\epsilon =1.9, \epsilon =1.85$, and $\epsilon =1.77$; (e) traveling wave state for $\epsilon =1.76$.

Figure 8

The width of the hysteresis as a function of the nonlocal coupling range $r_2$. The unfilled circles (black) denote the numerical data.

Figure 9

The space-time plots and snapshots of the variables $y_i$ as a function of coupling strength $\epsilon$ displaying (a),(b) amplitude chimera for $\epsilon =1.7$, (c),(d) cluster oscillation death (COD) for $\epsilon =2.0$, and (e),(f) cluster chimera death (CCD) for $\epsilon =2.3$. Other parameters: $r_1=0.4, r_2=0.34, \lambda =1.0, \omega =2.0$, and $N=100$.

Figure 10

Two-parameter diagrams in $(r_2,\epsilon )$ space, for the coupling ranges (a) $r_1=0.2$ and (b) $r_1=0.4$. Parts (c) and (d) show an exponential decrease of cluster size for $r_1=0.2$ and $r_1=0.4$ as a function of the coupling range $r_2$ for $\epsilon =1.98$. The corresponding power-law fit is shown in the inset in logarithmic scale. Unfilled circles represent the numerical data, and the corresponding best fit is shown by a red solid line.

Figure 11

Two-parameter phase diagram in ($r_1,r_2$) space for $\epsilon =1.7$. TW, AC, IOD, and SYN denote the traveling wave, amplitude chimera, incoherent oscillation death, and synchronized state, respectively. Other parameters are the same as in Fig. 9.

Figure 12

Stability curves of IOD regions for even and odd values of $P_1(r_1=\frac{P_1}{N})$ as a function of $P_2(r_2=\frac{P_2}{N})$. (a) $P_1=40$, (b) $P_1=5$, and (c) $P_1=45$. The shaded region corresponds to the numerically obtained IOD region.

Figure 13

The space-time plots and time average of the variable $y_i$ of each of the oscillators: (a),(b) MIOD for $c=0.1$, (c),(d) AC for $c=0.5$, and (e),(f) TW state for $c=0.8$. Other parameters: $r_1=0.01, r_2=0.22, \epsilon =2.7, \omega =2.0, \lambda =1.0$, and $N=100.$

Figure 14

Two-parameter bifurcation diagram in $(c,\epsilon )$ space. TW, AC, and MIOD represent the traveling wave, amplitude chimera, and multi-incoherent oscillation death state, respectively. Other parameter values are the same as in Fig. 1.

Figure 15

(a) Transient time ($t_{\text{tr}}$) of distinct dynamical states as a function of nonisochronicity parameter $c$. Transient amplitude chimera at $\epsilon =2.7$ for the nonisochronicity parameter, (a) $c=0.1$ and (b) $c=0.65$.

Figure 16

Space-time plot for $\epsilon =4.1$ for (a) $c=1.0$, imperfect breathing chimera I and (b) $c=2.5$, imperfect breathing chimera II. Other parameter values are the same as in Fig. 1.

Figure 17

Two-parameter bifurcation diagram in $(r_2,\epsilon )$ space for the noise intensity (a) $D=0.001$ and (b) $D=0.1$. TW, AC, and MIOD represent the traveling wave, amplitude chimera, and multi-incoherent oscillation death state, respectively. Other parameter values are the same as in Fig. 4.

Figure 18

Transient time of AC as a function of noise intensity $D$ for (a) $r_2=0.16$ and $\epsilon =1.8$, (b) $r_2=0.28$ and $\epsilon =1.9$. Other parameters are the same as in Fig. 4.

Figure 19

Snapshots of stable amplitude chimera emerging from the network of different sizes for $\epsilon =1.7$. (a) $N=500$, (b) $N=1000$, (c) $N=2500$, and (d) $N=5000$. Other parameters are the same as in Fig. 9.