Imperfectly synchronized states and chimera states in two interacting populations of nonlocally coupled Stuart-Landau oscillators

We investigate the emergence of different kinds of imperfectly synchronized states and chimera states in two interacting populations of nonlocally coupled Stuart-Landau oscillators. We find that the complete synchronization in population I and existence of solitary oscillators which escape from the synchronized group in population II lead to imperfectly synchronized states for sufficiently small values of nonisochronicity parameter. Interestingly, upon increasing the strength of this parameter further there occurs an onset of mixed imperfectly synchronized states where the solitary oscillators occur from both the populations. Synchronized oscillators from both the populations are locked to a common average frequency. In both cases of imperfectly synchronized states, synchronized oscillators exhibit periodic motion while the solitary oscillators are quasiperiodic in nature. In this region, for spatially prepared initial conditions, we can observe the mixed chimera states where the coexistence of synchronized and desynchronized oscillations occur from both the populations. On the other hand, imperfectly synchronized states are not always stable, and they can drift aperiodically due to instability caused by an increase of nonisochronicity parameter. We observe that these states are robust to the introduction of frequency mismatch between the two populations.

Figure 1

Space-time plots of the variables $x_j^{(1,2)}$ for imperfectly synchronized state (a) for population I and (b) for population II. Corresponding oscillator average frequencies of (c) population I and (d) population II. Parameter values: $c=2.3, \sigma =0.1, \eta =0.25, \omega =1.0$, and $r=0.1$.

Figure 2

Phase portraits of the oscillators: (a) periodic oscillation of synchronized oscillator $z_1^{\left(1\right)}(=x_1^{\left(1\right)}+i{y}_1^{\left(1\right)})$. (b) Periodic motion of the synchronized oscillator $z_1^{\left(2\right)}$ and quasiperiodic oscillation of solitary oscillators $z_3^{\left(2\right)}$. Their corresponding Poincaré surfaces of section: (c),(d) the red or gray dot represents the periodic oscillation of synchronized oscillator and black dots represent the quasiperiodic oscillation of solitary oscillator. Parameter values: $c=2.3, \sigma =0.1, \eta =0.25, \omega =1.0$, and $r=0.1$.

Figure 3

(a) and (b) Phase portraits of the solitary oscillators $z_{17}^{\left(1\right)}$ and $z_{21}^{\left(2\right)}$ which show the variation in the centers of rotation.

Figure 4

Space-time plots of the variables $x_j^{(1,2)}$ for mixed imperfectly synchronized state (a) for population I and (b) for population II. Corresponding oscillator average frequencies of (c) population I and (d) population II. Parameter values: $c=5, \sigma =0.1, \eta =0.25, \omega =1.0$, and $r=0.1$.

Figure 5

Phase portraits of the oscillators: (a) periodic oscillation of synchronized oscillator $z_1^{\left(1\right)}$ and quasiperiodic oscillation of solitary oscillator $z_6^{\left(1\right)}$. (b) Periodic motion of the synchronized oscillator $z_1^{\left(2\right)}$ and quasiperiodic oscillation of solitary oscillator $z_7^{\left(2\right)}$. Their corresponding Poincaré surfaces of section: in (c) and (d) red or gray dot represents the periodic oscillation of synchronized oscillator, and black dots represent the quasiperiodic oscillation of solitary oscillator. Parameter values: $c=5, \sigma =0.1, \eta =0.25, \omega =1.0$, and $r=0.1$.

Figure 6

Space-time plots of the variables $x_j^{(1,2)}$ for mixed chimera state for spatially prepared initial conditions for (a) population I and (b) population II. Other parameter values are $c=5, \sigma =0.1, \eta =0.25, \omega =1.0$, and $r=0.1$.

Figure 7

Space-time plots of the variables $x_j^{(1,2)}$ (a) for population I and (b) for population II. Snapshots for the variables $x_j^{\left(2\right)}$ of population II for the drifted imperfectly synchronized state [at two different times which are marked by the white line in Fig. 7] (c) $t=20$, (b) $t=35$. The other parameter values are $c=9, r=0.1, \sigma =0.1, \eta =0.15, \omega =1.0$.

Figure 8

Strength of the incoherence plot for three different imperfectly synchronized states with 10 bins of time: (a) first population and (b) second population, where ‘$\circ$’ represents the imperfectly synchronized state, ‘$•$’ the mixed imperfectly synchronized state, and ‘$▴$’ the drifted imperfectly synchronized state. The other parameter values are $c=9, r=0.1, \sigma =0.1, \eta =0.15$, and $\omega =1.0$.

Figure 9

Phase diagram of the system (1) in ($\eta ,c$) space for $r=0.1, \sigma =0.1$, and $\omega =1.0$. The regions $I, I{I}_1, I{I}_2, I{I}_3$, and $IV$ represent individual synchronization, the imperfectly synchronized state, the mixed imperfectly synchronized state, the drifted imperfectly synchronized state, and the globally synchronized state, respectively. Both the regions $II{I}_1$ (the cluster states are having oscillators within the populations) and $II{I}_2$ (the cluster states are having oscillators from both the populations) represent the cluster synchronized state. Here ‘$•$’, ‘$\circ$’, ‘$\blacksquare$’, ‘$▴$’, and ‘$⧫$’ mark the parameter values corresponding to Figs. 1, 4, 7, 10 and 10, and 10 and 10, respectively.

Figure 10

Time evolution for the variables $x_j^{(1,2)}$ of the cluster states for (a) population I and (b) population II with $c=1.5, \eta =0.35$ and for (c) population I and (d) population II with $c=3.3, \eta =0.8$. Other parameter values: $\sigma =0.1, \omega =1.0$. Snapshots for the variables $x_j^{(1,2)}$ of the cluster states are shown in insets of the corresponding figures.

Figure 11

Phase diagram of the system (1) in $(\eta ,r)$ space for $c=5, \sigma =0.1, \omega =1.0$. The regions $I, I{I}_2, II{I}_2$, and $IV$ represent individual synchronization, the mixed imperfectly synchronized state, the cluster synchronized state, and the globally synchronized state, respectively.

Figure 12

Space-time plots for the imperfectly synchronized state with frequency mismatch $\varepsilon =0.5$ (${\omega }_1=1.0$ and ${\omega }_2=1.5$) (a) for population I and (b) for population II with $c=2.3$ and corresponding average frequency of the oscillators in the imperfectly synchronized state (c) for population I and (d) population II. Other parameter values: $\sigma =0.1$ and $\eta =0.3.$

Figure 13

Space-time plots for the mixed imperfectly synchronized state with frequency mismatch $\varepsilon =0.5$ (${\omega }_1=1.0$ and ${\omega }_2=1.5$) (a) for population I and (b) for population II with $c=5$ and corresponding average frequency of the oscillators in the mixed imperfectly synchronized state (c) for population I and (d) population II. Other parameter values: $\sigma =0.1$ and $\eta =0.3$.

Figure 14

Space time plots for the imperfectly synchronized state for frequency mismatch $\varepsilon =1.7$ (${\omega }_1=1.0$ and ${\omega }_2=2.7$): (a) for population I and (b) for population II with $c=5$. Space time plots for the mixed imperfectly synchronized state for frequency mismatch $\varepsilon =1.7$ (${\omega }_1=1.0$ and ${\omega }_2=2.7$): (c) for population I and (d) for population II with $c=6$.

Figure 15

Phase diagram of the system (1) in the presence of frequency mismatch $\varepsilon$ (${\omega }_1=1.0$ and ${\omega }_2={\omega }_1+\varepsilon$) between the populations for the system parameter values $r=0.1, \sigma =0.1, c=5$. The regions $I, I{I}_1,I{I}_2,II{I}_2$, and $IV$ represent individual synchronization, the imperfectly synchronized state, the mixed imperfectly synchronized state, the cluster synchronized state, and the globally synchronized state, respectively. Here ‘$\blacksquare$’ and ‘$▴$’ mark the parameter values corresponding to Figs. 13–13 and Figs. 14 and 14, respectively.