Chimeralike states in two distinct groups of identical populations of coupled Stuart-Landau oscillators

We show the existence of chimeralike states in two distinct groups of identical populations of globally coupled Stuart-Landau oscillators. The existence of chimeralike states occurs only for a small range of frequency difference between the two populations, and these states disappear for an increase of mismatch between the frequencies. Here the chimeralike states are characterized by the synchronized oscillations in one population and desynchronized oscillations in another population. We also find that such states observed in two distinct groups of identical populations of nonlocally coupled oscillators are different from the above case in which coexisting domains of synchronized and desynchronized oscillations are observed in one population and the second population exhibits synchronized oscillations for spatially prepared initial conditions. Perturbation from such spatially prepared initial condition leads to the existence of imperfectly synchronized states. An imperfectly synchronized state represents the existence of solitary oscillators which escape from the synchronized group in population I and synchronized oscillations in population II. Also the existence of chimera state is independent of the increase of frequency mismatch between the populations. We also find the coexistence of different dynamical states with respect to different initial conditions, which causes multistability in the globally coupled system. In the case of nonlocal coupling, the system does not show multistability except in the cluster state region.

Figure 1

Space-time plots of the variables $x_j^{(1,2)}$ depicting (a) local synchronization for $\varepsilon =0.1$, (b) local cluster synchronization for $\varepsilon =0.8$, (c) chimeralike state for $\varepsilon =1.1$, (d) global cluster synchronization for $\varepsilon =1.8$, and (e) global synchronization for $\varepsilon =2.8$. In all these cases we have chosen $\mathrm{\Delta }\omega =0.5$ and $c=3.0$. Panels (f)–(j) represent the corresponding snapshots of the variables $x_j$ for panels (a)–(e). Note that in the above, $x_j^{(1,2)}, j=1,2...,100$ correspond to population I, that is, $x_j^{\left(1\right)}$, and $x_j^{(1,2)}, j=101,102,...,200$ correspond to population II, that is, $x_j^{\left(2\right)}$. One can obtain similar plots for $y_j^{(1,2)}$ also, which are not presented here.

Figure 2

Phase portraits of the representative oscillators in chimeralike states: (a) For synchronized oscillator ($z_{105}$) with segment ${\mathrm{\Lambda }}_{105}$ [red (gray) dots], (b) for desynchronized oscillator ($x_5$) with data set $A_5$ [red (gray) dots], (c) for desynchronized oscillator ($x_{55}$) with segment ${\mathrm{\Lambda }}_{55}$ [red (gray) dots], and (d) for desynchronized oscillator ($z_{51}$) with data set $A_{51}$ [red (gray) dots].

Figure 3

A. (a) Average frequencies of the oscillators as a function of the coupling strength for the parameter values $\mathrm{\Delta }\omega =0.5, c=3.0$. Enlarged structures for a specific value of $\varepsilon =2.05$: (b) Average frequencies of the synchronized oscillators (population II). (c) Frequencies of the desynchronized oscillators (population I). B. Corresponding to the evolution in the complex plane, snapshots of the variables $(x_j^{(1,2)},y_j^{(1,2)})$: (i)-a and -b correspond to desynchronization for $\varepsilon =0.01$ (Region I), (ii)-a and -b correspond to local synchronization for $\varepsilon =0.1$ (Region II), (iii)-a and -b represent local cluster states for $\varepsilon =0.8$ (Region III), (iv)-a and -b represent the chimeralike states for $\varepsilon =1.1$ (Region IV), (v)-a and -b represent the global cluster states for $\varepsilon =1.8$ (Region V), and (vi)-a and -b correspond to global synchronization for $\varepsilon =2.8$ (Region VI). Trajectories are shown by black solid lines, and red (gray) dots represent the snapshots of the variables $z_j^{(1,2)}$ in the complex plane.

Figure 4

The two-parameter phase diagram in the ($\varepsilon ,c$) parametric space: (A) for $\mathrm{\Delta }\omega =0.5$ and (B) for $\mathrm{\Delta }\omega =1.5$. Boundary ${\varepsilon }_c$ is obtained from the critical value of the coupling strength given in Eq. (5). The curve ${\varepsilon }_c^{\prime }$ is obtained from the Eq. (6), and the other boundaries are obtained numerically. Region LS represents the local synchronization, region GS represents the global synchronization, region LC indicates the local cluster states, region CS represents the chimeralike states, and region GC represents the global cluster states. Region I is the multistability region between the local synchronization and local cluster states. Region ${\mathrm{II}}_1$ is the multistability region between complete synchronization and local cluster states. Region ${\mathrm{II}}_2$ is the multistability region between complete synchronization and chimera states. Region ${\mathrm{II}}_3$ is the multistability region between complete synchronization and global cluster states. Panels (a)–(e) present the global synchrony order parameter R [blue (dark-gray) solid curve] and local order parameter $r^{\left(1\right)}$ (red dashed curve) and $r^{\left(2\right)}$ (black solid curve) for the local synchronization, local cluster states, chimeralike states, global cluster states, and global synchronization, respectively.

Figure 5

Phase diagrams for globally coupled system (1) by varying the values of $\mathrm{\Delta }\omega$ and $\varepsilon$ (a) for $c=1.5$ and (b) for $c=3.0$. Boundary ${\varepsilon }_c$ is obtained from the critical value of the coupling strength given in Eq. (5). The curve ${\varepsilon }_c^{\prime }$ is obtained from Eq. (6), and the other boundaries are obtained numerically. The region DS represents the desynchronized state, LS is the local synchronization region, GS shows the globally synchronized state, GC shows the state corresponding to global cluster region, CS shows chimera states, and LC shows the local cluster state. Region I is the multistability region between the local synchronization and local cluster states.

Figure 6

Log-log plots: (a) The value of $n$ (=$\frac{M_1}{M_2}$) as a function of the nonisochronicity parameter $c$ for fixed $\varepsilon$ values ($\varepsilon =0.7$ for local synchronization, $\varepsilon =0.9$ for local cluster states, $\varepsilon =1.5$ for chimeralike states, and $\varepsilon =2.3$ for global cluster states) and (b) the value of $n$ (=$\frac{M_1}{M_2}$) as a function of the coupling strength $\overline{\varepsilon }$ for a fixed $c=3.0$. Note that $\overline{\varepsilon }=\varepsilon -{\varepsilon }_e$, where ${\varepsilon }_e$ is the earliest coupling strength. Dots ($•$) represent the numerical data, and the corresponding best fit is represented by the (black) curve for the chimeralike states. Similarly the local synchronization, local cluster states, and global cluster states are represented by the solid line with ($▴$), ($⧫$), and ($\blacksquare$), respectively. Note that ${\varepsilon }_e=0.05$ for local synchronization, ${\varepsilon }_e=0.8$ for local cluster state, ${\varepsilon }_e=1.4$ for chimeralike state, and ${\varepsilon }_e=2.25$ for global cluster state. Other parameter value: $\mathrm{\Delta }\omega =0.5$

Figure 7

Space-time plots of the variables $x_j^{(1,2)}$: (a) local synchronization for $\varepsilon =0.2$, (b) imperfectly synchronized state (solitary state) for $\varepsilon =0.35$, (c) cluster state for $\varepsilon =0.4$, and (d) global synchronization for $\varepsilon =1.1$ with $c=3.0, r=0.4$, and $\mathrm{\Delta }\omega =0.5$. Panels (e)–(h) represent the snapshots of the variable $x_j^{(1,2)}$ for panels (a)–(d). Panels (i)–(l) represent the corresponding frequency profiles of the oscillators for panels (a)–(d). Again here oscillators $j=1,2,...,100$ correspond to group I ($x_j^{\left(1\right)}$) while oscillators numbered $j=101,102,...,200$ represent group II ($x_j^{\left(2\right)}$).

Figure 8

Phase portraits of the representative oscillators in imperfectly synchronized state: (a) Represents the nonphase-coherent oscillation of the solitary oscillator ($z_3$) in population I. (b) Represents the periodic oscillation of the oscillator ($z_{102}$) in population II.

Figure 9

Two phase diagrams in the ($\varepsilon ,c$) parametric space (a) for $\mathrm{\Delta }\omega =0.5$ and (b) for $\mathrm{\Delta }\omega =1.5$ with $r=0.4$ for the two population nonlocally coupled Stuart-Landau oscillators. DS represents the desynchronized region, LS represents the local synchronization region, IS shows the imperfectly synchronized states, CL is for cluster state region, and GS represents the globally synchronized region.

Figure 10

Two phase diagrams in the ($\varepsilon ,\mathrm{\Delta }\omega$) parametric space (a) for $c=1.5$ and (b) for $c=3.0$ with $r=0.4$ for a system of two distinct identical groups of nonlocally coupled Stuart-Landau oscillators. DS represents the desynchronized region, LS represents the local synchronization, IS shows the imperfectly synchronized states, CL is for cluster state region, and GS represents the globally synchronized region.

Figure 11

Two phase diagrams in the ($\varepsilon ,r$) parametric space (a) for $\mathrm{\Delta }\omega =0.5$ (b) for $\mathrm{\Delta }\omega =1.5$ with $c=3.0$ for the system (7). DS represents the desynchronized region, LS represents the local synchronization, IS shows the imperfectly synchronized states, CL is for cluster state region, and GS represents the globally synchronized region.