Abrupt symmetry-preserving transition from the chimera state

We consider two populations of the globally coupled Sakaguchi-Kuramoto model with the same intra- and interpopulations coupling strengths. The oscillators constituting the intrapopulation are identical whereas the interpopulations are nonidentical with a frequency mismatch. The asymmetry parameters ensure the permutation symmetry among the oscillators constituting the intrapopulation and a reflection symmetry among the oscillators constituting the interpopulation. We show that the chimera state manifests by spontaneously breaking the reflection symmetry and also exists in almost in the entire explored range of the asymmetry parameter without restricting to the near $\pi /2$ values of it. The saddle-node bifurcation mediates the abrupt transition from the symmetry breaking chimera state to the symmetry-preserving synchronized oscillatory state in the reverse trace, whereas the homoclinic bifurcation mediates the transition from the synchronized oscillatory state to synchronized steady state in the forward trace. We deduce the governing equations of motion for the macroscopic order parameters employing the finite-dimensional reduction by Watanabe and Strogatz. The analytical saddle-node and homoclinic bifurcation conditions agree well with the simulations results and the bifurcation curves.

Figure 1

The coexistence of the reflection symmetry preserving and reflection symmetry breaking self-organizing collective dynamical states. (a) and (b) The order parameters $r_{1,2}\left(t\right)$ corresponding to the two populations depicting the coexisting synchronized state and the static chimera state, respectively. (c) and (d) The order parameters ${\varphi }_{1,2}\left(t\right)$ depicting the reflection symmetry $[{\varphi }_1\left(t\right)=-{\varphi }_2\left(t\right)]$ preserving oscillatory nature of the synchronized state and the reflection symmetry broken $[{\varphi }_1\left(t\right)\ne -{\varphi }_2\left(t\right)]$ static chimera state, respectively. The values of the parameters are $\omega =2,K=0.7$, and $\alpha =0.3$.

Figure 2

Same as Fig. 1 for the coupling strength $K=1.5$. Here, the synchronized state is a stable steady state.

Figure 3

Phase space dynamics of (a) the oscillatory synchronized state for $K=0.4$, (b) the static synchronized state for $K=2.0$, and (c) the static chimera state for $K=2.0$. The unfilled inverted triangle corresponds to the unstable origin. The dashed and solid lines are the trajectories from two different initial conditions. Lines connected by filled triangles and squares correspond to the attractors of the first and second populations, respectively. The value of the other parameters are the same as in Fig. 1.

Figure 4

One parameter bifurcation diagrams, obtained from the evolution equation for the mean-field variable $z=X+iY$ using xppaut, for the asymmetry parameter (a) $\alpha =0.1$, (b) $\alpha =0.3$, and (c) $\alpha =1.0$ for $\omega =2.0$. The corresponding time averaged order parameter $R$ corroborating the dynamical transition is depicted in (d)–(f), respectively. Top row: The lines connected by filled circles correspond to the synchronized (OS) state, whereas the dashed (blue) and solid (red) lines for $K>K_{\mathrm{Hc}}$ correspond to the synchronized steady state in the forward trace. Dotted lines correspond to the unstable steady states. The lines connected by the filled triangles correspond to the synchronized steady state of the first population, whereas the lines connected by the filled squares correspond the symmetry breaking steady state of the second population in the reverse trace. Note that there is a transition from reflection symmetry breaking steady state to the symmetry preserving synchronized oscillatory state of the second population at $K=K_{SN}$. Bottom row: Forward trace is represented by lines connected by unfilled symbols, whereas the reverse trace is represented by lines connected by filled symbols. The lines connected by triangles illustrate the dynamical transitions of the first population, whereas the lines connected by squares elucidate the dynamical transitions of the second population. Note that there is an abrupt transition from the symmetry breaking state to the symmetry preserving state in the entire explored range of the asymmetry parameter. Solid and dashed vertical lines are the analytical critical curves corresponding to the homoclinic and saddle-node bifurcations, respectively. M1 and M2 are the bistable regions.

Figure 5

Two phase diagram on the ($K,\omega$) plane for the asymmetry parameter (a) $\alpha$=0.3 and (b) $\alpha =1$. The diagonal lines represent the symmetry preserving synchronized oscillatory state, the gray shaded region represents the symmetry breaking chimera state and the checked region represent the symmetry preserving synchronized steady state. The dashed and solid lines are the analytical critical curves corresponding to the saddle-node and homoclinic bifurcations, respectively.

Figure 6

Two phase diagram on the ($K,\alpha )$ plane for $\omega$=2. The collective dynamical states and the bifurcation transitions are similar to those observed in Fig. 5.

Figure 7

The two phase diagram for the ($A-\omega$) plane for asymmetry parameter, (a) $\alpha$=0.8 and (b) $\alpha$=1.5. Only the symmetry-preserving synchronized oscillatory state (indicated by diagonal lines) and steady state (indicated by checked region) is observed in the entire two parameter phase diagram.