Fig. 1. (a) Phase pattern for a chimera state, parameters: oscillators; (b) local phase coherence , computed from Eq. (3), locked oscillators satisfy ; (c) local average phase . Figure adapted from Ref. [20].

Fig. 2. Order parameter r versus time. In all three panels, and : (a) , stable chimera; (b) , breathing chimera; (c) , long period breather. Numerical integration began from an initial condition close to the chimera state, and plots shown begin after allowing a transient time of 2000 units. Figure adapted from Ref. [25].

Fig. 3. Stability diagram for chimera states. Bifurcation curves: saddle-node (dotted line) and supercritical Hopf (solid line), both found analytically; homoclinic (dashed line), found numerically. Figure adapted from Ref. [25].

Fig. 4. Alternating chimera states induced by external signals: (a) and ; (b) and ; (c) and ; (d) and . Figure adapted from Ref. [27].

Fig. 5. (a) Snapshot of the Kuramoto model; (b) absolute value of the local curvature obtained by applying the discrete Laplace operator on the data set shown in (a). Figure adapted from Ref. [36].

Fig. 6. Transition from a classical chimera state with one incoherent domain to multichimera states with two (a)–(d), and three (e)–(h) incoherent domains. In each panel the left column shows snapshot of variables uk, and the right column shows the corresponding mean phase velocities. Figure adapted from Ref. [37].

Fig. 7. Four typical behaviors in the cerebral cortex with where the up and down panels represent the two hemispheres, respectively, and the insets are their corresponding dynamics of at a moment t. The parameters are and in panels (a) and (e) of disorder; and in panels (b) and (f) of chimera state; and in panels (c) and (g) of an emergent state conceptually similar to the state of unihemispheric sleep; and and in panels (d) and (h) of synchronization. Figure adapted from Ref. [13].

Fig. 8. Graphic visualization of a hub network motif (star motif). Figure adapted from Ref. [57].

Fig. 9. Transition to PS for the hub motif. From the plot the onset of RS is clearly visible. The three annotations indicate synchronization between two, three, and four peripheral oscillators, respectively. Figure adapted from Ref. [57]

Fig. 10. Remote synchronization between node groups A and C mediated by incoherence in group B. The colors of the nodes schematically represent their states, indicating that nodes 1 and N are identically synchronized, while the dynamics of the nodes in B are incoherent. Figure adapted from Ref. [61].

Fig. 11. Six typical patterns of RS for and . Each pattern is chosen by the conditions: (i) There is no synchronization between the hub and its peripheral nodes; (ii) all the peripheral nodes are synchronized each other. Figure adapted from Ref. [62]

Fig. 12. A schematic figure of the new framework of RS with two huns, where the nodes with red, blue and pink numbers represent the hub, leaf and common leaf nodes, respectively. Figure adapted from Ref. [62].

Fig. 13. Examples of symmetries in networks: (a) A network of four identical oscillators coupled through three identical links; (b) the same network after a reflection opera-tion; (c) the same network after a rotation operation; (d) an 11-node network showing three clusters (blue, green, and white). Figure adapted from Ref. [73].

Fig. 14. Patterns of clusters in a five-node network. Left: All possible patterns displayed when the network connectivity is given by the adjacency matrix (Eq. (20)); right: Additional patterns displayed when the network connectivity is given by the Laplacian ma-trix (Eq. (21)). Figure adapted from Ref. [73].

Fig. 15. Grouping of symmetry clusters in a CS pattern for a 24-node network. Figure adapted from Ref. [68].