图 1  (a)奇异态的斑图, 参数: $A=0.995,\; \beta=0.18, $ N = 256个振子; (b)来自于方程(3)的局部相关联$R(x)$, 锁频的振子满足$R(x)\geqslant \varDelta$; (c) 局部平均相$\varTheta(x)$. 改编自文献[20]

Fig. 1.  (a) Phase pattern for a chimera state, parameters: $A=0.995,\; \beta=0.18, N=256$ oscillators; (b) local phase coherence $R(x)$, computed from Eq. (3), locked oscillators satisfy $R(x)\geqslant \varDelta$; (c) local average phase $\varTheta(x)$. Figure adapted from Ref. [20].

图 2  序参量r随时间的变化, 其中参数$N_1=N_2=128$, $\beta=0.1 $　(a) $ A=0.20 $, 稳定奇异态; (b) $ A=0.28 $, 呼吸奇异态; (c) $ A=0.35 $, 长周期呼吸子. 初值条件取为靠近奇异态, 结果为丢掉了2000个暂态时间单位后的情形. 改编自文献[25]

Fig. 2.  Order parameter r versus time. In all three panels, $N_1=N_2=128$ and $\beta=0.1$: (a) $ A=0.20 $, stable chimera; (b) $ A=0.28 $, breathing chimera; (c) $ A=0.35 $, 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].

图 3  奇异态的稳定性相图 分岔曲线: 鞍-节分岔(点划线)、超临界Hopf分岔(实线), 两者都是解析发现的; 同宿分岔(虚线), 数值发现的. 改编自文献[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].

图 4  外部信号诱导的交替奇异态 (a)$A=1, \varOmega=1$与$\tau_2 =1.5$; (b)$A=1, \varOmega=0.5$与$\tau_2 =1.5$; (c)$A=1.5, \varOmega=1$ 与$\tau_2 =1.5$; (d)$A=1, \varOmega=1$与$\tau_2 =3.0$. 改编自文献[27]

Fig. 4.  Alternating chimera states induced by external signals: (a) $A=1, \varOmega=1$ and $\tau_2 =1.5$; (b) $A=1, \varOmega=0.5$ and $\tau_2 =1.5$; (c) $A=1.5, \varOmega=1$ and $\tau_2 =1.5$; (d) $A=1, \varOmega=1$ and $\tau_2 =3.0$. Figure adapted from Ref. [27].

图 5  (a) 来自于Kuramoto模型的快照; (b) 将拉普拉斯算子用于(a)中的数据后得到的局部曲率的绝对值. 改编自文献[36]

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].

图 6  单一的非相关区域变成多个非相关区域的情形　(a)—(d)变成两个非相关区域: (e)—(h)变成三个非相关区域. 在每个子图中, 左列代表变量$u_k$的快照, 右列为对应的平均相速度. 改编自文献[37]

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].

图 7  大脑皮层网中的四个典型的行为. 参数为$\alpha={\text{π}}/2-0.1$, 上层为网络A、下层为网络B、子图为它们对应的动力学$u_{i}$ (a)与(e)的耦合参数为($\lambda_{\rm {in}}=0.1$, $\lambda_{\rm {out}}=0.3$); (b)与(f)的耦合参数为($\lambda_{\rm {in}}=0.1$, $\lambda_{\rm out}=1.8$)、(c)与(g)的耦合参数为($\lambda_{in}=0.4$, $\lambda_{\rm {out}}=3.5$); (d)与(h)的耦合参数为($\lambda_{\rm {in}}=4.0$, $\lambda_{\rm {out}}=3.5$). 改编自文献[13]

Fig. 7.  Four typical behaviors in the cerebral cortex with $\alpha={\text{π}}/2-0.1$ where the up and down panels represent the two hemispheres, respectively, and the insets are their corresponding dynamics of $u_{i}$ at a moment t. The parameters are $\lambda_{\rm {in}}=0.1$ and $\lambda_{\rm {out}}=0.3$ in panels (a) and (e) of disorder; $\lambda_{\rm {in}}=0.1$ and $\lambda_{\rm {out}}=1.8$ in panels (b) and (f) of chimera state; $\lambda_{\rm {in}}=0.4$ and $\lambda_{\rm {out}}=3.5$ in panels (c) and (g) of an emergent state conceptually similar to the state of unihemispheric sleep; and $\lambda_{\rm {in}}=4.0$ and $\lambda_{\rm {out}}=3.5$ in panels (d) and (h) of synchronization. Figure adapted from Ref. [13].

图 8  用于遥同步的星形网示意图. 改编自文献[57]

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

图 9  相同步的过程. 相同步的建立清晰可见, 其中的三个标注分别表明两个、三个与四个叶子节点间的同步化. 改编自文献[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]

图 10  A与C组之间的遥同步同步由B组的介导来完成. 节点的颜色代表了它们的状态, 可见节点1与N是完全同步的, 而B组中节点的动力学是无关的. 改编自文献[61]

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].

图 11  参数取$\tau=0.5$与$\varepsilon=0.1$时的六个典型的遥同步斑图. 每个斑图都是按如下条件挑选的: (i) 在中心节点与叶子节点间没有同步化; (ii)所有的叶子节点均相互同步化. 改编自文献[62]

Fig. 11.  Six typical patterns of RS for $\tau=0.5$ and $\varepsilon=0.1$. 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]

图 12  具有两个中心节点的遥同步新框架示意图, 其中红、蓝、粉红数字分别代表中心节点、叶子节点与公共叶子节点. 改编自文献[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].

图 13  网络中对称性的例子　(a) 四个全同振子通过三根连线耦合的网络; (b) 一个反射操作后的同一网络; (c) 一个旋转操作后的同一网络; (d) 一个$11$个节点的网络有三个集团(蓝、绿、白). 改编自文献[73]

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].

图 14  五个节点的网络中的集团斑图 左边: 当网络连接为邻接矩阵(方程(20))时所有可能的斑图; 右边: 当网络连接为拉普拉斯矩阵(方程(21))时额外的斑图. 改编自文献[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].

图 15  24个节点构成的网络中对称集团的分组. 改编自文献[68]

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