测度链上复杂网络同步的多重稳定性及基于阻碍集的同步控制

The synchronization and synchronous control of complex networks have complicated dynamical properties and wide-ranging application from which quite a lot of research achievements have been obtained.However,most of research focuses on continuous-time or discrete-time complex networks,we know little about synchronous behavior,geometric features of synchronization manifold and synchronous control on general time scales.By this project,we will investigate synchronous behavior of complex networks on measure chains and its time-scale characteristic.From the distribution of state space of each node,we will discuss global synchronization and local synchronization of networks in order to establish inherent relationship between dynamic system of each node and multistable synchronization. Hence, we can improve the theory of synchronization control of complex networks based on calculus on measure chains. Finally,we will attain key characteristic quantity about synchronization and synchronization control of complex networks by obstruction sets.The connectedness,separation axion and compactness of obstruction sets will be analyzed by using tangent space of synchronization manifold and immersed submanifold theory. Hence, we can establish geomeric synchronization control approach for complex networks in the view of set of points.Our project incorporates calculus on measure chains and hybrid method combining geometry and functional analysis into synchronization control of complex networks. It has innovative significance and theoretical importance for revealing synchronization mechanism of complex networks on general time scales.

复杂网络的同步与同步控制具有复杂的动力学性态和广泛的应用价值，研究成果十分丰富。然而绝大多数研究集中在连续型或离散型复杂网络上，在更一般的时间标尺上，对复杂网络的同步行为、同步流形及同步控制的几何特征等问题知之甚少。本项目将在测度链上研究复杂网络同步的动力学行为及其时标特性；从节点动态演化的状态空间分布研究网络的全局与局部同步行为，力图得出网络各节点演化系统与同步的多重稳定性之间的内在关系，完善基于测度链上微积分的复杂网络同步控制理论。最后，我们将基于阻碍集把握复杂网络同步及同步控制的关键特征量，利用同步流形的切空间和浸入子流形相关理论刻画阻碍集的连通性、分离性以及紧致性等几何性质，发展基于“点集分析”意义上的几何同步控制方法。本项目引入测度链微积分、几何和泛函相结合的研究手段不仅对复杂网络同步控制方向具有创新性意义，而且对揭示一般时间标尺上的复杂网络同步机理具有重要的理论价值。

复杂网络的同步与同步控制具有复杂的动力学性态和广泛的应用价值。绝大多数研究集中在连续型或离散型复杂网络上，本项目在更一般的时间标尺上对复杂网络的同步行为、同步流形及同步控制策略的研究具有重要的学术价值和理论意义。本项目从节点动态演化的状态空间分布研究复杂时域上网络的全局与局部同步行为，得出网络各节点演化系统与同步的多重稳定性之间的内在关系；基于一些同步流形来把握复杂网络同步及不同同步控制方案下的关键特征量，给出了复杂网络具有复杂时域的同步准则，这些结果改进了传统复杂网络同步控制理论。引入测度链微积分、几何和泛函相结合的研究手段研究复杂网络同步控制，不仅揭示了一般时间标尺上的复杂网络同步机理而且完善基于测度链上微积分的复杂网络同步控制理论。