不连续动力系统理论及其在复杂网络中的应用

This proposal presents a study on the discontinuous dynamical systems and complex networks. When discontinuities occur in the dynamics of a single node or the coupling of a group of nodes, it is proposed to describe the network models by using differential equations with discontinuities on the right-hand side. By adopting the concepts and methods of discontinuous dynamic systems, the tendency of the evolution is investigated, and the emergent behavior of such networked systems in terms of the new built models is analyzed. This research emphasizes on the synchronization of complex networks with discontinuous characteristics. Appropriate control protocols will be designed and the intrinsic mechanism, which induces synchronization, will be explored. In addition, the analysis on the essential of why discontinuous complex networks can lead to finite-time convergence will be conducted and the difference between discontinuous networks and other normal ones will be explained. The key factors which may affect networks reaching finite-time synchronization will be found. Furthermore, the differential inclusions theory of discontinuous dynamic systems and the optimal control theory will be applied to the analysis of finite-time synchronization for complex networks, and the effective methods and the optimal control strategies will be obtained to realize the finite-time synchronization of such networks.

本项目以不连续动力系统及复杂网络为研究对象，考虑当节点的动力学演化方式或节点间的耦合作用不连续时，以右端不连续的微分方程来刻画网络模型，研究复杂网络整体行为的变化趋势。运用不连续动力系统的概念和方法，在新建模型的基础上分析这类网络化系统的特殊演化机制。重点研究不连续复杂网络的同步过程：通过设计适当的同步控制协议，探寻这类网络发生同步的内在机理；分析网络发生有限时间同步的本质及与通常的复杂网络产生同步的区别，确定影响网络有限时间同步的关键因素。将不连续动力系统的微分包含理论和最优控制理论引入到复杂网络的有限时间同步分析中来，寻求使网络有限时间内达到同步的有效方法和最优控制策略。

本项目以不连续动力系统及复杂网络为研究对象，运用不连续动力系统的概念和方法，在新建模型的基础上分析了这类网络化系统的特殊演化机制。重点研究了不连续复杂网络的同步过程：通过设计适当的同步控制协议，探寻了这类网络发生同步的内在机理；分析了网络发生有限时间同步的本质及与通常的复杂网络产生同步的区别，确定了影响网络有限时间同步的关键因素。将不连续动力系统的微分包含理论和最优控制理论引入到复杂网络的有限时间同步分析中来，获得了使网络有限时间内达到同步的有效方法和最优控制策略。