Analysis and Control of Synchronization in Coupled Nonlinear Systems with Time-delays

本课题分两个阶段进行。(A)第一阶段研究两个变时滞耦合非线性系统的同步问题。在这一阶段，我们引入了严格半耗散的概念，并利用小增益定理证明了耦合系统满足严格半被动的收敛性质。利用这一结果，可以估计轨迹的界，然后以线性矩阵不等式(LMI)的形式推导出同步的充分条件。此外，在这一阶段，我们提出了一种新的状态观测器期望同步的结构。(B)第二阶段是将第一阶段的结果推广到N个耦合系统。首先，我们得到了同步误差动力学有原点为平衡解的必要条件。所得到的条件意味着网络必须具有对称结构。在此条件下，得到了具有时滞耦合的N个混沌系统的单向或双向网络同步的充分条件。文中推导的同步条件是基于Lyapunov-Krasovskii定理和Lyapunovs间接法。所得到的条件通过数值模拟得到了验证。在本文中，我们只考虑耦合系统的完全同步，并揭示了网络中的部分同步现象。因此，对部分同步的分析将是进一步研究的主题。

This project has been carried out in two phases.(a) The first phase was to study on synchronization of two coupled nonlinear systems with time-varying delay. In this phase, we introduced the notion of strict semi-dissipativity and showed the convergence property of coupled systems that each system satisfies the strict semi-passive by using the small gain theorem. By using this result, the bounds of the trajectories can be estimated and then a sufficient condition for synchronization can be derived in the form of Linear Matrix Inequalities (LMIs). In addition, in this phase, we proposed a new configuration of anticipating synchronization with the state observer.(b) The second phase was to extend the results in the first phase into N coupled systems. First, we derived a necessary condition under which the synchronization error dynamics has the origin as an equilibrium solution. The obtained condition means that the networks must have symmetric structures. Under the condition, we obtained sufficient conditions for synchronization in unidirectional or bidirectional networks of N chaotic systems with time-delay coupling. The synchronization conditions derived in this project are based on the Lyapunov-Krasovskii theorem in combination with the LyapunoVs indirect method. The obtained conditions were verified by numerical simulations.In this paper we targeted on only complete synchronization of coupled systems and have uncovered a phenomenon of partial synchronization in networks. Therefore the analysis of partial synchronization will be the subject of further research.