Synchronization and Pattern Formation of Nonlinear Systems with Delay Couplings

在这个项目中，我们考虑了具有时滞耦合的全同非线性系统网络的同步问题。具体地，我们得到了以下结果：(1)提出了一种新的基于频域稳定性条件的同步判据，并证明了该判据比基于Lyapunov-Krasovskii方法的传统同步判据具有更低的保守性。此外，我们还给出了网络结构与同步条件之间的关系，并基于两个耦合系统的同步条件和图的拉普拉斯矩阵的特征值，提出了同步和部分同步的估计方法。(2)应用谐波平衡方法，提出了一种分析和设计具有时延耦合的相同非线性系统网络的模式形成方法。

In this project, we considered the synchronization problems in networks of identical nonlinear systems with delayed couplings. In particular, we obtained the following results:(1) We proposed a new synchronization criterion based on a frequency domain stability condition and showed that the proposed criterion is less conservative than the conventional synchronization criteria based on the Lyapunov-Krasovskii approach. Furthermore, we showed the relationship between the network structure and the synchronization condition and developed estimation methods of synchronization and partial synchronization based on the synchronization condition for two coupled systems and the eigenvalues of the graph Laplacian.(2) We proposed an analysis and a design method for pattern formation in networks of identical nonlinear systems with delay couplings by applying the harmonic balance methods.