In this article, we search for polynomial Lyapunov functions beyond the quadratic form to investigate the synchronization problems of nonlinearly coupled complex networks. First, with a relaxed assumption than the quadratic condition, a synchronization criterion is established for nonlinearly coupled networks with asymmetric coupling matrices. Compared with the existing synchronization criteria, our results are less conservative and have a wider application. Second, the synchronization problem for polynomial networks is characterized as the sum-of-squares (SOS) optimization one. In this way, polynomial Lyapunov functions can be obtained efficiently with SOS programming tools. Furthermore, it is shown that the local synchronization of certain nonpolynomial networks can also be analyzed by using the SOS optimization method through the Taylor series expansion. Finally, three numerical examples are presented to verify the effectiveness and less conservatism of our analytical results.

中文翻译：

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用于非线性耦合复杂网络同步的多项式 Lyapunov 函数


在本文中，我们搜索二次形式之外的多项式李雅普诺夫函数，以研究非线性耦合复杂网络的同步问题。首先，采用比二次条件宽松的假设，为具有不对称耦合矩阵的非线性耦合网络建立同步准则。与现有的同步标准相比，我们的结果保守性较小，并且具有更广泛的应用范围。其次，多项式网络的同步问题被描述为平方和（SOS）优化问题。这样，利用SOS编程工具就可以高效地得到多项式Lyapunov函数。此外，结果表明，某些非多项式网络的局部同步也可以通过泰勒级数展开使用SOS优化方法进行分析。最后，给出了三个数值例子来验证我们的分析结果的有效性和较少的保守性。