Abstract: In order to fit a broader application scope, the outer synchronization problem of a complex dynamical network with double non-delayed and double delayed coupling is investigated. Unlike most studies, the paper do not add any restrictions to coupling matrix. Based on Lyapunov stability theory combined with the descriptor model transformation, the decomposition technique of coefficient matrix and the methods of the free-weighting matrix, a novel synchronization condition is derived and expressed in the form of matrix inequalities. The linear feedback synchronization controllers are designed. Two examples with different topological structures are presented to the demonstrate the feasibility and effiectiveness of the results.

Key words: double non-delayed and double delayed coupling, synchronization, network split, linear matrix inequality

摘要： 为了更真实地仿真现实的网络世界和提高模型的适应性，研究一类具有双重时滞和非时滞耦合的复杂网络同步问题。不同于大多数研究中限定耦合矩阵满足耗散耦合条件，对耦合矩阵未添加任何限制。基于李雅普诺夫稳定性定理，结合线性矩阵不等式，利用广义模型中的等价转换和系数矩阵分解方法，引入自由矩阵，在驱动系统中设置非线性控制器，得到驱动系统与响应系统同步的充分条件。两个仿真例子选定了两个不同的拓扑结构，实验结果证明了定理的可行性和有效性。

关键词: 双重时滞和双重非时滞, 同步, 矩阵分解, 线性矩阵不等式