Local and global exponential synchronization of complex delayed dynamical networks with general topology
Jin-Liang Wang Zhi-Chun Yang Tingwen Huang Mingqing Xiao
Discrete & Continuous Dynamical Systems - B 2011, 16(1): 393-408 doi: 10.3934/dcdsb.2011.16.393
In this paper, we consider a generalized complex network possessing general topology, in which the coupling may be nonlinear, time-varying, nonsymmetric and the elements of each node have different time-varying delays. Some criteria on local and global exponential synchronization are derived in form of linear matrix inequalities (LMIs) for the complex network by constructing suitable Lyapunov functionals. Our results show that the obtained sufficient conditions are less conservative than ones in previous publications. Finally, two numerical examples and their simulation results are given to illustrate the effectiveness of the derived results.
keywords: Complex networks time-varying delays exponential synchronization.
Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case
Zhiming Guo Zhi-Chun Yang Xingfu Zou
Communications on Pure & Applied Analysis 2012, 11(5): 1825-1838 doi: 10.3934/cpaa.2012.11.1825
This paper deals with a class of non-local second order differential equations subject to the homogeneous Dirichlet boundary condition. The main concern is positive steady state of the boundary value problem, especially when the equation does not enjoy the monotonicity. Nonexistence, existence and uniqueness of positive steady state for the problem are addressed. In particular, developed is a technique that combines the method of super-sub solutions and the estimation of integral kernels, which enables us to obtain some sufficient conditions for the existence and uniqueness of a positive steady state. Two examples are given to illustrate the obtained results.
keywords: Dirichlet boundary value problems uniqueness. Non-local second order differential equation positive solutions existence

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