Nonlinear Dyn (2012) 69:1751–1764 DOI 10.1007/s11071-012-0383-5 ORIGINAL PAPER Outer synchronization of complex networks with delay via impulse Wen Sun · Zhong Chen · Jinhu Lü · Shihua Chen Received: 31 May 2011 / Accepted: 20 February 2012 / Published online: 16 March 2012 © Springer Science+Business Media B.V. 2012 Abstract Synchronization between the driving net- work and the responding network (outer synchroniza- tion) has attracted increasing attention from various fields of science and engineering. In this paper, we ad- dress outer synchronization of complex networks with delays. Both the cases of coupling delay and node de- lay are considered. Employing the impulsive control method which is simple, efficient, low cost, and easy to implement in practical applications, we obtain some sufficient conditions of outer synchronization. It in- dicates that outer synchronization can be achieved if the maximal impulsive intervals are less than a critical value. Numerical simulations are also given to demon- strate the effectiveness of the proposed impulsive con- trol scheme. Keywords Complex network · Outer synchronization · Impulsive control · Comparison theorem W. Sun ( ) · Z. Chen School of Information and Mathematics, Yangtze University, Jingzhou 434023, P.R. China e-mail: sunwen_2201@163.com W. Sun · J. Lü Institute of Systems Science Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P.R. China S. Chen College of Mathematics and Statistics, Wuhan University, Wuhan 430072, P.R. China 1 Introduction Many large-scale and complicated systems in na- ture and society can be modeled as complex net- works, where nodes are the elements of the sys- tems and edges represent the interactions between them [1]. Complex networks with the well-known small-world property [2] and scale-free property [3] have been received much attention from researchers across a variety of disciplines including physics, soci- ology, biology, mathematics, engineering science, and so on. As one of the basic characteristics of a dynami- cal network, complete synchronization inside a net- work which named “inner synchronization” is a hot topic ranging from physical to chemical, biologi- cal, information technology, mathematical, and even to social sciences [3–6]. Moreover, another type of synchronization inside a network: antisynchroniza- tion (AS) characterized by the vanishing of the sum of relevant variables has been investigated exten- sively [7–13]. It is noted that in the practical cases time delays are caused by finite signal transmission speeds or memory effects, and often encountered in various complex net- works, such as communication networks, neural net- works, and metabolic networks, in either the state vari- ables or the coupling coefficients. For example, a dis- tributed computer network, in which time delays in dynamical nodes represent computing time and cou- pling delays represent communication delays. Ignor-