ORIGINAL ARTICLE Stability of Hopfield neural networks with time delays and variable-time impulses Chao Liu • Chuandong Li • Tingwen Huang • Chaojie Li Received: 2 May 2011 / Accepted: 5 July 2011 / Published online: 23 July 2011 Ó Springer-Verlag London Limited 2011 Abstract The stability of Hopfield neural networks with fixed-time impulses has been intensively investigated in recent years. However, few existing publications addressed the stability of delayed neural networks with variable-time impulses. In this paper, we consider the case of variable- time impulses and attempt to establish the general stability criteria. It shows that the proposed results can also be applied to the case of fixed-time impulses, which provide a new stability condition for the case of fixed-time impulses. To illustrate the effectiveness of our theoretical results, numerical examples and simulations are also presented. Keywords Hopfield neural network (HNN)  Time delays  Variable-time impulse  Stability 1 Introduction Since Hopfield neural network [8] was referred by Hopfield in 1984, it has attracted attentions of many scientists and has been applied in various realms such as pattern recog- nition, associative memory, combinatorial optimization and so on. Among them, stability of HNNs is a crucial dynamic feature and demanded heavily in many applica- tions. Hence, the stability analysis has been intensively investigated, and some sufficient criteria ensuring the sta- bility of the HNNs has been obtained [7, 10, 14, 17, 25]. It is well known that time delays are unavoidable due to finite switching speeds of the amplifiers and they may cause oscillations or instability of dynamic systems. Therefore, delay effects on the performance of neural networks have attracted interests of many researchers, and many profound results for the stability of delayed neural networks have been proposed [3, 5, 21, 22]. Meanwhile, impulsive phenomena exists in a wide variety of evolu- tionary process, such as financial systems and nanoscale electronic circuit in which many state variables change instantaneously, in the form of impulses. On the other hand, impulsive control is also applied widely in many fields of information science, electronics, automatic control systems, computer networking, artificial intelligence, robotics and telecommunications, etc. In neural networks, system state may jump instantaneously because of envi- ronmental changes (such as external noise and distur- bance). Moreover, one will also introduced impulses deliberately to stabilize the oscillating neural networks or to synchronize the chaotic neural networks. As a matter of fact, impulse not only stabilizes [16] but also destabilizes neural networks [15]. Therefore, the effect of impulse in neural networks is worthy of investigation. Many researchers introduced impulse into Hopfield neural net- works and obtained many interesting and significant sta- bility results [1, 6, 9, 20, 24]. The mathematical model of an evolving process with impulsive effects consists of three components: a contin- uous-time subsystem, a switching set, and a jumping operator. Based on different characteristics of impulsive events, we usually encounter two types of impulsive sys- tems: (i) in which impulses occur at fixed time; (ii) in which impulses occur when the trajectory hits a hyper- surface in the extended phase space, i.e., impulses occur at variable time. Generally, HNN with impulse at fixed time C. Liu  Chuandong Li (&)  Chaojie Li College of Computer, Chongqing University, Chongqing 400044, People’s Republic of China e-mail: licd@cqu.edu.cn T. Huang Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatar e-mail: tingwen.huang@qatar.tamu.edu 123 Neural Comput & Applic (2013) 22:195–202 DOI 10.1007/s00521-011-0695-2