ORIGINAL ARTICLE Dynamic analysis for high-order Hopfield neural networks with leakage delay and impulsive effects R. Rakkiyappan • C. Pradeep • A. Vinodkumar • Fathalla A. Rihan Received: 30 May 2011 / Accepted: 23 May 2012 / Published online: 20 June 2012 Ó Springer-Verlag London Limited 2012 Abstract This paper considers existence, uniqueness, and the global asymptotic stability for a class of High-order Hopfield neural networks with mixed delays and impulses. The mixed delays include constant delay in the leakage term (i.e., ‘‘leakage delay’’) and time-varying delays. Based on the Lyapunov stability theory, together with the linear matrix inequality approach and free-weighting matrix method, some less conservative delay-dependent sufficient conditions are presented for the global asymp- totic stability of the equilibrium point of the considered neural networks. These conditions are expressed in terms of LMI and can be easily checked by MATLAB LMI toolbox. In addition, two numerical examples are given to illustrate the applicability of the result. Keywords Impulsive high-order Hopfield neural networks  Stability  Leakage delays  Linear matrix inequality 1 Introduction Due to the fact that the high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks, high-order neural networks have been the object of intensive analysis by numerous authors in recent years. In particular, there have been extensive results on the problem of the existence and sta- bility of equilibrium points of high-order neural networks in the literature [1, 2, 3]. By using M-matrix and linear matrix inequality (LMI) techniques, the authors in [4] and [5] investigated the problems of the global exponential stability and robust stability of the equilibrium point of high-order Hopfield-type neural networks without the delays. For high- order Hopfield neural networks with constant time delays, the existence and global asymptotic stability conditions were obtained in [2] which were based on the LMI approach and with the assumption that the activation functions are monotonic nondecreasing. There are numerous articles have been published for the study of dynamic behaviors of high- order neural networks with different time delays, see for example [6] and references therein. Although the convergence dynamics of impulsive neural networks have been considered, see for example [7, 8, 9] and references therein. The problem of the exponential stability analysis for impulsive high-order Hopfield-type neural net- works with time-varying delays was studied in [10], where the delays are not required to be differentiable. We note that the conditions in [10] were obtained based on simple Lyapunov functionals, which may lead to conservatism to some extent when using them to study the exponential stability of delayed high-order neural networks without an impulse. To the best of authors’ knowledge, few authors have considered high-order Hopfield neural networks with impulses. For instance, [10, 11] R. Rakkiyappan (&) Department of Mathematics, Bharathiar University, Coimbatore 641046, Tamilnadu, India e-mail: rakkigru@gmail.com C. Pradeep Department of Science and Humanities, Sri Ramakrishna Institute of Technology, Pachapalayam, Coimbatore 641010, Tamilnadu, India A. Vinodkumar Department of Mathematics and Computer Applications, P.S.G College of Technology, Coimbatore 641004, Tamilnadu, India F. A. Rihan Department of Mathematics, Faculty of Science, UAE University, Al-Ain 17551, UAE 123 Neural Comput & Applic (2013) 22 (Suppl 1):S55–S73 DOI 10.1007/s00521-012-0997-z