Anti-periodic solution for impulsive high-order Hopfield neural networks with time-varying delays in the leakage terms
© Wang; licensee Springer 2013
Received: 3 May 2013
Accepted: 21 August 2013
Published: 11 September 2013
This paper presents new results on anti-periodic solutions for impulsive high-order Hopfield neural networks with time-varying delays in the leakage terms. By employing a novel proof, some criteria are derived for guaranteeing the existence and exponential stability of the anti-periodic solution, which are new and complement previously known results. Moreover, a numerical simulation is given to show the effectiveness.
Keywordsimpulsive high-order Hopfield neural networks anti-periodic solution exponential stability time-varying delay leakage term
where and n is the number of units in a neural network, corresponds to the state vector of the i th unit at time t, represents the rate at which the i th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs, and are the first- and second-order connection weights of the neural network, denotes the leakage delay and for all . , , correspond to the transmission delays, denotes the external inputs at time t, and is the activation function of signal transmission. , , , , , , , , are continuous functions on R. is a positive constant. , , , , . are impulsive moments satisfying and . is the initial condition and denotes real-valued continuous functions defined on , .
The impulsive differential equations have been proposed in many fields such as control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, etc. [1–3]. High-order neural networks have been the object of intensive analysis by numerous authors since high-order neural networks have stronger approximation property, faster convergence rate, greater storage capacity, and higher fault tolerance than lower-order neural networks [4–8]. Thus, many high-order Hopfield neural networks with impulses have been studied extensively, and a great deal of literature focuses on the existence and stability of equilibrium points, periodic solutions, almost periodic solutions and anti-periodic solutions [9–16]. However, to the best of our knowledge, few authors have considered the existence and stability of an anti-periodic solution of system (1.1) with the leakage delay . We mention that arguments in [9–16] are not applicable to system (1.1).
The purpose of this paper is to discuss the existence and exponential stability of an anti-periodic solution for IHHNNs with time-varying delays in the leakage terms of system (1.1). The outline of the paper is as follows. In Section 2, some preliminaries and basic results are established. In Section 3, we give sufficient conditions for the existence and exponential stability of an anti-periodic solution for system (1.1). In Section 4, we give an example and numerical simulation to illustrate our results.
2 Preliminaries and basic results
Throughout this paper, we assume that the following conditions hold.
() for and .
For convenience, let be the set of all real vectors. We use to denote a column vector, in which the symbol (T) denotes the transpose of a vector. As usual in the theory of impulsive differential equations, at the points of discontinuity of the solution , we assume that . It is clear that, in general, the derivative does not exist. On the other hand, according to system (1.1), there exists the limit . In view of the above convention, we assume that .
where the smallest positive number ω is called the anti-period of function .
In what follows, we shall prove the lemmas which will be used to prove our main results in Section 3.
It is a contradiction and shows that (2.5) holds. The proof is now completed. □
Remark 2.1 After conditions ()-(), the solution of system (1.1) always exists (see [1, 2]). In view of the boundedness of this solution, from the theory of impulsive differential equations in , it follows that the solution of system (1.1) can be defined on .
Now, we consider two cases.
Remark 2.2 If is an ω-anti-periodic solution of system (1.1), it follows from Lemma 2.2 that is globally exponentially stable.
3 Main results
In this section, we study the existence and exponential stability for an anti-periodic solution of system (1.1).
Theorem 3.1 Suppose that all conditions in Lemma 2.2 are satisfied. Then system (1.1) has exactly one ω-anti-periodic solution . Moreover, is globally exponentially stable.
Combining (3.3)-(3.4) with (3.5)-(3.6), we know that converges uniformly to a piecewise continuous function on any compact set of R.
Thus, is an ω-anti-periodic solution of system (1.1).
Finally, by Lemma 2.2, we can prove that is globally exponentially stable. This completes the proof. □
In this section, we give an example to demonstrate the results obtained in previous sections.
Remark 4.1 Since [9–16] only dealt with IHHNNs without leakage delays, one can observe that all the results in these works and the references therein cannot be applicable to prove the existence and exponential stability of 1-anti-periodic solution for IHHNNs (4.1). This implies that the results of this paper are essentially new.
The author has made this manuscript independently. The author read and approved the final version.
I am grateful to the referees for their suggestions that improved the writing of the paper. This work was supported by the National Natural Science Foundation of China (grant no. 11201184), the Natural Scientific Research Fund of Zhejiang Provincial of P.R. China (grant no. LY12A01018), and the Natural Scientific Research Fund of Zhejiang Provincial Education Department of P.R. China (grant no. Z201122436).
- Lakshmikantham V, Bainov DD, Simeonov PS: Theory of Impulsive Differential Equations. World Scientific, Singapore; 1989.View ArticleGoogle Scholar
- Samoilenko AM, Perestyuk NA: Impulsive Differential Equations. World Scientific, Singapore; 1995.Google Scholar
- Akhmet MU: On the general problem of stability for impulsive differential equations. J. Math. Anal. Appl. 2003, 288: 182–196. 10.1016/j.jmaa.2003.08.001MathSciNetView ArticleGoogle Scholar
- Giles CL, Maxwell T: Learning, invariance and generalization in high-order neural networks. Appl. Opt. 1987, 26: 4972–4978. 10.1364/AO.26.004972View ArticleGoogle Scholar
- Karayiannis NB: On the training and performance of higher-order neural networks. Math. Biosci. 1995, 129: 143–168. 10.1016/0025-5564(94)00057-7View ArticleGoogle Scholar
- Schmidt WAC, Davis JP: Pattern recognition properties of various feature spaces for higher order neural networks. IEEE Trans. Pattern Anal. Mach. Intell. 1993, 15: 795–801. 10.1109/34.236250View ArticleGoogle Scholar
- Reid, MB, Spirkovska, L, Ochoa, E: Rapid training of higher-order neural networks for invariant pattern recognition. In: Proc. Int. Joint Conf. Neural Networks, Washington, 1989Google Scholar
- Zhang J, Gui Z: Existence and stability of periodic solutions of high-order Hopfield neural networks with impulses and delays. J. Comput. Appl. Math. 2009, 224: 602–613. 10.1016/j.cam.2008.05.042MathSciNetView ArticleGoogle Scholar
- Shi P, Dong L: Existence and exponential stability of anti-periodic solutions of Hopfield neural networks with impulses. Appl. Math. Comput. 2010, 216: 623–630. 10.1016/j.amc.2010.01.095MathSciNetView ArticleGoogle Scholar
- Guan Z, Chen G: On delayed impulsive Hopfield neural networks. Neural Netw. 1999, 12: 273–280. 10.1016/S0893-6080(98)00133-6View ArticleGoogle Scholar
- Zhang A: Existence and exponential stability of anti-periodic solutions for HCNNs with time-varying leakage delays. Adv. Differ. Equ. 2013., 2013: Article ID 162 10.1186/1687-1847-2013-162Google Scholar
- Liu B, Gong S: Periodic solution for impulsive cellar neural networks with time-varying delays in the leakage terms. Abstr. Appl. Anal. 2013., 2013: Article ID 701087Google Scholar
- Jiang Y, Yang B, Wang J, Shao C: Delay-dependent stability criterion for delayed Hopfield neural networks. Chaos Solitons Fractals 2009, 39: 2133–2137. 10.1016/j.chaos.2007.06.039MathSciNetView ArticleGoogle Scholar
- Xiao B, Meng H: Existence and exponential stability of positive almost periodic solutions for high-order Hopfield neural networks. Appl. Math. Model. 2009, 33: 532–542. 10.1016/j.apm.2007.11.027MathSciNetView ArticleGoogle Scholar
- Zhang F, Li Y: Almost periodic solutions for higher-order Hopfield neural networks without bounded activation functions. Electron. J. Differ. Equ. 2007, 99: 1–10.View ArticleGoogle Scholar
- Ou CX: Anti-periodic solutions for high-order Hopfield neural networks. Comput. Math. Appl. 2008, 56: 1838–1844. 10.1016/j.camwa.2008.04.029MathSciNetView ArticleGoogle Scholar
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