The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects

doi:10.1016/j.neucom.2013.09.071

Neurocomputing

Volume 149, Part A, 3 February 2015, Pages 22-28

https://doi.org/10.1016/j.neucom.2013.09.071 Get rights and content

Abstract

In this paper, we investigate the existence and global exponential stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects. First, based on the Lyapunov functional theory and by applying inequality technique, we give some new and useful sufficient conditions to ensure existence and exponential stability of the anti-periodic solutions. Then, we present an example with numerical simulations to illustrate our results.

Introduction

Since Cohen–Grossberg neural networks (CGNNs) have been first introduced by Cohen and Grossberg in 1983 [1], they have been intensively studied due to their promising potential applications in classification, parallel computation, associative memory and optimization problems. In these applications, the dynamics of networks such as the existence, uniqueness, Hopf bifurcation and global asymptotic stability or global exponential stability of the equilibrium point, periodic and almost periodic solutions for networks plays a key role, see [2], [3], [4], [5], [6], [7] and references cited therein.

Time delays unavoidably exist in the implementation of neural networks due to the finite speed of switching and transmission of signals. Besides delay effects, it has been observed that many evolutionary processes, including those related to neural networks, may exhibit impulsive effects. In these evolutionary processes, the solutions of system are not continuous but present jumps which could cause instability in the dynamical systems. Since the existence of delays and impulses is frequently a source of instability, bifurcation and chaos for neural networks, it is important to consider both delays and impulsive effects when investigating the stability of CGNNs, see [8], [9], [10], [11], [12].

Over the past decades, the anti-periodic solutions of Hopfield neural networks, recurrent neural networks and cellular neural networks have actively been investigated by a large number of scholars. For details, see [13], [14], [15], [18] and references therein. In [13], the author considered the existence and exponential stability of the anti-periodic solutions for a class of recurrent neural networks with time-varying delays and continuously distributed delays. In [14], by constructing fundamental function sequences based on the solution of networks, the authors studied the existence of anti-periodic solutions for Hopfield neural networks with impulses. In [15], by establishing impulsive differential inequality and using Krasnoselski׳s fixed point theorem together with Lyapunov function method, the authors investigated the existence and exponential stability of anti-periodic solution for delayed cellular neural networks with impulsive effects. In [16], by constructing fundamental function sequences based on the solution of networks, the authors studied the existence and exponential stability of anti-periodic solutions for a class of delayed CGNNs. However, till now, there are very few or even no results on the problems of anti-periodic solutions for delayed CGNNs with impulsive effects, while the existence of anti-periodic solutions plays a key role in characterizing the behavior of nonlinear differential equations (see [17]). Thus, it is worth investigating the existence and stability of anti-periodic solutions for CGNNs with both time-varying delays and impulsive effects.

Motivated by the above discussions, in this paper, we are concerned with the existence and the exponential stability of anti-periodic solutions for a class of impulsive CGNNs model with periodic coefficients and time-varying delays. By using analysis technique and constructing suitable Lyapunov function, we establish some simple and useful sufficient conditions on the existence and exponential stability of anti-periodic solutions for CGNNs with impulsive effects.

The rest of the paper organized as follows. In Section 2, the proposed model is presented together with some related definitions. In addition, a preliminary lemma is given. Next section is devoted to investigate the existence and exponential stability of anti-periodic solution of the addressed networks. An illustrative example ends Section 4.

Section snippets

Preliminaries

In this paper, we consider the following impulsive CGNNs model with time-varying delays: ${\begin{matrix} x'_{i} (t) = a_{i} (x_{i} (t)) [- b_{i} (t, x_{i} (t)) + \sum_{j = 1}^{n} c_{ij} (t) f_{j} (x_{j} (t)) \\ + \sum_{j = 1}^{n} d_{ij} (t) g_{j} (x_{j} (t - τ_{ij} (t))) + I_{i} (t)], t \geq 0, t \neq t_{k}, \\ Δ x_{i} (t_{k}) = x_{i} (t_{k}^{+}) - x_{i} (t_{k}) = γ_{ik} x_{i} (t_{k}), k \in N ≜ {1, 2, \dots}, \end{matrix}$ where n denotes the number of neurons in the network, x_i(t) corresponds to the state of the ith unit at time t, $a_{i} (x_{i} (t))$ represents an amplification function, $b_{i} (t, x_{i} (t))$ is an appropriate behaved function, $f_{j} (x_{j} (t))$ and $g_{j} (x_{j} (t - τ_{ij} (t)))$ denote, respectively, the

Main results

In this section, we consider the existence and global exponential stability of anti-periodic solutions for system $(1)$ .

Suppose that $x^{⁎} (t) = {(x_{1}^{⁎} (t), x_{2}^{⁎} (t), \dots, x_{n}^{⁎} (t))}^{T}$ is a solution of system $(1)$ with initial conditions $x_{i}^{⁎} (s) = φ_{i}^{⁎} (s), | φ_{i}^{⁎} (s) | < p_{i}, s \in [- τ, 0],$ where p_i are defined in Lemma 1. Then, on the stability of system $(1)$ , we have the following results.

Theorem 1

Let $(H_{1})$ – $(H_{3})$ and $(H_{5})$ hold. Assume that the following inequality is satisfied: $(λ - β_{i} (t)) p_{i} + \sum_{j = 1}^{n} | c_{ij} (t) | L_{j}^{1} p_{j} + \sum_{j = 1}^{n} | d_{ij} (t) | L_{j}^{2} p_{j} e^{λ τ} < 0, t \in [0, ω],$ where $λ$

Numerical simulations

In this section, an example is given to illustrate the effectiveness of our results obtained in this paper.

For n=2, consider the following delayed Cohen–Grossberg neural networks system: ${\begin{matrix} x_{i}^{'} (t) = a_{i} (x_{1} (t)) [- b_{i} (t, x_{i} (t)) + \sum_{j = 1}^{2} b_{ij} (t) f_{j} (x_{j} (t)) + \sum_{j = 1}^{2} c_{ij} (t) g_{j} (x_{j} (t - τ_{ij} (t))) + I_{i} (t)], \\ t \geq 0, t \neq t_{k}, i = 1, 2, \end{matrix}$ with impulses $Δ x_{1} (t_{k}) = 0.6 x_{1} (t_{k}), Δ x_{2} (t_{k}) = 0.4 x_{2}, k = 1, 2, \dots,$ where $f_{i} (u) = \tanh (u / 2), g_{i} (u) = \frac{(| u + 1 | - | u - 1 |)}{2}, τ_{i 1} (t) = τ_{i 2} (t) = 1, i = 1, 2$ and $a_{1} (u) = 2 - \frac{0.5}{1 + u^{2}}, I_{1} (t) = 3.2 \cos (π t), b_{1} (t, u) = (6.1 - | \cos (π t) |) u,$ $a_{2} (u) = 1 + \frac{0.5}{1 + u^{2}}, I_{2} (t) = 1.9 \sin (π t +$

Conclusion

In this paper, we study the existence and global exponential stability of the anti-periodic solution for impulsive Cohen–Grossberg neural networks with periodic coefficients and time-varying delays. Based on the Lyapunov functional theory, we give some new and useful sufficient conditions for the existence and exponential stability of the anti-periodic solutions by applying mathematical induction and inequality technique. Finally, we give an example with its numerical simulations to verify the

Acknowledgements

This work was supported by the National Natural Science Foundation of PR China (Grant no. 61164004), the Natural Science Foundation of Xinjiang (Grant no. 2010211A07) and the Excellent Doctor Innovation Program of Xinjiang University (Grant no. XJUBSCX-2010003). The authors are grateful to the Editor and anonymous reviewers for their kind help and constructive comments.

Abdujelil Abdurahman was born in Xinjiang Uyghur Autonomous Region, China, in 1987. He received the B.S. degree in mathematics and applied mathematics from the College of Mathematics and System Sciences, Xinjiang University, Urumqi, China, in 2010. In July 2013, he received the M.S. degree in operations research and control theory from Xinjiang University. Currently, he is working towards the Ph.D. degree at Xinjiang University. His current research interests include chaotic systems, neural

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Haijun Jiang was born in Hunan, China, in 1968. He received the B.S. degree from the Mathematics Department, Yili Teacher College, Yili, Xinjiang, China; the M.S. degree from the Mathematics Department, East China Normal University, Shanghai, China, and the Ph.D. degree from the College of Mathematics and System Sciences, Xinjiang University, China, in 1990, 1994, and 2004, respectively. He was a Post doctoral Research Fellow in the Department of Southeast University, Nanjing, China, from 2004.9 to 2006.9. He is a professor and Doctoral Advisor of Mathematics and System Sciences of Xinjiang University, Xinjiang, China. His current research interests include nonlinear dynamics, delay differential equations, dynamics of neural networks, mathematical biology.

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The existence and stability of the anti-periodic solution for delayed Cohen–Grossberg neural networks with impulsive effects

Abstract

Introduction

Section snippets

Preliminaries

Main results

Numerical simulations

Conclusion

Acknowledgements

Phys. Lett. A

J. Comput. Appl. Math.

Appl. Math. Comput.

Phys. Lett. A

Nonlinear Anal. RWA

Chaos Solitons Fractals

Neurocomputing

Nonlinear Anal. RWA

Nonlinear Anal. RWA