张 红



不具备全局利普希茨条件的时滞分流抑制型细胞神经网络的伪概周期解

张 红*

(湖南文理学院 数学与计算科学学院, 湖南 常德, 415000)

研究了不具备全局利普希茨条件的时滞分流抑制型细胞神经网络系统, 得到其伪周期解存在性及其唯一性的充分条件, 推广并改进了早期有关这方面研究的结果. 通过构造Lyapunov函数并利用Banach压缩映像原理, 得到本系统具有指数型稳定性的伪概周期解的充分条件.

指数型稳定; Lyapunov函数; 伪概周期解; 分流抑制系统

自从文献[1]提出分流抑制系统(SICNNs), 分流型神经网络已被广泛应用于心理物理、演讲、感知、机器人、自适应模式识别、视觉图像处理[2—4]. 研究神经动力系统不仅涉及到其稳定性性能, 而且涉及很多其它动态特性, 如周期振荡、概周期振荡特性、混沌现象和分歧问题等[5—6]. 最近, 文献[7—8]研究了如下SICNNs系统的伪概周期解的存在性:

1 伪概周期函数的基础知识和基本结论

这里是维向量.

2 伪概周期解的存在性和稳定性

定理1 若假设

在R上具有指数二分性. 结合引理2可知式(6)存在唯一的伪概周期解:

证

于是有:

定理2证毕.

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[3] Zhou Q, Xiao B, Yu Y, et al. Existence and exponential stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays[J]. Chaos Solitons Fract, 2007, 34: 860—866.

[4] Liu B. Almost periodic solutions for shunting inhibitory cellular neural networks without global Lipschitz activation functions[J]. Journal of Computational and Applied Mathematics, 2007, 203: 159—168.

[5] 何崇佑. 概周期微分方程[M]. 北京: 高等教育出版社, 1992.

[6] Fink A M. Almost periodic differential equation[M]. Berlin: Spring-Verlag, 1974.

[7] Farouk Ch érif. Existence and global exponential stability of pseudo almost periodic solution for SICNNs with mixed delays[J]. Appl Math Comput, 2012, 39: 235—251.

[8] 孙献德. 具变时滞分流抑制型细胞神经网络的伪概周期解及吸引性[J]. 福州大学学报: 自然科学版, 2011, 39(2): 180—185.

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Pseudo almost periodic solutions of delayed shunting inhibitory cellular neural networks without Global Lipschitz Activaty Functions

ZHANG Hong

(college of Mathematics and Computer Science, Hunan University of Arts and Science, Changde, 415000, China)

In this paper, shunting inhibitory cellular neural networks are studied. Without assuming the global Lipschitz conditions of activaty functions, some new sufficient conditions are obtained for ensuring the existence and uniqueness of pseudo almost periodic solution of this system. Our results improve and generalize those of the previous results. Furthermore,several methods are applied to establish sufficient criteria for the exponential stability of this system. The approaches are based on con-structing suitable Lyapunov functionals and the well-known Banach contraction mapping principle.

exponential stability; Lyapunov functional; pseudo almost periodic solu-tion; SICNNs

10.3969/j.issn.1672-6146.2013.03.001

O 175.14

1672-6146(2013)03-0001-05

email: hongzhang320@aliyun.com.

2013-09-01

湖南省教育厅科研资助项目(11C0915)

(责任编校:刘晓霞)