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Existence and Global Uniform Asymptotic Stability of Almost Periodic Solutions for Cellular Neural Networks with Discrete and Distributed Delays  [PDF]
Zongyi Hou,Hongying Zhu,Chunhua Feng
Journal of Applied Mathematics , 2013, DOI: 10.1155/2013/516293
Abstract: This paper discusses the existence and global uniform asymptotic stability of almost periodic solutions for cellular neural networks (CNNS). By utilizing the theory of the almost periodic differential equation and the Lyapunov functionals method, some sufficient conditions are obtained to ensure the existence and global uniform asymptotic stability. An example is given to illustrate the effectiveness of the main results. 1. Introduction Cellular neural networks (CNNS) are composed of a large number of simple processing units (called neurons), widely interconnected to form a complex network system. It reflects many basic features of the human brain functions. It is a highly complicated nonlinear dynamics system and has successful applications in many fields such as associative, signal, and image processing, pattern recognition, and optimization. In 1984, Hopfield proposed that the dynamic behavior of neurons should be described with a set of ordinary differential equations or functional differential equations. Since then, a lot of research achievements have been published in the world. Recently, many scholars have paid much attention to the research on the dynamics and applications of CNNS. Specially, some scholars have studied the existence and stability of almost periodic solution for neural networks, which can be seen from [1–10] and therein references. In [4], without product systems, by utilizing the generalized Halanay inequality technique and combining the theory of exponential dichotomy with fixed point method, Huang et al. study the existence and global exponential stability of almost periodic solutions for recurrent neural network with continuously distributed delays as follows: where are incentive functions, which satisfy . ? are all almost periodic functions. In [5], Xiang and Cao discuss the following system: Without product systems, by using the Lyapunov functionals method and analytical skills, the results about the existence, attractivity, and exponential stability of almost periodic solutions for the system (2) are obtained. However, a more general system than the systems above is discussed in this paper. We consider the existence and global uniform asymptotic stability of almost periodic solutions to the CNNS with discrete and continuously distributed delays. The system is as follows: By using Lemmas 3 and 4 in the next section and under the less restrictive conditions, some sufficient conditions are obtained to ensure the existence and global uniform asymptotic stability of almost periodic solutions to the system (3). An example is
Weighted pseudo-almost periodic functions on time scales with applications to cellular neural networks with discrete delays  [PDF]
Yongkun Li,Lili Zhao
Mathematics , 2015,
Abstract: In this paper, we first propose a concept of weighted pseudo-almost periodic functions on time scales and study some basic properties of weighted pseudo-almost periodic functions on time scales. Then, we establish some results about the existence of weighted pseudo-almost periodic solutions to linear dynamic equations on time scales. Finally, as an application of our results, we study the existence and global exponential stability of weighted pseudo-almost periodic solutions for a class of cellular neural networks with discrete delays on time scales. The results of this paper are completely new.
Stability and Existence of Periodic Solutions for Cellular Neural Networks with State Dependent Delays on Time Scales  [PDF]
Yaping Ren,Yongkun Li
Discrete Dynamics in Nature and Society , 2012, DOI: 10.1155/2012/386706
Abstract: We study delayed cellular neural networks on time scales. Without assuming the boundedness of the activation functions, we establish the exponential stability and existence of periodic solutions. The results in this paper are completely new even in case of the time scale or and improve some of the previously known results. 1. Introduction Consider the following cellular neural networks with state-dependent delays on time scales: where , is an -periodic time scale which has the subspace topology inherited from the standard topology on , , will be defined in the next section, , corresponds to the number of units in the neural network, corresponds to the state of the th unit at time , denotes the output of the th unit on th unit at time , denotes the strength of the th unit on the th unit at time , denotes the external bias on the th unit at time , corresponds to the transmission delay along the axon of the th unit, represents the rate with which the th unit will reset its potential to the resting state in isolation when disconnected from the network and external inputs. It is well known that the cellular neural networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories, especially in image processing and solving nonlinear algebraic equations. They have been widely studied both in theory and applications [1–3]. Many results for the existence of their periodic solutions and the exponential convergence properties for cellular neural networks have been reported in the literatures. See, for instance, [4–17] and references cited therein. In fact, continuous and discrete systems are very important in implementation and applications. It is well known that the theory of time scales has received a lot of attention which was introduced by Stefan Hilger in order to unify continuous and discrete analysis. Therefore, it is meaningful to study dynamic systems on time scales which can unify differential and difference systems see [18–28]. When , , (1.1) reduces to where . By using Mawhin’s continuation theorem and Liapunov functions, the authors [6, 14] obtained the existence and stability of periodic solutions of (1.2), respectively. Furthermore, (1.1) also covers discrete system (for when ; see [15]) where , . In [15], the author firstly obtained the discrete-time analogue of (1.3) by the semidiscretization technique [29, 30], and then some sufficient conditions for the existence and global asymptotical stability of periodic solutions of (1.3) were established by using Mawhin’s continuation theorem and
Existence of Periodic Solutions for Shunting Inhibitory Cellular Neural Networks with Neutral Delays  [PDF]
Ninghua Chen
Discrete Dynamics in Nature and Society , 2013, DOI: 10.1155/2013/143706
Abstract: This paper considers the existence of periodic solutions for shunting inhibitory cellular neural networks (SICNNs) with neutral delays. By applying the theory of abstract continuation theorem of -set contractive operator and some analysis technique, a new result on the existence of periodic solutions is obtained. 1. Introduction It is well known that SICNNs have been extensively applied in psychophysics, speech, perception, robotics, adaptive pattern recognition, vision, and image processing. The dynamical behaviors of SICNNs with delays have been widely investigated in recent years. A large number of important results on the dynamical behaviors of SICNNs have been established and successfully applied to signal processing, pattern recognition, associative memories, and so on. In particular, there exist many results on the existence and stability of periodic and almost periodic solutions for SICNNs with delays. We refer the reader to [1–4] and references cited therein. On the other hand, the stability analysis of various neutral delay-differential systems has drawn much research attention [5–7]. The theory of neutral delay-differential systems is of both theoretical and practical interest. For a large class of electrical networks containing lossless transmission lines, the describing equations can be reduced to neutral delay-differential equations; such networks arise in high speed computers where nearly lossless transmission lines are used to interconnect switching circuits. Also, the neutral systems often appear in the study of automatic control, population dynamics, and vibrating masses attached to an elastic bar. Motivated by the above, we consider shunting inhibitory cellular neural networks with neutral delays described by where , is the cell at the position of the lattice and the -neighborhood of is given as is the activity of the cell, , is the passive decay rate of the cell activity, is the external input to , is the connection or coupling strength of postsynaptic activity of the cell transmitted to the cell , and the activity function is a continuous function representing the output or firing rate of cell . By using the Lyapunov functions and linear matrix inequality approach (LMI), most authors studied the asymptotic stability or exponential stability of the equilibrium point (see [7–10]). However, few papers have been published on the existence of periodic solutions or almost periodic solutions for neutral type neural networks ([11–14]). Thus, it is worthwhile to study the existence of periodic solutions for neutral type neural networks with
Existence and exponential stability for anti-periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays
Yuanheng Wu,Zhan Zhou
Electronic Journal of Differential Equations , 2009,
Abstract: This article concerns anti-periodic solutions for shunting inhibitory cellular neural networks (SICNNs), with continuously distributed delays, arising from the description of the neurons state in delayed neural networks. Without assuming global Lipschitz conditions of activation functions, we obtain existence and local exponential stability of anti-periodic solutions.
Existence and exponential stability of periodic solution for continuous-time and discrete-time generalized bidirectional neural networks
Yongkun Li
Electronic Journal of Differential Equations , 2006,
Abstract: We study the existence and global exponential stability of positive periodic solutions for a class of continuous-time generalized bidirectional neural networks with variable coefficients and delays. Discrete-time analogues of the continuous-time networks are formulated and the existence and global exponential stability of positive periodic solutions are studied using the continuation theorem of coincidence degree theory and Lyapunov functionals. It is shown that the existence and global exponential stability of positive periodic solutions of the continuous-time networks are preserved by the discrete-time analogues under some restriction on the discretization step-size. An example is given to illustrate the results obtained.
Existence and global stability of periodic solution for delayed discrete high-order Hopfield-type neural networks  [PDF]
Hong Xiang,Ke-Ming Yan,Bai-Yan Wang
Discrete Dynamics in Nature and Society , 2005, DOI: 10.1155/ddns.2005.281
Abstract: By using coincidence degree theory as well as a priori estimates and Lyapunov functional, we study the existence and global stability of periodic solution for discrete delayed high-order Hopfield-type neural networks. We obtain some easily verifiable sufficient conditions to ensure that there exists a unique periodic solution, and all theirs solutions converge to such a periodic solution.
Existence and stability of almost periodic solutions for shunting inhibitory cellular neural networks with continuously distributed delays
Qiyuan Zhou,Bing Xiao,Yuehua Yu
Electronic Journal of Differential Equations , 2006,
Abstract: In this paper, we consider shunting inhibitory cellular neural networks (SICNNs) with continuously distributed delays. Sufficient conditions for the existence and local exponential stability of almost periodic solutions are established using a fixed point theorem, Lyapunov functional method, and differential inequality techniques. We illustrate our results with an example for which our conditions are satisfied, but not the conditions in [4,6,8].
Dynamic properties of cellular neural networks  [cached]
Angela Slavova
International Journal of Stochastic Analysis , 1993, DOI: 10.1155/s1048953393000103
Abstract: Dynamic behavior of a new class of information-processing systems called Cellular Neural Networks is investigated. In this paper we introduce a small parameter in the state equation of a cellular neural network and we seek for periodic phenomena. New approach is used for proving stability of a cellular neural network by constructing Lyapunov's majorizing equations. This algorithm is helpful for finding a map from initial continuous state space of a cellular neural network into discrete output. A comparison between cellular neural networks and cellular automata is made.
Existence and Exponential Stability of Almost Periodic Solutions for Cellular Neural Networks with Delays
时滞细胞神经网络概周期解的存在性与全局指数稳定性

Liu Bingwen,Hunang Lihong,
刘炳文
,黄立宏

数学物理学报(A辑) , 2007,
Abstract: In this paper cellular neural networks with delays are considered. Sufficient conditions for the existence and exponential stability of the almost periodic solutions are established by using a fixed-point theorem and Lyapunov functional method. The results of this paper are new and they complement previously known results.
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