Multiplicity of almost periodic solutions for multidirectional associative memory neural network with distributed delays
Introduction
Generally, the multistability of neural networks, that is the existence and stability of multiple equilibrium points, periodic solutions or almost periodic solutions, is prerequisite when the neural network is applied to solve some problems of many-to-many association memory. It has attracted much attention in the recent years [1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16]. According to the different characteristics of active function, such as piecewise linear nondecreasing, Mexican-hat-type, discontinuous or concave–convex, the coexistence and local stability of multiple equilibria have been extensively investigated [1], [2], [3], [4], [5], [6], [7], [8], [9], [10]. As is known to all, the cyclic behaviors or rhythmic activities are widespread in nature. Therefore, the existence and stability of multiple periodic or almost periodic solutions have also been analyzed [11], [12], [13], [14], [15], [16], [17]. From a phenomenological point of view, the almost periodic functions can represent more exactly rhythmic activities [15]. However, to the best of our knowledge, the investigation of multiple almost periodic solutions of neural networks is seldom reported [15], [17], [16]. In [15], a general methodology that involves geometric configuration of the n-neuron network structure for studying multistability was developed. Through applying the contraction mapping principle, some sufficient conditions guaranteeing the existence of 2n exponentially stable almost periodic solutions were given, when the connection strengths, time lags, and external bias are almost periodic functions of time. In [16], by considering two classes of activation functions, the authors revealed that under some conditions, there are 2n locally exponentially stable almost-periodic solutions of a delayed n-neuron neural network. The authors of [17] investigated the dynamics of 2n almost periodic attractors for cellular neural networks with variable and distributed delays, they obtained exponential attracting domains.
The multidirectional associative memory (MAM) neural network, which was proposed by the Japanese scholar M. Hagiwara in 1990 [18], is an extension of BAM neural network model [19]. In an MAM neural network, its neurons are arranged in three or more fields. The neurons in the same field of an MAM neural network are not connected to each other, but the neurons between every two different fields are fully interconnected. By using of MAM neural networks, one can achieve the many-to-many association memory. Therefore, this class of networks possesses wide application fields such as pattern recognition, image denoising and intelligent information processing [18], [20], [21], [22], [23]. In order to achieve the many-to-many association memory by MAM neural networks, it is necessary to ensure their multistability. In [24], we studied the multistability issue for a delayed MAM neural network with m fields and nk neurons in the field m as follows.where , , denotes the membrane voltage of the ith neuron in the field k at time t, denotes the decay rate of the ith neuron in the field k, is a neuronal activation function of the jth neuron in the field p, is the connection weight from the jth neuron in the field p to the ith neuron in the field is the external input of the ith neuron in the field is the time delay of the synapse from the j neuron in the field p to the ith neuron in the field k. By using the Brouwer fixed point theorem and Dini upper right derivative, we obtained that some sufficient conditions which ensure the multidirectional associative memory neural network has 3l equilibria and 2l equilibria of them are stable, where l is a parameter associated with the number of neurons. System (1) is with discrete delay. It can only reflect the relationship between the state of the neuron at time t and the state of any other neuron at the time ahead of it by a fixed time length .
Since a neural network usually has a spatial extent due to the presence of a multitude of parallel pathways of a variety of axon sizes and lengths, there is a distribution of propagation delays over a duration of time [25]. That is to say, the state of the neural network in the past every time affects its current state. Obviously, the distributed delays can reflect the relationship more really than discrete delays. Hence, we also studied the MAM neural network with the continuously distributed delays [26] as follows.where is the delay kernel function of the synapse from the jth neuron in field p to the ith neuron in field k. By constructing a suitable Liapunov function and a Poincaré mapping, we proved the existence and the exponential stability of multiple periodic solutions of the MAM neural network (2).
From the viewpoint of biological nervous system, there exists memory vibration in human׳s cerebrum. And from the viewpoint of networks implementation, the neural network׳s external inputs , the neuronal signal decay rates , and the connection weights often almost periodically vary with time because there are almost periodic phenomenons in the electrical power system. Therefore, the discussion on existence and stability of multiple almost periodic solutions of MAM neural networks is also very meaningful. Motivated by the above, in this paper, we study the existence and the exponential stability of multiple almost periodic solutions of an MAM neural network with almost periodic coefficients and continuously distributed delays as follows.
The initial conditions associated with (3) are of the formwhere , and are continuous functions.
In this paper, our main object is to obtain the sufficient conditions ensuring the existence and the exponential stability of multiple almost periodic solutions of the MAM neural network (3) with initial condition (4). After constructing an invariant basin of system (3), we split it to multiple subsets. In every subset, there exists a almost periodic solution. And in the invariant subset, the almost periodic solution is exponentially stable. This paper is organized as follows. In the next section, we introduce some lemmas of exponential dichotomy method and almost periodicity. In Section 3, we construct an invariant basin of MAM neural network (3) and split it into multiple invariant subsets. In Section 4, we investigate the existence of multiple almost periodic solutions of (3) by using exponential dichotomy and Schauder׳s fixed point theorem. Meanwhile, we construct exponential attracting domain of each almost periodic solution. In Section 5, we investigate the exponential stability of multiple almost periodic solutions of (3) by constructing exponential attracting domain of each almost periodic solution. In Section 6, an example is given to illustrate the effectiveness of our results.
Section snippets
Preliminary
We firstly introduce two lemmas on exponential dichotomy [27].
Set the vector Lemma1 Let . If are almost periodic functions and , then systemadmits an exponential dichotomy. Lemma 2 If system (5) admits an exponential dichotomy, the almost periodic system has a unique almost periodic solution , and
Invariant basins
In this section, we will prove that the set Φ as following is an invariant basin of MAM neural network (3). Under some assumptions on activation functions and system, we construct invariant subsets, which methods used in this section are similar to the literatures [24], [26].
Set Theorem 1 If conditions (H1)–(H3) hold, then Φ is an invariant basin of (3). proof Let be the solution of (3) with initial condition . From (3),we
The existence of almost periodic solutions
In this section, we discuss the existence of almost periodic solution of system (3). We firstly define a vectorwhere is defined as follows. If and , then or 3; if or then . And define the following subsets of Ψwhere the vector q is defined as (16).
We make further the assumption (H7).
(H7) There exist constants such that for any
The exponential stability of almost periodic solutions
In this section, in order to discuss the stability of almost periodic solutions of MAM neural network (3), we restrict in vector (16), and thus there exist vectors . Set where , , and define vectors .
We relax the condition (21) to the following assumption (H8).
(H8) There exist constants such that Theorem 5 Suppose that conditions (H1)~(H8) hold.
An example
Consider an MAM neural network with four fields as follows:where , the neuronal signal decay rates , , the external inputs , the connection weights
Conclusions
In this paper, the multistability has been studied for the MAM neural network (3) with almost periodic coefficients and continuously distributed delays. Sufficient conditions are obtained which ensure the existence of almost periodic solutions. It is proved that the almost periodic solutions among them are exponentially stable.
We know that constants and periodic functions are all special almost periodic functions. The discrete time delay can be obtained from distributed delays with
Tiejun Zhou is a professor of mathematics at the College of Science, Hunan Agricultural University, Changsha, China. He received the M.Sc. degree in applied mathematics from the University of Hunan, Changsha, China, in 2003, and received the Ph.D. degree in mathematics from Center South University, Changsha, China, in 2007. His research interests focus on the theory of differential and difference equations, neural networks and population dynamical systems.
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Tiejun Zhou is a professor of mathematics at the College of Science, Hunan Agricultural University, Changsha, China. He received the M.Sc. degree in applied mathematics from the University of Hunan, Changsha, China, in 2003, and received the Ph.D. degree in mathematics from Center South University, Changsha, China, in 2007. His research interests focus on the theory of differential and difference equations, neural networks and population dynamical systems.
Yi Wang received the master degree in computer application technology from Central South University, Changsha, China,in 2006. Since 1995, she has been in the College of Information Science and Technology at Hunan Agricultural University, Changsha, China, where she is currently an associate professor of computer science and technology. Her research interests include crop information acquisition and neural network dynamical systems.
Min Wang received the M.Sc. in mathematics from the Hunan Normal University of Changsha, Hunan, China, in 2006, and received the Ph.D. degree in agriculture from Hunan Agriculture University of Changsha, Hunan, China, in 2014. Since 2006, she has been working in the College of Science, Hunan Agriculture University. Her research interests include dynamical systems and their applications.