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Abstract and Applied Analysis
Volume 2014 (2014), Article ID 682592, 9 pages
Dynamics of Almost Periodic BAM Neural Networks with Neutral Delays
Department of Mathematics, Kunming University, Kunming 650031, China
Received 30 December 2013; Revised 6 April 2014; Accepted 6 April 2014; Published 28 April 2014
Academic Editor: Sabri Arik
Copyright © 2014 Yaqin Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
The paper investigates the almost periodic oscillatory properties of neutral-type BAM neural networks with time-varying delays. By employing the contracting mapping principle and constructing suitable Lyapunov functional, several sufficient conditions are established for the existence, uniqueness, and global exponential stability of almost periodic solution of the system. The results of this paper are new and a simple example is given to illustrate the effectiveness of the new results.
Recent years have witnessed rapid development of bidirectional associative memory (BAM) neural networks due to the vast applications in pattern recognition, artificial intelligence, automatic control engineering, and optimization because of its better abilities of information memory and information association [1–6]. It is well known that studies on neural dynamical systems not only involve a discussion of stability properties, but also involve many dynamic behaviors such as periodic oscillatory behavior, almost periodic oscillatory properties, chaos, and bifurcation. In applications, almost periodic oscillatory is more accordant with fact [7–9]. A great number of results for BAM neural networks concerning the existence and global stability of (almost) periodic solution have been derived (see, e.g., [10–16]).
In addition, owing to the complicated dynamic properties of the neural cells in the real world, the existing neural network models in many cases cannot characterize the properties of a neural reaction process precisely. It is natural and important that systems will contain some information about the derivative of the past state to further describe and model the dynamics for such complex neural reactions. This new type of neural networks is called neutral neural networks or neural networks of neutral type. The motivation for us to study neural networks of neutral type comes from three aspects. First, based on biochemistry experiments, neural information may transfer across chemical reactivity, which results in a neutral-type process. Second, in view of electronics, it has been shown that neutral phenomena exist in large-scale integrated (LSI) circuits. Last, the key point is that cerebra can be considered as a super LSI circuit with chemical reactivity, which reasonably implies that the neutral dynamic behaviors should be included in neural dynamic systems [17, 18]. However, up to date, there are hardly any articles concerning the almost periodic oscillation analysis for neural networks of neutral type.
Motivated by the above reason, in this paper, we consider the following neutral-type BAM neural networks with time-varying delays: where and denote the rate with which the cells and reset their potential to the resting states when isolated from the other cells and inputs, , , , and are the connection weights at the time , and denote the constant external inputs, and are time delays in the state, and are neutral delays, , , , and are the activation functions, and , , , , , , , , , , , , , , , and are all almost periodic functions, , .
Recently, there are many papers concerning the existence and exponential stability of (almost) periodic solution for BAM neural networks [10–16]. However, the research on the neutral-type BAM neural networks is few. Therefore, the main purpose of this paper is to establish some new sufficient conditions on the existence, uniqueness, and exponential stability of almost periodic solution of neutral-type BAM neural networks (1). First, by using the exponential dichotomy and the contracting mapping principle, the existence and uniqueness of almost periodic solution of system (1) is considered. Besides, by constructing a new Lyapunov functional, the stability criterion with system (1) is introduced. The methods used in this paper provide a possible method to study the existence and exponential stability of almost periodic solutions of neutral-type neural networks.
Let and be the space of continuous functions and continuously differential functions which map into , respectively. In particular, and . For any bounded function , and .
Let . The initial conditions associated with system (1) are of the form
Now we list some assumptions which will be used in this paper.(H1).(H2)There exist some positive constants , , , and such that where and .(H3), where
The organization of this paper is as follows. In Section 2, we give some basic definitions and necessary lemmas which will be used in later sections. In Sections 3 and 4, by using a fixed point theorem and constructing suitable Lyapunov functional, we obtain some sufficient conditions ensuring existence, uniqueness, and global exponential stability of almost periodic solution of system (1). Finally, an example is given to illustrate that our results are feasible.
Now, let us state the following definitions and lemmas, which will be useful in proving our main result.
Definition 1 (see, ). is called almost periodic, if, for any , it is possible to find a real number and, for any interval with length , there exists a number in this interval such that , for all . The collection of those functions is denoted by .
Definition 2 (see, ). Let and be a continuous matrix defined on . The linear system is said to be an exponential dichotomy on if there exist constants , projection , and the fundamental matrix satisfying
Lemma 3 (see, ). If the linear system has an exponential dichotomy, then almost periodic system has a unique almost periodic solution which can be expressed as follows:
Lemma 4 (see, ). Let be an almost periodic function and Then the linear system admits an exponential dichotomy, where = .
Definition 5. The almost periodic solution of system (1) with the initial value is said to be globally exponentially stable, if there exist constants and , for any solution of system (1) with initial value such that where
Lemma 6 (see, ). Assume that is a Banach space with norm ; is a contraction mapping; that is, there exists , such that Then has a unique fixed point in .
Let and with the norm where , , and , for all . Then is a Banach space with the norm .
Let be defined by
3. Existence and Uniqueness
In this section, we study the existence and uniqueness of almost periodic solution of system (1).
Theorem 7. Assume that (H1)–(H3) hold, then system (1) has a unique almost periodic solution.
Proof. Consider the following nonlinear operator:
For all , , it follows from the definitions of and that
where and .
Further, we also obtain that where and .
Hence, Together with the above results, one has where . By Lemma 6, there exists a unique fixed point satisfying , which implies that system (1) has unique almost periodic solution. This completes the proof.
4. Global Exponential Stability
Theorem 9. Assume that (H1)–(H3) hold and suppose further the following.(H4), , , and are differential functions, with , , , , where and .(H5)There exists a positive constant such that Then system (1) has a unique almost periodic solution, which is globally exponentially stable.
Proof. It follows from Theorem 7 that system (1) has a unique almost periodic solution with initial value . We next show that the almost periodic solution is globally exponentially stable.
Make a transformation for system (1): , , , , where is arbitrary solution of system (1) with initial value .
By (H4), there exists a small enough positive constant such that
Define In view of system (1), we have
Let So Similar to the arguments as that in (30) and (31), we obtain
Define . From (28)–(33), it follows that which implies that , for all . Obviously, On the other hand, we have which implies from (35) that where Thus, the almost periodic solution of system (1) is globally exponentially stable. This completes the proof.
5. An Example
Example 1. Consider the following neutral BAM neural networks with time-varying delays: where , Then system (39) has a unique almost periodic solution, which is globally exponentially stable.
Proof. Corresponding to system (1), , , , , , and . Taking , it is easy to verify that (H1)–(H5) hold and the results follow from Theorems 7–9(see Figures 1, 2, 3, 4, 5, 6, and 7). This completes the proof.
For numerical simulation, Figures 1–3 depict the time responses of state variables of and with step of system (39), respectively. Figures 4–7 depict the phase responses of state variables , , , and , respectively. It confirms that the proposed conditions in our results are effective for system (39).
In this paper, the neutral BAM neural network is considered. By employing fixed point theory and constructing suitable Lyapunov functional some new sufficient conditions are obtained for the existence and global exponential stability of almost periodic solution of the system. Conditions (H3) and (H5) in Theorems 7 and 9 indicate that the neutral terms and time delays are harm for the existence, uniqueness, and global exponential stability of almost periodic solution of the neutral-type system. The method used in this paper provides a possible method to study the existence and global exponential stability of almost periodic solution of other neutral neural networks (with impulses [20–23]).
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
The author thanks the referee for his/her careful reading of the original paper and valuable comments and suggestions that greatly improved the presentation of this work. This work is supported by the Research Program of Kunming University in China (XJL13007).
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