New Results of a Class of Two-Neuron Networks with Time-Varying Delays

Abstract

With the help of the continuation theorem of the coincidence degree, a priori estimates, and differential inequalities, we make a further investigation of a class of planar systems, which is generalization of some existing neural networks under a time-varying environment. Without assuming the smoothness, monotonicity, and boundedness of the activation functions, a set of sufficient conditions is given for checking the existence of periodic solution and global exponential stability for such neural networks. The obtained results extend and improve some earlier publications.

1. Introduction

Neural networks are complex and large-scale nonlinear dynamics, while the dynamics of the delayed neural network are even richer and more complicated [1]. To obtain a deep and clear understanding of the dynamics of neural networks, one of the usual ways is to investigate the delayed neural network models with two neurons, which can be described by differential systems (see [2–8]). It is hoped that, through discussing the dynamics of two-neuron networks, we can get some light for our understanding about the large networks. In [7], Táboas considered the system of delay differential equations

(1.1)

which arises as a model for a network of two saturating amplifiers (or neurons) with delayed outputs, where is a constant, and are bounded functions on satisfying

(1.2)

and the negative feedback conditions: , ; , . Táboas showed that there is an such that for , there exists a nonconstant periodic solution with period greater than . Further study on the global existence of periodic solutions to system (1.1) can be found in [2, 3]. All together there is only one delay appearing in both equations. Ruan and Wei [6] investigated the existence of nonconstant periodic solutions of the following planar system with two delays

(1.3)

where , , and are constants, the functions and satisfy , , ; for ; for ; , .

Recently, Chen and Wu [9] considered the following system:

(1.4)

where and are positive constants, and are nonnegative constants with , and are constants, and and are bounded functions with , and for . By discussing a two-dimensional unstable manifold of the transformation of the system (1.4), they got some results about slowly oscillating periodic solutions.

However, delays considered in all above systems (1.1)–(1.4) are constants. It is well known that the delays in artificial neural networks are usually time-varying, and they sometime vary violently with time due to the finite switching speed of amplifiers and faults in the electrical circuit. They slow down the transmission rate and tend to introduce some degree of instability in circuits. Therefore, fast response must be required in practical artificial neural-network designs. As pointed out by Gopalsamy and Sariyasa [10, 11], it would be of great interest to study the neural networks in periodic environments. On the other hand, drop the assumptions of continuous first derivative, monotonicity, and boundedness for the activation might be better candidates for some purposes, (see [12]). Motivated by above mentioned, in this paper, we continue to consider the following planar system:

(1.5)

where , , are periodic with a common period , , , being -periodic.

Under the help of the continuation theorem of the coincidence degree, a priori estimates, and differential inequalities, we make a further investigation of system (1.5). Without assuming the smoothness, monotonicity, and boundedness of the activation functions, a family of sufficient conditions are given for checking the existence of periodic solution and global exponential stability for such neural networks. Our results extend and improve some earlier publications. The remainder of this article is organized as follows. In Section 2, the basic notations and assumptions are introduced. After giving the criteria for checking the existence of periodic solution and global exponential stability for the neuron networks in Section 3, one illustrative example and simulations are given in Section 4. We also conclude this paper in Section 5.

2. Preliminaries

In this section, we state some notations, definitions, and lemmas. Assume that nonlinear system (1.5) is supplemented with initial values of the type

(2.1)

in which are continuous functions. Set , if is a solution of system (1.5), then we denote

(2.2)

Definition 2.1.

The solution is said to be globally exponentially stable, if there exist and such that for any solution of (1.5), one has

(2.3)

where is called to be globally exponentially convergent rate.

To establish the main results of the model (1.5), some of the following assumptions will be applied:

(H₁) for all , where are constants;

(H₂) there exist constants such that for any .

For , one denotes

(2.4)

To obtain the existence of the periodic solution of (1.5), we will introduce some results from Gaines and Mawhin [13].

Consider an abstract equation in a Branch space In this section, we use the coincidence degree theory to obtain the existence of an -periodic solution to (1.5). For the sake of convenience, we briefly summarize the theory as below.

Let and be normed spaces, be a linear mapping, and be a continuous mapping. The mapping will be called a Fredholm mapping of index zero if and is closed in . If is a Fredholm mapping of index zero, then there exist continuous projectors and such that and . It follows that is invertible. We denote the inverse of this map by . If is a bounded open subset of , the mapping is called -compact on if is bounded and is compact. Because is isomorphic to , there exists an isomorphism .

Let be open and bounded, and , that is, is a regular value of . Here, , the critical set of , and is the Jacobian of at . Then, the degree is defined by

(2.5)

with the agreement that the above sum is zero if . For more details about the degree theory, one refers to the book of Deimling [14].

Now, with the above notation, we are ready to state the continuation theorem.

Lemma 2.2 (Continuation theorem [13, page 40]).

Let be a Fredholm mapping of index zero and let be -compact on . Suppose that

(a) for each , every solution of is such that ;

(b) for each and

(2.6)

Then, the equation has at least one solution lying in . For more details about degree theory, we refer to the book by Deimling [14].

For the simplicity of presentation, in the remaining part of this paper, we also introduce the following notation:

(2.7)

3. Main results

Theorem 3.1.

Suppose holds and , , then system (1.5) has a periodic solution.

Proof.

Take and denote

(3.1)

Equipped with the norm , is a Banach space. For any , because of the periodicity, it is easy to check that

(3.2)

Let

(3.3)

where for any , we identify it as the constant function in X with the value vector . Define given by

(3.4)

Then, system (1.5) can be reduced to the operator equation . It is easy to see that

(3.5)

and P, Q are continuous projectors such that

(3.6)

It follows that L is a Fredholm mapping of index zero. Furthermore, the generalized inverse (to ) is given by

(3.7)

Thus,

(3.8)

Clearly, and are continuous. For any bounded open subset , is obviously bounded. Moreover, applying the Arzela-Ascoli theorem, one can easily show that is compact. Note that is a compact operator and is bounded, therefore, is -compact on with any bounded open subset . Since , we take the isomorphism of onto to be the identity mapping. Corresponding to equation , we have

(3.9)

Now we reach the position to search for an appropriate open bounded subset for the application of the Lemma 2.2. Assume that is a solution of system (3.9) for some . Then, the components of are continuously differentiable. Thus, there exists such that . Hence, . This implies

(3.10)

Since

(3.11)

we get

(3.12)

Set , , we find that

(3.13)

Now, we choose a constant number and take

(3.14)

where , . We will show that satisfies all the requirements given in Lemma 2.2. In fact, we will prove that if then for . Therefore, it means that is uniformly bounded with respect to when the initial value function belongs to . It follows from (3.12) and (3.13) that

(3.15)

. Therefore,

(3.16)

Clearly, , are independent of . It is easy to see that there are no and such that . If , then is a constant vector in with for . Note that , we have

(3.17)

We claim that

(3.18)

Contrarily, suppose that there exists some such that , that is,

(3.19)

So, we have

(3.20)

this is a contradiction. Therefore, (3.17) holds, and hence,

(3.21)

Now, consider the homotopy , defined by

(3.22)

where and . When and , is a constant vector in with for . Thus

(3.23)

We claim that

(3.24)

Contrarily, suppose that , then,

(3.25)

Thus,

(3.26)

This is impossible. Thus, (3.22) holds. From the property of invariance under a homotopy, it follows that

(3.27)

We have shown that satisfies all the assumptions of Lemma 2.2. Hence, has at least one -periodic solution on . This completes the proof.

Corollary 3.2.

Suppose that there exist positive constants such that for . Then system (1.5) has at least an -periodic solution.

Proof.

As () implies that , hence, the conditions in Theorem 2.1 are all satisfied.

Remark 3.3.

From Corollary 3.2, we can find that the condition in [4, Theorem 2.1] can be dropped out. Therefore, Theorem 3.1 is greatly generalized results to [4, Theorem 2.1]. Furthermore, we should point out that the Theorem 3.1 is different from the the existing work in [5] when without assuming the boundedness, monotonicity, and differentiability of activation functions. In fact, the explicit presence of in Theorem 2.1 (see [5]) may impose a very strict constraint on the coefficients of (1.5) (e.g., when is very large or small). Since our results are presented independent of , it is more convenient to design a neural network with delays.

Theorem 3.4.

Suppose that satisfy the hypotheses . If , , then system (1.5) has exactly one periodic solution . Moreover, it is globally exponentially stable.

Proof.

From , we can conclude is true. It is obvious that all the hypotheses in Theorem 2.1 hold with Thus, system (1.5) has at least one periodic solution, say . Let be an arbitrary solution of system (1.5). For , a direct calculation of the upper left derivative of along the solutions of system (1.5) leads to

(3.28)

where denotes the upper left derivative, . Let . Then, (3.27) can be transformed into

(3.29)

From , and (3.13), we have

(3.30)

Then, there exists a constant such that

(3.31)

Thus, we can choose a constant such that

(3.32)

Now, we choose a constant such that

(3.33)

Set

(3.34)

for . From (3.32) and (3.34), we obtain

(3.35)

In view of (3.33) and (3.34), we have

(3.36)

We claim that

(3.37)

Contrarily, there must exist and such that

(3.38)

It follows that

(3.39)

From (3.29), (3.35), and (3.38), we obtain

(3.40)

which contradicts (3.39). Hence, (3.37) holds. Letting and , from (3.34) and (3.37), we have

(3.41)

This completes the proof.

Remark 3.5.

From Theorem 3.4, it is easy to see that our results independent of and , we have dropped out the condition (b) of Theorem 3.1 in [4] and the condition: of Theorem 3.1 in [5]. Therefore, it is generalized the corresponding results in [4, 5].

4. Example and Numerical Simulation

In this section, we will give a example to illustrate our results. Using the method of numerical simulation in [15], we will find that the theoretical conclusions are in excellent agreement with the numerically observed behavior.

Example 4.1.

Consider a two-neuron networks with delays as follows:

(4.1)

where and are any -periodic continuous functions for .

Obviously, the requirements of smoothness, monotonicity, and boundedness on the activation functions are relaxed in our model. By some simple calculations, we obtain , , , , , , . Therefore, , . Applying Theorem 3.4, there exists a unique -periodic solution for (4.1) and it is globally exponentially stable. These conclusions are verified by the following numerical simulations in Figures 1 and 2.

Figure 1

Numerical solution of system (4. 1), where for .

Figure 2

Phase trajectories of system (4. 1).

5. Conclusion

In this paper, we have derived sufficient algebraic conditions in terms of the parameters of the connection and activation functions for periodic solutions and global exponential stability of a two-neuron networks with time-varying delays. The obtained results extend and improve some earlier publications, which are all independent of the delays and the period () and may be highly important significance in some applied fields.