New Results of a Class of Two-Neuron Networks with Time-Varying Delays
© Chuangxia Huang et al. 2008
Received: 19 September 2008
Accepted: 4 December 2008
Published: 5 March 2009
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.
where , , and are constants, the functions and satisfy , , ; for ; for ; , .
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.
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.
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 .
To obtain the existence of the periodic solution of (1.5), we will introduce some results from Gaines and Mawhin .
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 .
with the agreement that the above sum is zero if . For more details about the degree theory, one refers to the book of Deimling .
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 ;
Then, the equation has at least one solution lying in . For more details about degree theory, we refer to the book by Deimling .
3. Main results
Suppose holds and , , then system (1.5) has a periodic solution.
We have shown that satisfies all the assumptions of Lemma 2.2. Hence, has at least one -periodic solution on . This completes the proof.
Suppose that there exist positive constants such that for . Then system (1.5) has at least an -periodic solution.
As ( ) implies that , hence, the conditions in Theorem 2.1 are all satisfied.
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  when without assuming the boundedness, monotonicity, and differentiability of activation functions. In fact, the explicit presence of in Theorem 2.1 (see ) 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.
Suppose that satisfy the hypotheses . If , , then system (1.5) has exactly one periodic solution . Moreover, it is globally exponentially stable.
This completes the proof.
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  and the condition: of Theorem 3.1 in . 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 , we will find that the theoretical conclusions are in excellent agreement with the numerically observed behavior.
where and are any -periodic continuous functions for .
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.
This work was supported in part by the High-Tech Research and Development Program of China under Grant no. 2006AA04A104, China Postdoctoral Science Foundation under Grants no. 20070410300 and no. 200801336, the Foundation of Chinese Society for Electrical Engineering, and the Hunan Provincial Natural Science Foundation of China under Grant no. 07JJ4001.
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