Anti-periodic Solutions for Neural Networks with Delays and Impulses

In this paper we investigate a class of artificial neural networks with delays subject to periodic impulses. By exploiting Lyapunov functions, we analyze the global exponential stability of an arbitrary solution with initial value being bounded by . Further, we discuss the existence of anti-periodic solutions by constructing fundamental function sequences based on a solution with initial value being bounded by . We also establish sufficient conditions to ensure the existence, uniqueness and exponential stability of anti-periodic solutions, which are new and easily verifiable. At last, we present a network with its time-series and phase graphics to demonstrate our results.


INTRODUCTION
It is a significant fact that we spend a lot time in investigating artificial neural networks.We explore and study artificial neural networks, and we intend to reproduce products such as Robots, which possess functions similar to human's and can process signals and information.Mathematical models for artificial neural networks have been established.Neural networks, which are described as continuous differential dynamical systems, have been discussed, and there are extensive results about their dynamical behaviors including equilibriums, periodic solutions, anti-periodic solutions and their respective stability ( [1], [2], [3], [4], [6], [10], [11]).But, in artificial networks for signal and image processing, the finites switching speed of amplifiers may cause delays in the transmission of signals, in addition, faulty elements may experience abrupt changes of state voltage and therefore the normal transient behaviors in processing signal and information are influenced.Consequently an artificial neural network with delays and impulses should be modeled.The dynamics of autonomous neural networks with delays and impulses have been considered recently, and the research are mainly focus on the existence and stability of an equilibrium state and periodic solution ( [5], [7], [8], [19]).To the best of our knowledge, few authors have considered the problems of anti-periodic solutions for non-autonomous delayed networks with impulses.In the neural networks, the performance of the global stable anti-periodic solution may reveal the characteristic and stability of the mode of the signal.Sometimes, the existence and stability of a unique anti-periodic solution is a requirement in designing artificial neural networks, especially the existence of destabilizing factors such as delays and impulses.So we discuss anti-periodic problems on ANNs with delays and impulses.In this paper, we first consider anti-periodic solutions for the following delayed differential system with impulses: Here n is the number of units in a neural network and and there exists a constant , and are continuous and there exist nonnegative constants j L and From inequality (2), we easily know

MAIN RESULTS
Lemma 2.1.Let (H1)-(H6) hold.Assume that Proof.Suppose that (3) does not hold.Since and by (H2), then . Thus, we may assume that there exist Calculating the upper left derivative of ) (t x i , together with (H4), ( 2), ( 4), and , it is followed by And it is easily known that Obviously, (7) holds for 0  t .We first prove that ( 7) is true for Combining ( 8) with ( 6), we have which contradicts to (H6).Therefore, (7) holds.From (7), we know that we may repeat the above procedure, and obtain Further, we have .By hypotheses (H1), we have, for Further, by hypothesis of (H3), we have Further, for any natural number m , we have In view of ( 11), ( 12) and (13), we know that ) ) will converge uniformly to a piecewise continuous function on any compact set of R .Now we are in the position of proving that ) Further, from (9), and by hypothesis (H4) and (H5), we have, for on both side of ( 9) and ( 10), we have is a T -anti-periodic solution of system (1).

AN EXAMPLE
In this section, we illustrate our results by numerical simulation.Example 3.1.Consider the following neural network consisting of two neurons , which is described by Here, it is assumed that the activation functions are the external inputs.Obviously, network ( 14) is an anti-periodic system with anti-period T being 2 and impulsive jumps occur at time , from the result of paper [11], we know that there exists unique 2-anti-periodic solution, which is globally exponentially stable.In Fig. 1  and 2 .Further, we discuss the influence of impulsive effects on the existence of antiperiodic solutions.We choose the magnitude of impulsive jump for all N k  .It is easily verified that all conditions in Theorem 2.1 are satisfied.Thus, system (14) has exactly one 2-anti-periodic solution, and it is globally asymptotically stable.This result is demonstrated by the following numerical simulation in Fig. 3  .When t tends to the positive infinity, this solution will tends to the anti-periodic solution with anti-period being 2.
Figure 1.The time series graph of ``t-x(t)" and ``t-y(t)"for system (14) without impulses Figure 4.The time series graph of ``t-x(t)" and ``t-y(t)"for system (14) with impulses

CONCLUDING REMARKS
A set of sufficient conditions are derived in this work to guarantee the existence and global exponential stability of a unique anti-periodic solution for an artificial neural networks with delays and impulses.One of our conclusions is that every solution eventually converges exponentially to the anti-periodic solution under the impulsive jumps of the form ) ( ) ( , if the frequency and the magnitude of the impulsive effects satisfy the hypotheses (H2) and (H3).Although continuous differential ANNs systems are studied extensively, but there are few results on antiperiodic solution for non-autonomous ANNs with delays and impulses.

Figure 2 .Figure 3 .
Figure 2. The graph of ``x-y" for (14) without impulses a contradiction, which shows that (3) holds.The proof is now completed.
and the delay kernels , we draw the time-series of "