Estimate of Multiple Attracing Domains for Cohen-Grossberg Neural Network with Distributed Delays

In this paper, we present multiplicity results of exponential stability and attracting domains for CohenGrossberg neural network (CGNN) with distributed delays. We establish new criteria for the coexistence of equilibrium points and estimate their attracting domains. Moreover, we base our criteria on coefficients of the networks and the derivative of activation functions within the attracting domains. It is shown that our results are new and complement corresponding results existing in the previous literature. 2N


Introduction
Cohen-Grossberg neural network (see [1,2]) is usually described by the following differential equations system where , is the number of neurons in the network; describes the state variable of neuron i at time t; i    represents an amplification function and the function can include a constant term indicating a fixed input to the network; weights the strength of the j unit on the unit at time t; the activation function th i   j g  shows how the neurons react to the input.CGNN not only has a wide range of applications in pattern recognition, associative memory and combinatorial optimization but also includes a number of models from neurobiology, population biology and evolution theory.Hence studies on stability of CGNN with or without delays have been vigorously done and many criteria have been obtained so far [3][4][5][6][7][8][9][10][11][12][13][14].
In the applications of neural network to associative memory storage or pattern recognitions, the coexistence of multiple stable equilibrium points is an important feature [15][16][17][18][19][20][21].However, few papers focus on the existence of multiple equilibrium points of CGNN and their complex convergence analysis.Hence, we should consider multistability of the following CGNN with distributed delays where the delay kernel function is assumed to be

 
 is a positive constant.In this paper, we not only de new criteria for the existence of 2 rive N equilibrium points of CGNN (1.1) but also estimate at acting domains for these equilibrium points.When we relax our conditions to be common assumptions, our results improve corresponding results in [12].Moreover, our results can extend the corresponding results in [3][4][5][6][7][8][9][10][11][12][13] to local exponential stability of multiple equilibrium points of Cohen-Grossberg networks.It is shown that our results are new and complement the existing results in the literature.
The rest tr of this paper is organized as follows.In Section 2, we should make some preparations by giving some notations, assumptions and a basic lemma.Meanwhile, we discuss the existence of 2 N equilibrium points of CGNN (1.1).In Section Throughout this paper we always assume that   For each and there where i and l networks [5

Rem
For ra ,8-10], we have   , where 0 where .Then it follows ma 2 Assu nd the following assumption , there exist only tw points and with The proof is complete.
Now, we consid wi er the follo ng additional assumption: ere exists a unique ,by similar argument, we de- Due to the monotonicity of following theorem With these notations, w h e have t Theorem 2.1.Under the assumptions     , there exist at least 2 N e point .1).Proof.For each , , , . For ea (2.5) . Therefore, there exist at least 2 N equilibriu CGNN (1.1).The proof is comp e.
Next we should make some preparations for the comin m points of let g section.For each i   , we define the following subsets of , we defi subsets for each ne semi-close

ility a imation of Attracting Domains
Th that assumptions

Stab
s for all such that 0 which leads to a contradiction.Since the choice of holds for all 0 t  .Hence, for any , we have that , if there exist and hold.For each and Then is locally exponentially stable and .In view of (3.2), we obtain where From (3.5), it is easy for us to estimate From (3.4), (3.5) and by simple calculations, we obtain It follows from (3.3) that .Hence for any Therefore, we have Then is locally exponentially stable and u  H  is containe he attracting domain of Rem 1 For each d in t ark 3.
ous with a pschitz constant j L and there exists nstant 0

S
For each i   ,   i g  is globally Lipschitz continuous with a Lipschitz constant i L and there exists a constant 0 If there exist positive constants T en there exists at least a unique f CGN (1.1).which is globally exponential Proof.
Corollary 3.2 leads to Theorem 3.2 and Theorem 3.3 in [17].It is obvious that Theorem 3.2-3.5 in [9] are only ou vior of C

Concluding Remarks
In this paper, some new c istence of 2N equilibrium also given for each e results are new and co [7, [9][10][11] and (3.7), we obtain thatCopyright © 2011 SciRes.AM attracting domains H  can be estimated as Example 4.1. )