Stability and Hopf Bifurcation of an n-Neuron Cohen-Grossberg Neural Network with Time Delays

A Cohen-Grossberg neural network with discrete delays is investigated in this paper. Sufficient conditions for the existence of local Hopf bifurcation are obtained by analyzing the distribution of roots of characteristic equation. Moreover, the direction and stability of Hopf bifurcation are obtained by applying the normal form theory and the center manifold theorem. Numerical simulations are given to illustrate the obtained results.


Introduction
In recent years, more and more mathematicians, biologists, physicists, and computer scientists focus on artificial neural networks.It is well known that the analysis of the dynamical behaviors is a necessary step for practical design of neural networks since their applications heavily depend on the dynamical behaviors; many important results on dynamical behaviors of neural networks have been obtained .The neural networks are large-scale and complex systems, and the dynamical behaviors of neural networks with delays are more complicated; in order to obtain a deep and clear understanding of the dynamics of complicated neural networks with time delays, researchers have focused on the studying of simple systems [12][13][14][15][16][17][18][19][20][21][22].This is indeed very useful since the complexity found may be carried over to large neural networks.
The research on dynamical behaviors of neural networks involves not only the dynamic analysis of equilibrium but also that of periodic solution, bifurcation, and chaos; especially, the periodic oscillatory behavior of the neural networks is of great interest in many applications [2,3].Since periodic oscillatory can arise through the Hopf bifurcation in different system with or without time delays, it is very important to discuss the Hopf bifurcation of neural networks.
In 1983, Cohen-Grossberg [1] proposed a kind of neural networks, which are now called Cohen-Grossberg neural networks.The networks have been successfully applied to signal processing, pattern recognition, optimization, and associative memories.Recently, some results on the existence and globally asymptotical stability of periodic Cohen-Grossberg neural networks have been obtained [7][8][9][10][11][12][13][14][15].However, up to now, to the best of the author's knowledge, bifurcation of Hopfield neural networks has been discussed by many researchers [12][13][14][15][16][17][18][19], but few results on the bifurcation of Cohen-Grossberg neural networks have been obtained.Zhao discussed the bifurcation of a two-neuron discrete-time Cohen-Grossberg neural network in [20] and the bifurcation of a two-neuron continuous-time Cohen-Grossberg neural network with distributed delays in which kernel function is  − in [21].We discussed the bifurcation of a two-neuron Cohen-Grossberg neural network with discrete delays in [22].The objective of this paper is to study the following neuron continuous-time Cohen-Grossberg neural network with discrete delays and ring architecture: where   () denote the state variable of the th neuron;   (⋅) represent amplification functions which are positive for ;   (⋅) denote the signal functions of the th neuron;   (⋅) are appropriately behaved functions;   are connection weights of the neural networks; discrete delays   correspond to the finite speed of the axonal signal transmission:  = 1, 2, . . ., ,  ≥ 2.
Ring architectures have been found in variety of neural structures, and they are investigated to gain insight into the mechanisms underlying the behaviors of recurrent neural networks [23].
The rest of this paper is organized as follows.Stability property and existence of Hopf bifurcation for system (1) are obtained in Section 2. Based on the normal form method and the center manifold, the formulas for the direction of Hopf bifurcation and stability of the bifurcating periodic solutions are derived in Section 3.An example is given in Section 4 to illustrate the main results, and conclusions are drawn in Section 5.
We know from the second equation in (9) that ℎ() = .Denote   = ℎ −1 (),  = 1, 2, . ..; then we have from the first equation in (9) that When  = 0,  =  0 = ∏  =1   > 0. Furthermore, from the value   given above, we have Obviously, if  is a root of equation ( 7), − is also the root of equation ( 7).This implies that ± is a pair of purely imaginary roots of equation (7).On the other hand, we have from ( 7) that where Hence we have Since the roots of the characteristic equation ( 7) are −  < 0,  = 1, 2, . . .,  when  = 0, so the equilibrium (0, 0, . . ., 0) of system ( 5) is asymptotically stable.As the parameter  varies on the open right half plane can change only if a zero appears on or across the imaginary axis.According to Lemma 1 and ( 14), we obtain that the equilibrium (0, 0, . . ., 0) of system ( 5) is asymptotically stable if and only if  ∈ ( 1 ,  0 ).When  =  0 = ∏  =1   , the characteristic equation of system (5) has a simple root  = 0, and all the other roots have negative real parts.A pitchfork bifurcation may occur at the origin in system (5) [16].
From the above discusses, Lemma 1, and the Hopf bifurcation theorem in [24] for functional differential equations, we have the following results.

Direction and Stability of Hopf Bifurcation for the Network
In this section, we will derive explicit formulas for determining the properties of the Hopf bifurcation at critical  =  1 by using the normal form theory and the center manifold theorem [25], and we always make vary with a parameter and the other 3 − 1 ones are fixed.We still discuss system (5).For the sake of generality, let () = ().Denote  =  and  = , dropping the bars for simplification of notation; then system (5) can be written as functional differential equation in  = ([−1, 0],   ) as where () = ( 1 (),  2 (), . . .,   ())  ,   () = ( + ), and   :  → ,  :  ×  →  are given, respectively, by where where in which From the discussions in Section 2, we know that if  =  1 , system (19) undergoes a Hopf bifurcation at the equilibrium (0, 0, . . ., 0), and the associated characteristic equation of system (19) has a pair simple imaginary roots ± 1 .
Denote the th element of (0) by   and the th element of (, , ) by Then if follows that Substitute ( 46) into (43) and comparing the coefficients in (40) with those in (43), we have where   is th element of (0) shown in (32).Since there are  20 () and  11 () in  21 , we still need to figure them out.Note that on the center manifold  0 , we have We have from ( 41), (42), and (48) that Equations ( 39) and (41) mean for  ∈ [−1, 0).

A Numerical Example
Example 1.Consider the following Cohen-Grossberg neural network with discrete delays: We can obtain that  1 = 0.3423 and furthermore we obtain that  1 = −6.2979 in view of bisection method by using MATLAB.It is easy to know  0 = 5.4.We also know from (3) that  = −7.2 1 .
According to Theorem 2, the zero solution of system (64) is asymptotically stable when  ∈ (−6.2979, 5.4), and when  =  1 , the Hopf bifurcation occurs at the origin.
The presented numerical simulations illustrate the theoretical results.

Conclusions
An -neuron Cohen-Grossberg neural network with discrete delays and ring architecture is analyzed in this paper.By using  = ∏  =1   = ∏  =1   (0)     (0) as a bifurcation parameter, we show that this system undergoes a Hopf bifurcations at a critical parameter: where  1 ∈ (0,/) and satisfies the equation  = ∑  =1 cot −1 (  /) + ,  =  1 +  2 + ⋅ ⋅ ⋅ +   .The direction of Hopf bifurcation and the stability of the bifurcating periodic solutions are investigated by applying the normal form theory and the center manifold theorem for continuous time system.The phenomena of bifurcating periodic solutions for Cohen-Grossberg neural networks coincide with the fact that learning usually requires repetition [2], and periodic sequences of neural impulse are also of fundamental significance for the control of dynamic functions of the body such as heart beat which occurs with great regularity and breathing [19].In this paper, we extend the results about the existence of local Hopf bifurcation in [22] to the case of a discrete-time -neuron Cohen-Grossberg system with discrete delays.In the future, the problem for the existence of global Hopf bifurcation will be expected to be solved.
and  are local coordinates for the center manifold  0 in the direction of  * and  * .Note that  is real if   is real.We only consider real solutions.For solution   ∈  0 of (19), since  =  1 , we have def=  1  +  * (0)  0 (, ) .