Hopf Bifurcation Analysis for a Four-Dimensional Recurrent Neural Network with Two Delays

A four-dimensional recurrent neural network with two delays is considered. The main result is given in terms of local stability and Hopf bifurcation. Sufficient conditions for local stability of the zero equilibrium and existence of the Hopf bifurcation with respect to both delays are obtained by analyzing the distribution of the roots of the associated characteristic equation. In particular, explicit formulae for determining the direction of the Hopf bifurcation and the stability of the bifurcating periodic solutions are established by using the normal form theory and center manifold theory. Some numerical examples are also presented to verify the theoretical analysis.


Introduction
In recent years, neural networks have attracted many scholars' attention all over the world and have been applied in different areas such as signal processing [1], pattern recognition [2][3][4], optimization [5], and automatic control [6][7][8].In particular, the appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter has been widely investigated [9][10][11][12][13][14][15][16][17].In [18], Ruiz et al. studied the following recurrent neural network for the first time: where () ∈   is the state,   ∈ ,  = 1, . . .,  − 1 are the network parameters or weights, () is the input, () is the output, and (⋅) is the transfer function of the neurons.The three-node network of system (1) in the feedback configuration, with () = (), has been studied in [12,18,19]; that is ( It is well known that time delays can play a complicated role on neural networks.They can be the source of instabilities and bifurcation in neural networks.Based on this fact, Hajihosseini et al. [11] considered system (2) with distributed delays and (⋅) = tanh(⋅).It is shown that a Hopf bifurcation takes place in the delayed system as the mean delay passes a critical value where a family of periodic solutions bifurcate from the equilibrium.The existence and stability of such solutions are determined by the Hopf bifurcation theorem in the frequency domain and the generalized Nyquist stability criterion.
As far as we know, there are some papers on the bifurcations of neural network with two or multiple delays [20][21][22].Motivated by the work in [11,[20][21][22] and considering that when the number of neurons is large, the simplified model can reflect the really large neural networks more closely, we consider the following four-dimensional recurrent neural network with two discrete delays that occur in the interaction between the neurons: where  1 ≥ 0,  2 ≥ 0 are time delays that occur in the interaction between the neurons.This paper is organized as follows.In Section 2, the stability of the zero equilibrium of system (3) and the existence of local Hopf bifurcation with respect to possible combinations of the two delays are investigated.In Section 3, the properties of the Hopf bifurcation such as the direction and the stability are determined by using the normal form theory and center manifold theory.Some numerical simulations are also included in Section 4 to illustrate the validity of the main results.
Clearly,  0 = (0, 0, 0, 0)  is the zero equilibrium of system (3).Linearization of system (3) at the zero equilibrium is The characteristic equation of the linearized system ( 4) is where In order to study the local stability of the zero equilibrium of system (3), we investigate the distribution of the roots of (5) in the following.
(I) if sin  1  1 = √1 − cos 2  1  1 , then (14) takes the following form: which is equivalent to where ( Let  = cos  1  1 , and denote that Thus, Let Let  =  + ( 2 /4 1 ).Then, (20) becomes where Define Then, we can get Then, we can get the expression of cos  1  1 and we denote 12), we can get the expression of sin  1  1 and we denote  2 ( 1 ) = sin  1  1 .Thus, a function with respect to  1 can be established by We assume that ( 21 ), (25), has finite positive roots, which are denoted by  11 , . . .,  1 .For every fixed  1 (1 ≤  ≤ ), the corresponding critical value of time delay is Then, ± 1 are a pair of purely imaginary roots of (11) with can be transformed into the following form: (28) Thus, similar as the process in case (I), we can get the expression of cos  1  1 and sin  1  1 .Let Therefore, Then, we can get the critical value of time delay corresponding to every fixed positive root   1 of (30):

Journal of Applied Mathematics
Let Next, we verify the transversality.Taking the derivative of  with respect to  1 in (11), we obtain Thus, where Namely, if the condition ( 22 ) holds, then the transversality condition is satisfied.By the discussion above and the Hopf bifurcation theorem in [23], it is easy to obtain the following results.= 0 holds, then the zero equilibrium  0 of system (3) is asymptotically stable for  1 ∈ [0,  10 ), system (3) undergoes a Hopf bifurcation at  0 when  1 =  10 , and a branch of periodic solutions bifurcates from the zero equilibrium near  1 =  10 .Case 3 ( 2 > 0,  1 = 0).When  1 = 0, (5) becomes the following form: Let  =  2 ( 2 > 0) be a root of (36).Substituting it into (36) and separating the real and imaginary parts, we obtain It follows that where Let  2 2 = , then (38) can be transformed into Next, we make the following assumption.
(H 31 ) means that (40) has at least one positive root.
Case 4 ( 1 =  2 =  > 0).For  1 =  2 =  > 0, (5) can be transformed into the following form: Multiplying  2 on both sides of (48), we obtain Let  =  be a root of (49); then we have (51) Then, we get Similar as in Case 2, we can obtain the expression of cos 2 and sin 2, which is denoted as  1 () and  2 (), respectively.Further we can get a function with respect to Next, we make the following assumption.( 41 ): Equation (53) has finite positive real roots, which are denoted by  1 , . . .,   , respectively.For every fixed positive root of (53), the corresponding critical value of time delay is Differentiating both sides of (49) with respect to , we can obtain Thus, where Obviously, if the condition ( 42 ):       +       ̸ = 0 holds, then Re[/ 2 ] −1 = 0 ̸ = 0. Namely, if the condition ( 42 ) holds, the transversality condition is satisfied.Thus, by the Hopf bifurcation theorem in [23] we have the following results.
Let  =  2 * ( 2 * > 0) be a root of (5).Then, we can get In the following, we make the following assumption.
Through the analysis above and by the Hopf bifurcation theorem in [23], we have the following results.

Conclusion
In this paper, we have investigated a four-dimensional recurrent neural network with two discrete delays.Compared with the literature [11], we consider the neural network model which can reflect the really large neural networks more closely.By regarding the possible combinations of the two delays as the bifurcation parameter, sufficient conditions for the local stability of the zero equilibrium and the existence of Hopf bifurcation are obtained.If the conditions are satisfied,