#### Abstract

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.

#### 1. 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 [2–4], optimization [5], and automatic control [6–8]. 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 [9–17]. In [18], Ruiz et al. studied the following recurrent neural network for the first time: where is the state, , 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 . 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 [20–22]. Motivated by the work in [11, 20–22] 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 , 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.

#### 2. Stability of the Zero Equilibrium and Local Hopf Bifurcation

Throughout this paper we make the following assumption on the transfer function :(*H*), , .

Clearly, 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.

*Case 1 (). *Equation (5) reduces to
where

Obviously, . Therefore, by the Routh-Hurwitz criterion, the zero equilibrium of system (3) is locally asymptotically stable if the following condition holds:

*Case 2 (, ). *On substituting , (5) becomes
Multiplying on both sides of (10), it is easy to obtain
Let be a root of (11). Then, we can get
where
Squaring both sides of the two equations in (12) and adding them up we obtain
According to , we consider the two cases:

(I) if , then (14) takes the following form:
which is equivalent to
where
Let , and denote that
Thus,
Let
Let . Then, (20) becomes
where
Define
Then, we can get
Then, we can get the expression of and we denote . Substituting into (12), we can get the expression of and we denote . Thus, a function with respect to can be established by
We assume that , (25), has finite positive roots, which are denoted by . For every fixed , the corresponding critical value of time delay is
Then, are a pair of purely imaginary roots of (11) with . Let

(II) If , then (14) can be transformed into the following form:

Thus, similar as the process in case (I), we can get the expression of and . Let Therefore,

Then, we can get the critical value of time delay corresponding to every fixed positive root of (30): Let Next, we verify the transversality. Taking the derivative of with respect to in (11), we obtain Thus, where

Obviously, if the condition : holds, then . Namely, if the condition 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.

Theorem 1. *If the condition means that (25) has finite positive roots and means that holds, then the zero equilibrium of system (3) is asymptotically stable for , system (3) undergoes a Hopf bifurcation at when , and a branch of periodic solutions bifurcates from the zero equilibrium near .*

*Case 3 (, ). *When , (5) becomes the following form:
Let be a root of (36). Substituting it into (36) and separating the real and imaginary parts, we obtain
It follows that
where
Let , then (38) can be transformed into
Next, we make the following assumption.(*H*_{31}) means that (40) has at least one positive root.

Without loss of generality, we assume that (40) has four positive roots, which are denoted by , , , and . Thus, (38) has four positive roots , . The corresponding critical value of time delay is

Then, are a pair of purely imaginary roots of (36) with . Let Taking the derivative of with respect to in (36), we can get Then, we can get From (38), we have Thus, where Therefore, if the condition : , then Re. From the analysis above and by the Hopf bifurcation theorem in [23], we have the following results.

Theorem 2. *If the condition means that (40) has at least one positive root and means that holds, then the zero equilibrium of system (3) is asymptotically stable for , system (3) undergoes a Hopf bifurcation at when , and a branch of periodic solutions bifurcates from the zero equilibrium near .*

*Case 4 (). *For , (5) can be transformed into the following form:
Multiplying on both sides of (48), we obtain
Let be a root of (49); then we have
where
Then, we get

Similar as in Case 2, we can obtain the expression of and , which is denoted as and , respectively. Further we can get a function with respect to

Next, we make the following assumption. : Equation (53) has finite positive real roots, which are denoted by , respectively. For every fixed positive root of (53), the corresponding critical value of time delay is Then, are a pair of purely imaginary roots of (49) with . Let Differentiating both sides of (49) with respect to , we can obtain Thus, where

Obviously, if the condition : holds, then . Namely, if the condition holds, the transversality condition is satisfied. Thus, by the Hopf bifurcation theorem in [23] we have the following results.

Theorem 3. *If the condition means that (53) has finite positive real roots and means that holds, then the zero equilibrium of system (3) is asymptotically stable for , system (3) undergoes a Hopf bifurcation at when , and a branch of periodic solutions bifurcates from the zero equilibrium near .*

*Case 5 ( and ). *We consider (5) with in its stable interval and is considered as a parameter. Without loss of generality, we consider (5) under Case 2.

Let be a root of (5). Then, we can get Suppose that means that (59) has finite positive real roots, which are denoted as . For every positive real root , their exists a sequence , such that (59) has a pair of purely imaginary roots when .

Let , and when (59) has a pair of purely imaginary roots . In the following, we make the following assumption.(*H*_{52}): .

Through the analysis above and by the Hopf bifurcation theorem in [23], we have the following results.

Theorem 4. *If the condition means that (59) has finite positive real roots and means that holds, and , then the zero equilibrium of system (3) is asymptotically stable for , system (3) undergoes a Hopf bifurcation at when , and a branch of periodic solutions bifurcates from the zero equilibrium near .*

#### 3. Stability of Bifurcated Periodic Solutions

In this section, the formulae for determining the direction of Hopf bifurcation and the stability of bifurcating periodic solutions of system (3) with respect to for are derived by using the normal form method and center manifold theorem introduced by Hassard et al. [23]. Throughout this section, it is considered that system (3) undergoes Hopf bifurcation at and . Without loss of generality, we assume that , where .

For convenience, let , . Drop the bars for simplification of notations. Then system (3) becomes where and , are given, respectively, by with

Therefore, according to the Riesz representation theorem, there exists a matrix function whose elements are of bounded variation such that In fact, we choose For , we define Then system (60) can be transformed into the following operator equation: where for .

For , where is the 4-dimensional space of row vector, we define the adjoint operator of : and a bilinear inner product where .

Then and are adjoint operators. From the discussion above, we know that are eigenvalues of and they are also eigenvalues of . Let be the eigenvector of corresponding to the eigenvalue , and let be the eigenvector of corresponding to the eigenvalue . Then, we have By a simple computation, we can obtain and , .

From (68), we can get

Following the algorithms given in [23] and using similar computation process in [24], we can get the coefficients which can be used to determine direction of the Hopf bifurcation and stability of the bifurcating periodic solutions: with where and can be computed by the following equations, respectively: with

Therefore, we can calculate the following values:

Based on the discussion above, we can obtain the following results.

Theorem 5. *For system (3),*(i)* determines the direction of the Hopf bifurcation. If ; then the Hopf bifurcation is supercritical (subcritical);*(ii)* determines the stability of the bifurcating periodic solutions. If ; then the bifurcating periodic solutions are stable (unstable);*(iii)* determines the period of the bifurcating periodic solutions. If ; then the period of the bifurcating periodic solutions increases (decreases).*

#### 4. Numerical Simulation

In this section, we present some numerical simulations to support the theoretical analysis in Sections 2 and 3. As an example, we consider the following special case of system (3) with the parameters , , , and . Then , , and system (3) becomes

Obviously, is the equilibrium of system (77). By a simple computation, we get , , and . That is, the condition holds.

For . We can obtain by some complicated computations. From Theorem 1, we know that is asymptotically stable when as illustrated by Figures 1 and 2. When passes through, the critical value , becomes unstable and a Hopf bifurcation occurs and a branch of periodic solutions bifurcate from , which can be seen from Figures 3 and 4. Similarly, we have for . The corresponding waveforms and the phase plots are shown in Figures 5, 6, 7, and 8.

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