Abstract

A four-dimensional neural network model with delay is investigated. With the help of the theory of delay differential equation and Hopf bifurcation, the conditions of the equilibrium undergoing Hopf bifurcation are worked out by choosing the delay as parameter. Applying the normal form theory and the center manifold argument, we derive the explicit formulae for determining the properties of the bifurcating periodic solutions. Numerical simulations are performed to illustrate the analytical results.

1. Introduction

The interest in the periodic orbits of a delay neural networks has increased strongly in recent years and substantial efforts have been made in neural network models, for example, Wei and Zhang [1] studied the stability and bifurcation of a class of -dimensional neural networks with delays, Guo and Huang [2] investigated the Hopf bifurcation behavior of a ring of neurons with delays, Yan [3] discussed the stability and bifurcation of a delayed trineuron network model, Hajihosseini et al. [4] made a discussion on the Hopf bifurcation of a delayed recurrent neural network in the frequency domain, and Liao et al. [5] did a theoretical and empirical investigation of a two-neuron system with distributed delays in the frequency domain. For more information, one can see [6–23]. In 1986 and 1987, Babcock and Westervelt [24, 25] had analyzed the stability and dynamics of the following simple neural network model of two neurons with inertial coupling: where denotes the input voltage of the th neuron, is the output of the th neuron, is the damping factor and is the overall gain of the neuron which determines the strength of the nonlinearity. For a more detailed interpretation of the parameters, one can see [24, 25]. In 1997, Lin and Li [26] made a detailed investigation on the bifurcation direction of periodic solution for system (1).

Considering that there exists a time delay (we assume that it is ) in the response of the output voltages to changes in the input, then system (1) can be revised as follows: As is known to us that the research on the Hopf bifurcation, especially on the stability of bifurcating periodic solutions and direction of Hopf bifurcation is very critical. When delays are incorporated into the network models, stability, and Hopf bifurcation analysis become much complex. To obtain a deep and clear understanding of dynamics of neural network model with delays, we will make a investigation on system (2), that is, we study the stability, the local Hopf bifurcation for system (2).

The remainder of the paper is organized as follows. In Section 2, local stability for the equilibrium state of system (2) is discussed. We investigate the existence of the Hopf bifurcations for system (2) choosing time delay as the bifurcation parameter. In Section 3, the direction and stability of the local Hopf bifurcation are analyzed by using the normal form theory and the center manifold theorem by Hassard et al. [27]. In Section 4, numerical simulations for justifying the theoretical results are illustrated.

2. Stability of the Equilibrium and Local Hopf Bifurcations

The object of this section is to investigate the stability of the equilibrium and the existence of local Hopf bifurcations for system (2). It is easy to see that if the following condition:(H1)

holds, then (2) has a unique equilibrium . To investigate the local stability of the equilibrium state we linearize system (2). We expand it in a Taylor series around the orgin and neglect the terms of higher order than the first order. The linearization of (2) near can be expressed as: whose characteristic equation has the form namely, In order to investigate the distribution of roots of the transcendental equation (5), the following Lemma is necessary.

Lemma 1 (see [28]). For the transcendental equation: as vary, the sum of orders of the zeros of in the open right half plane can change, and only a zero appears on or crosses the imaginary axis.

For , (5) becomes In view of the Routh-Hurwitz criteria, we know that all roots of (7) have a negative real part if the following condition:(H2)

is satisfied.

For is a root of (5) if and only if Separating the real and imaginary parts gives It follows from (9) that which is equivalent to where Without loss of generality, we assume that (11) has eight positive roots, denoted by . Then by (9), we derive where , then are a pair of purely imaginary roots of (5) when . Define The above analysis leads to the Lemma as follows.

Lemma 2. If (H1) and (H2) hold, then all roots of (5) have a negative real part when and (5) admits a pair of purely imaginary roots when .

Let be a root of (5) near , and , and . Due to functional differential equation theory, for every , there exists such that is continuously differentiable in for . Substituting into the left hand side of (5) and taking derivative with respect to , we have Then where We assume that the following condition holds:(H3).

According to above analysis and the results of Kuang [29] and Hale [30], we have the following theorem

Theorem 3. If (H1) and (H2) hold, then the equilibrium of system (2) is asymptotically stable for . Under the conditions (H1) and (H2), if the condition (H3) holds, then system (2) undergoes a Hopf bifurcation at the equilibrium when .

3. Direction and Stability of the Hopf Bifurcation

In this section, we discuss the direction, stability and the period of the bifurcating periodic solutions. The used methods are based on the normal form theory and the center manifold theorem introduced by Hassard et al. [27]. From the previous section, we know that if . any root of (5) of the form satisfies and .

For convenience, let and , where is defined by (13) and , drop the bar for the simplification of notations, then system (3) can be written as an FDE in as where and ,  ,  , and are given by respectively, where .

From the discussion in Section 2, we know that if , then system (18) undergoes a Hopf bifurcation at the equilibrium and the associated characteristic equation of system (18) has a pair of simple imaginary roots .

By the representation theorem, there is a matrix function with bounded variation components such that In fact, we can choose where is the Dirac delta function.

For , define Then (18) is equivalent to the abstract differential equation where . For , define For and , define the bilinear form where , the and are adjoint operators. By the discussions in Section 2, we know that are eigenvalues of , and they are also eigenvalues of corresponding to and respectively. By direct computation, we can obtain where Furthermore, and .

Next, we use the same notations as those in Hassard et al. [27] and we first compute the coordinates to describe the center manifold at . Let be the solution of (18) when .

Define on the center manifold , and we have where and and are local coordinates for center manifold in the direction of and . Noting that is also real if is real, we consider only real solutions. For solutions of (18), That is, where Hence we have and we obtain Therefore, we can compute the following values: which determine the quantities of bifurcation periodic solutions on the center manifold at the critical value , determines the direction of the Hopf bifurcation: If   , then the Hopf bifurcation is supercritical (subcritical); determines the stability of the bifurcating periodic solutions: the periodic solutions are stable (unstable) if   ; and determines the period of the bifurcating periodic solutions: the period is increases (decreases) if   .

4. Numerical Examples

To illustrate the analytical results found, let , then system (2) becomes which has a unique equilibrium and satisfies the conditions indicated in Theorem 3. The equilibrium is asymptotically stable for . For , using the software Matlab, we derive . Thus by the algorithm (36) derived in Section 3, we get . Thus the equilibrium is stable when . Figures 1(a)–1(j) show that the equilibrium is asymptotically stable when . When passes through the critical value , the equilibrium loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcate from the equilibrium . Since and , the direction of the Hopf bifurcation is , and these bifurcating periodic solutions from at are stable. Figures 1(j)–2(d) suggest that Hopf bifurcation occurs from the equilibrium when .