Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms

and Applied Analysis 3 In order to investigate the distribution of roots of the transcendental equation 2.2 , the following Lemma that is stated in 15 is useful. Lemma 2.1 see 15 . For the transcendental equation P ( λ, e−λτ1 , . . . , e−λτm ) λ p 0 1 λ n−1 · · · p 0 n−1λ p 0 n [ p 1 1 λ n−1 · · · p 1 n−1λ p 1 n ] e−λτ1 · · · [ p m 1 λ n−1 · · · p m n−1λ p m n ] e−λτm 0, 2.3 as τ1, τ2, τ3, . . . , τm vary, the sum of orders of the zeros of P λ, e−λτ1 , . . . , e−λτm in the open right half plane can change and only a zero appears on or crosses the imaginary axis. Now we make the following assumptions: H2 α1f ′ 0 < 0 and α1 − α2 − a2 b2 > 0; H3 |α1 − a2| < |α2 − b2|. Lemma 2.2. If (H1)–(H3) hold, then one has the following. i When τ τ def 1 ω0 [ arccos 2α1ω0 ( α2 − b2 ) f ′2 0 2jπ ] , j 0, 1, 2, . . .. 2.4 Equation 2.2 has a simple pair of imaginary roots ±iω0, where ω0 √( α1 a 2 )2 − [( α1 − a2 )2 − α22 − b2 )2] f ′4 0 − α21 a2 ) . 2.5 ii For τ ∈ 0, τ0 , all roots of 2.2 have strictly negative real parts. iii When τ τ0, 2.2 has a pair of imaginary roots ±iω0 and all other roots have strictly negative real parts. Proof. Obviously, by assumption H2 , λ 0 is not the root of 2.2 . When τ 0, then 2.2 becomes λ2 − 2α1f ′ 0 λ [ α1 − α2 − a2 b2 ] f ′2 0 0. 2.6 It is easy to see that all roots of 2.6 have negative real parts. ±iω ω > 0 is a pair of purely imaginary roots of 2.2 if and only if ω satisfies −ω2 − 2α1f ′ 0 ωi ( α1 − a2 ) f ′2 0 − ( α2 − b2 ) f ′2 0 cosωτ − i sinωτ 0. 2.7 4 Abstract and Applied Analysis Separating the real and imaginary parts, we get ( α2 − b2 ) f ′2 0 cosωτ ( α1 − a2 ) f ′2 0 −ω2, ( α2 − b2 ) f ′2 0 sinωτ 2α1f ′ 0 ω. 2.8 It follows from 2.8 that ω4 2 ( α1 a 2 ) f ′2 0 ω2 [( α1 − a2 )2 − ( α2 − b2 )2] f ′4 0 0. 2.9


Introduction
Based on the assumption that the elements in the network can respond to and communicate with each other instantaneously without time delays, Hopfield proposed Hopfield neural networks HNNs model in 1980s 1, 2 . During the past several years, the dynamical phenomena of neural networks have been extensively studied because of the widely application in various information processing, optimization problems, and so forth. In particular, the appearance of a cycle bifurcating from an equilibrium of an ordinary or a delayed neural network with a single parameter, which is known as a Hopf bifurcation, has attracted much attention see 3-13 . In 2008, Yang et al. 14 investigated the Bautin bifurcation of the two-neuron networks with resonant bilinear terms and without delay: where x i t i 1, 2 represents the state of the ith neuron at time t, f x i i 1, 2 is the connection function between two neurons, and α 1 , α 2 , a, b, c, d are real parameters, and obtained a sufficient condition for a Bautin bifurcation to occur for system 1.1 by using the standard normal form theory and with Maple software. It is well known that in the implementation of networks, time delays are inevitably encountered because of the finite switching speed of signal transmission. Motivated by the viewpoint, in the following, we assume that the time delay from the first neuron to the second neuron is τ 2 and back to the first neuron is τ 1 , then we have the following neural networks whose delays are introduced: where x i t i 1, 2 represents the state of the i-th neuron at time t, f x i i 1, 2 is the connection function between two neurons, α 1 , α 2 , a, b, c, d are real parameters, and τ 1 , τ 2 are positive constants. We all know that time delays that occurred in the interaction between neurons will affect the stability of a network by creating instability, oscillation, and chaos phenomena. The purpose of this paper is to discuss the stability and the properties of Hopf bifurcation of model 1.2 . To the best of our knowledge, it is the first to deal with the stability and Hopf bifurcation of the system 1.2 .
This paper is organized as follows. In Section 2, the stability of the equilibrium and the existence of Hopf bifurcation at the equilibrium are studied. In Section 3, the direction of Hopf bifurcation and the stability and periods of bifurcating periodic solutions on the center manifold are determined. In Section 4, numerical simulations are carried out to illustrate the validity of the main results. Some main conclusions are drawn in Section 5.

Stability of the Equilibrium and Local Hopf Bifurcations
Throughout this paper, we assume that the function f satisfies the following conditions: H1 f ∈ C 3 R , f 0 0, and uf u > 0, for u / 0. Hypothesis H1 implies that E * 0, 0 is an equilibrium of the system 1.2 and linearized system of 1.2 takes the forṁ

2.1
The associated characteristic equation of 2.1 is In the section, we consider the sum of two delays as the parameter to give some conditions that separate the first quadrant of the τ 1 , τ 2 plane into two parts, one is the stable region another is the unstable region, and the boundary is the Hopf bifurcation curve.

Abstract and Applied Analysis 3
In order to investigate the distribution of roots of the transcendental equation 2.2 , the following Lemma that is stated in 15 is useful. Now we make the following assumptions:
iii When τ τ 0 , 2.2 has a pair of imaginary roots ±iω 0 and all other roots have strictly negative real parts.
Proof. Obviously, by assumption H2 , λ 0 is not the root of 2.2 . When τ 0, then 2.2 becomes It is easy to see that all roots of 2.6 have negative real parts.
±iω ω > 0 is a pair of purely imaginary roots of 2.2 if and only if ω satisfies Abstract and Applied Analysis Separating the real and imaginary parts, we get

2.12
From 2.8 , we know that 2.2 with τ τ j j 0, 1, 2, . . . has a pair of imaginary roots ±iω 0 , which are simple. According, the discussion and applying the Lemma 2.1 and Cooke and Grossman 16 , we obtain the conclusion ii and iii . This completes the proof.
Let λ j τ α j τ iω j τ be a root of 2.2 near τ τ j , and α j τ j 0, ω j τ j ω 0 , j 0, 1, 2 . . . . Due to functional differential equation theory, for every τ j , k 0, 1, 2 . . ., there exists ε > 0 such that λ j τ is continuously differentiable in τ for |τ −τ j | < ε. Substituting λ τ into the left-hand side of 2.2 and taking derivative with respect to τ, we have Abstract and Applied Analysis 5 which leads to 2.14 By 2.8 , we get So we have From the above analysis, we have the following results.

Lemma 2.3.
Let τ τ j , then the following transversality condition: i If τ ∈ 0, τ 0 , then the equilibrium point of system 1.2 is asymptotically stable.
ii If τ > τ 0 , then the equilibrium point of system 1.2 is unstable.

Direction and Stability of the Hopf Bifurcation
In the previous section, we obtained some conditions which guarantee that the two-neuron networks with resonant bilinear terms undergo the Hopf bifurcation at some values of τ τ 1 τ 2 . In this section, we shall derived the explicit formulae determining the direction, stability, and period of these periodic solutions bifurcating from the equilibrium E * 0, 0 at this critical value of τ, by using techniques from normal form and center manifold theory 17 , Throughout this section, we always assume that system 2.1 undergoes Hopf bifurcation at the equilibrium E * 0, 0 for τ τ 0 and then ±iω 0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium E * 0, 0 .

3.2
We expand the nonlinear part of the system 1.2 and derive the following expression:

3.4
By the representation theorem, there is a matrix function with bounded variation components η θ, μ , θ ∈ −τ 0 1 , 0 such that Abstract and Applied Analysis 7 In fact, we can choose where δ is the Dirac delta function.
Then 1.2 is equivalent to the abstract differential equatioṅ 3.9 For φ ∈ C −τ 0 1 , 0 , R 2 and ψ ∈ C 0, τ 0 1 , R 2 * , define the bilinear form where η θ η θ, 0 . We have the following result on the relation between the operators A A 0 and A * .

3.11
This shows that A A 0 and A * are adjoint operators and the proof is complete.
By the discussions in the Section 2, we know that ±iω 0 are eigenvalues of A 0 and they are also eigenvalues of A * corresponding to iω 0 and −iω 0 , respectively. We have the following result.

3.15
Abstract and Applied Analysis 9 is the eigenvector of A * corresponding to the eigenvalue −iω 0 , moreover, q * s , q θ 1, where Proof. Let q θ be the eigenvector of A 0 corresponding to the eigenvalue iω 0 and q * s be the eigenvector of A * corresponding to the eigenvalue −iω 0 , namely, A 0 q θ iω 0 q θ and A * q * T s −iω 0 q * T s . From the definitions of A 0 and A * , we have A 0 q θ dq θ /dθ and A * q * T s −dq * T s /ds. Thus, q θ q 0 e iω 0 θ and q * s q * 0 e iω 0 s . In addition, That is, Therefore, we can easily obtain and so On the other hand, Namely, Abstract and Applied Analysis Therefore, we can easily obtain and so In the sequel, we will verify that q * s , q θ 1. In fact, from 3.10 , we have

3.27
Next, we use the same notations as those in Hassard et al. 17 and we first compute the coordinates to describe the center manifold C 0 at μ 0. Let x t be the solution of 1.2 when μ 0.
Define and z and z are local coordinates for center manifold C 0 in the direction of q * and q * . Noting that W is also real if x t is real, we consider only real solutions. For solutions x t ∈ C 0 of 1.2 ,

3.54
From 3.43 , we have

3.56
Noting that That is, 3.60 Hence,

Conclusions
In this paper, we have analyzed a two-neuron networks with resonant bilinear terms. Firstly, we obtained the sufficient conditions to ensure local stability of the equilibrium E * 0, 0 and the existence of local Hopf bifurcation. Moreover, we note also that, if the two-neuron networks with resonant bilinear terms begin with a stable equilibrium, but then become unstable due to delay, then it will likely be destabilized by means of a Hopf bifurcation which leads to periodic solutions with small amplitudes. Finally, the direction of Hopf bifurcation and the stability of the bifurcating periodic orbits are discussed by applying normal form theory and center manifold theorem.