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Abstract and Applied Analysis
Volume 2011, Article ID 697630, 21 pages
http://dx.doi.org/10.1155/2011/697630
Research Article

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

1Guizhou Key Laboratory of Economics System Simulation, Guizhou College of Finance and Economics, Guiyang 550004, China
2Department of Mathematics, Zhangjiajie College of jishou University, Zhangjiajie 427000, China

Received 8 January 2011; Revised 27 February 2011; Accepted 27 April 2011

Academic Editor: Nobuyuki Kenmochi

Copyright © 2011 Changjin Xu and Xiaofei He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

A class of two-neuron networks with resonant bilinear terms is considered. The stability of the zero equilibrium and existence of Hopf bifurcation is studied. It is shown that the zero equilibrium is locally asymptotically stable when the time delay is small enough, while change of stability of the zero equilibrium will cause a bifurcating periodic solution as the time delay passes through a sequence of critical values. Some explicit formulae for determining the stability and the direction of the Hopf bifurcation periodic solutions bifurcating from Hopf bifurcations are obtained by using the normal form theory and center manifold theory. Finally, numerical simulations supporting the theoretical analysis are carried out.

1. 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 [313]).

In 2008, Yang et al. [14] investigated the Bautin bifurcation of the two-neuron networks with resonant bilinear terms and without delay:̇𝑥1𝛼(𝑡)=1𝑓𝑥+𝑎1+𝛼2𝑓𝑥+𝑏2+𝑐𝑥1𝑥2,̇𝑥2(𝛼𝑡)=2𝑓𝑥𝑏1+𝛼1𝑓𝑥𝑎2+𝑑𝑥1𝑥2,(1.1) where 𝑥𝑖(𝑡)(𝑖=1,2) represents the state of the 𝑖th neuron at time 𝑡, 𝑓(𝑥𝑖)(𝑖=1,2) is the connection function between two neurons, and 𝛼1,𝛼2,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 𝜏2 and back to the first neuron is 𝜏1, then we have the following neural networks whose delays are introduced:̇𝑥1𝛼(𝑡)=1𝑓𝑥+𝑎1+𝛼2𝑓𝑥+𝑏2𝑡𝜏1+𝑐𝑥1𝑥2,̇𝑥2(𝛼𝑡)=2𝑓𝑥𝑏1𝑡𝜏2+𝛼1𝑓𝑥𝑎2+𝑑𝑥1𝑥2,(1.2) where 𝑥𝑖(𝑡)(𝑖=1,2) represents the state of the 𝑖-th neuron at time 𝑡, 𝑓(𝑥𝑖)(𝑖=1,2) is the connection function between two neurons, 𝛼1,𝛼2,𝑎,𝑏,𝑐,𝑑 are real parameters, and 𝜏1,𝜏2 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.

2. Stability of the Equilibrium and Local Hopf Bifurcations

Throughout this paper, we assume that the function 𝑓 satisfies the following conditions:

(H1) 𝑓𝐶3(𝑅),𝑓(0)=0, and 𝑢𝑓(𝑢)>0, for 𝑢0.

Hypothesis (H1) implies that 𝐸(0,0) is an equilibrium of the system (1.2) and linearized system of (1.2) takes the forṁ𝑥1𝛼(𝑡)=1𝑓+𝑎(0)𝑥1+𝛼2𝑓+𝑏(0)𝑥2𝑡𝜏1,̇𝑥2(𝛼𝑡)=2𝑓𝑏(0)𝑥1𝑡𝜏2+𝛼1𝑓𝑎(0)𝑥2.(2.1) The associated characteristic equation of (2.1) is𝜆22𝛼1𝑓𝛼(0)𝜆+21𝑎2𝑓2𝛼(0)22𝑏2𝑓2(0)𝑒𝜆𝜏=0,(2.2) where 𝜏=𝜏1+𝜏2.

In the section, we consider the sum of two delays as the parameter to give some conditions that separate the first quadrant of the (𝜏1,𝜏2) plane into two parts, one is the stable region another is the unstable region, and the boundary is the Hopf bifurcation curve.

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 𝑃𝜆,𝑒𝜆𝜏1,,𝑒𝜆𝜏𝑚=𝜆𝑛+𝑝1(0)𝜆𝑛1++𝑝(0)𝑛1𝜆+𝑝𝑛(0)+𝑝1(1)𝜆𝑛1++𝑝(1)𝑛1𝜆+𝑝𝑛(1)𝑒𝜆𝜏1+𝑝+1(𝑚)𝜆𝑛1++𝑝(𝑚)𝑛1𝜆+𝑝𝑛(𝑚)𝑒𝜆𝜏𝑚=0,(2.3) as (𝜏1,𝜏2,𝜏3,,𝜏𝑚) vary, the sum of orders of the zeros of 𝑃(𝜆,𝑒𝜆𝜏1,,𝑒𝜆𝜏𝑚) 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 𝛼21𝛼22𝑎2+𝑏2>0;

(H3) |𝛼21𝑎2|<|𝛼22𝑏2|.

Lemma 2.2. If (H1)–(H3) hold, then one has the following.
(i)When 𝜏=𝜏𝑗def=1𝜔0arccos2𝛼1𝜔0𝛼22𝑏2𝑓2(0)+2𝑗𝜋,𝑗=0,1,2,.(2.4) Equation (2.2) has a simple pair of imaginary roots ±𝑖𝜔0, where 𝜔0=𝛼21+𝑎22𝛼21𝑎22𝛼22𝑏22𝑓4𝛼(0)21+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 𝜆22𝛼1𝑓𝛼(0)𝜆+21𝛼22𝑎2+𝑏2𝑓2(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 𝜔22𝛼1𝑓𝛼(0)𝜔𝑖+21𝑎2𝑓2𝛼(0)2𝑏2𝑓2(0)(cos𝜔𝜏𝑖sin𝜔𝜏)=0.(2.7) Separating the real and imaginary parts, we get 𝛼22𝑏2𝑓2𝛼(0)cos𝜔𝜏=21𝑎2𝑓2(0)𝜔2,𝛼22𝑏2𝑓2(0)sin𝜔𝜏=2𝛼1𝑓(0)𝜔.(2.8) It follows from (2.8) that 𝜔4𝛼+221+𝑎2𝑓2(0)𝜔2+𝛼21𝑎22𝛼22𝑏22𝑓4(0)=0.(2.9) Thus, we obtain 𝜔=𝛼21+𝑎22𝛼21𝑎22𝛼22𝑏22𝑓4𝛼(0)21+𝑎2.(2.10) It is clear that 𝜔 is well defined if condition (H3) holds.
Denote 𝜔0=𝛼21+𝑎22𝛼21𝑎22𝛼22𝑏22𝑓4𝛼(0)21+𝑎2.(2.11) Let 𝜏𝑗=1𝜔0arccos2𝛼1𝜔0𝛼22𝑏2𝑓2(0)+2𝑗𝜋,𝑗=0,1,2,.(2.12) From (2.8), we know that (2.2) with 𝜏=𝜏𝑗(𝑗=0,1,2,) has a pair of imaginary roots ±𝑖𝜔0, 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 𝜆𝑗(𝜏)=𝛼𝑗(𝜏)+𝑖𝜔𝑗(𝜏) be a root of (2.2) near 𝜏=𝜏𝑗, and 𝛼𝑗(𝜏𝑗)=0,𝜔𝑗(𝜏𝑗)=𝜔0,(𝑗=0,1,2). Due to functional differential equation theory, for every 𝜏𝑗,𝑘=0,1,2, there exists 𝜀>0 such that 𝜆𝑗(𝜏) is continuously differentiable in 𝜏 for |𝜏𝜏𝑗|<𝜀. Substituting 𝜆(𝜏) into the left-hand side of (2.2) and taking derivative with respect to 𝜏, we have 𝑑𝜆𝑑𝜎1=2𝜆2𝛼1𝑓(0)𝛼22𝑏2𝑓2(0)𝑒𝜆𝜏𝜆𝜏𝜆,(2.13) which leads to Re𝑑𝜆𝑑𝜎1𝜏=𝜏𝑗=2𝛼1𝑓𝛼(0)22𝑏2𝑓2(0)𝜔0sin𝜔0𝜏𝑗2𝜔20𝛼22𝑏2𝑓2(0)cos𝜔0𝜏𝑗𝛼22𝑏2𝑓2(0)𝜔0sin𝜔0𝜏𝑗2+𝜔0𝛼22𝑏2𝑓2(0)cos𝜔0𝜏𝑗2.(2.14) By (2.8), we getRe𝑑𝜆𝑑𝜎1𝜏=𝜏𝑗=2𝜔20+3𝛼21+𝑎2𝑓2(0)2𝜔0𝛼1𝑓(0)2+𝛼21𝑎2𝑓2(0)𝜔202>0.(2.15) So we havesignRe𝑑𝜆𝑑𝜏𝜏=𝜏𝑗=signRe𝑑𝜆𝑑𝜏1𝜏=𝜏𝑗>0.(2.16)

From the above analysis, we have the following results.

Lemma 2.3. Let 𝜏=𝜏𝑗, then the following transversality condition: 𝑑𝜆𝑑𝜏Re𝑗|||(𝜏)𝜏=𝜏𝑗>0(2.17) is satisfied.

From Lemma 2.3, we can obtain the following lemma.

Lemma 2.4. Assume that (H3) holds. If 𝜏>𝜏0, then (2.2) has at least one root with strictly positive real part.

Remark 2.5. In fact, Applying the lemma in Cooke and Grossman [16] and Lemma 2.3, we can easily see that if 𝜏(𝜏𝑗,𝜏𝑗+1), (2.2) has 2(𝑗+1)(𝑗=0,1,2,) roots with positive real parts.
From Lemma 2.22.4, we have the following results on the local stability and Hopf bifurcation for system (1.2).

Theorem 2.6. For system (1.2), let 𝜏0 be defined by (2.4) and assume that (H1)–(H3) hold.
(i)If 𝜏[0,𝜏0), then the equilibrium point of system (1.2) is asymptotically stable.(ii)If 𝜏>𝜏0, then the equilibrium point of system (1.2) is unstable.(iii)𝜏=𝜏𝑗(𝑗=0,1,2,) are Hopf bifurcation values for system (1.2).

3. 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=𝜏1+𝜏2. 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=𝜏0 and then ±𝑖𝜔0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium 𝐸(0,0).

For convenience, let 𝜏=𝜏0+𝜇,𝜇𝑅. Then 𝜇=0 is the Hopf bifurcation value of (1.2). Thus, we shall study Hopf bifurcation of small amplitude periodic solutions of (1.2) from the equilibrium point for 𝜇 close to 0. Without loss of generality, we assume that 𝜏01>𝜏02 and let |𝜇|𝜏01𝜏02. Since our analysis is local, where 𝜏0=𝜏01+𝜏02 and 𝜏=𝜏01+(𝜏02+𝜇). We can consider the fixed phase space 𝐶=𝐶([𝜏01,0],𝑅2).

For (𝜙1,𝜙2)𝐶, define𝐿𝜇𝜙=𝐴1𝜙(0)+𝐵𝜙𝜏2+𝐶𝜙𝜏1,(3.1) where 𝐴1=𝛼1𝑓+𝑎00𝛼(0)1𝑓𝑎𝛼(0),𝐵=002𝑓𝑏0𝛼(0)0,𝐶=2𝑓+𝑏.(0)00(3.2) We expand the nonlinear part of the system (1.2) and derive the following expression:𝑓𝑓(𝜇,𝜙)=1(𝑓𝜇,𝜙)2(𝜇,𝜙),(3.3) where𝑓1𝛼(𝜇,𝜙)=1𝑓+𝑎(0)2𝜙21𝑓(0)+(0)𝜙3!31+𝛼(0)2𝑓+𝑏(0)2𝜙22𝜏1+𝑓(0)𝜙3!32𝜏1+𝑐𝜙1(0)𝜙2𝑓(0)+h.o.t.,2𝛼(𝜇,𝜙)=2𝑓𝑏(0)2𝜙21𝜏2+𝑓(0)𝜙3!31𝜏2+𝛼1𝑓𝑎(0)2𝜙22𝑓(0)+(0)𝜙3!32(0)+𝑑𝜙1(0)𝜙2(0)+h.o.t..(3.4) By the representation theorem, there is a matrix function with bounded variation components 𝜂(𝜃,𝜇),𝜃[𝜏01,0] such that𝐿𝜇𝜙=0𝜏01𝑑𝜂(𝜃,𝜇)𝜙(𝜃),for𝜙𝐶.(3.5) In fact, we can choose𝐴𝜂(𝜃,𝜇)=1,𝜃=0,𝐵𝛿𝜃+𝜏2,𝜃𝜏2,,0𝐶𝛿𝜃+𝜏01,𝜃𝜏01,𝜏2,(3.6) where 𝛿 is the Dirac delta function.

For 𝜙𝐶([𝜏01,0],𝑅2), define𝐴(𝜇)𝜙=𝑑𝜙(𝜃)𝑑𝜃,𝜏01𝜃<0,0𝜏01𝑅𝑑𝜂(𝑠,𝜇)𝜙(𝑠),𝜃=0,(𝜇)𝜙=0,𝜏01𝜃<0,𝑓(𝜇,𝜙),𝜃=0.(3.7) Then (1.2) is equivalent to the abstract differential equatioṅ𝑥𝑡=𝐴(𝜇)𝑥𝑡+𝑅(𝜇)𝑥𝑡,(3.8) where 𝑥=(𝑥1,𝑥2)𝑇,𝑥𝑡(𝜃)=𝑥(𝑡+𝜃),𝜃[𝜏01,0].

For 𝜓𝐶([0,𝜏01],(𝑅2)), define𝐴𝜓(𝑠)=𝑑𝜓(𝑠)𝑑𝑠,s0,𝜏01,0𝜏01𝑑𝜂𝑇(𝑡,0)𝜓(𝑡),s=0.(3.9)

For 𝜙𝐶([𝜏01,0],𝑅2) and 𝜓𝐶([0,𝜏01],(𝑅2)), define the bilinear form𝜓,𝜙=𝜓(0)𝜙(0)0𝜏01𝜃𝜉=0𝜓(𝜉𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉,(3.10) where 𝜂(𝜃)=𝜂(𝜃,0). We have the following result on the relation between the operators 𝐴=𝐴(0) and 𝐴.

Lemma 3.1. 𝐴=𝐴(0) and 𝐴 are adjoint operators.

Proof. Let 𝜙𝐶1([𝜏01,0],𝑅2) and 𝜓𝐶1([0,𝜏01],(𝑅2)). It follows from (3.10) and the definitions of 𝐴=𝐴(0) and 𝐴 that 𝜓(𝑠),𝐴(0)𝜙(𝜃)=𝜓(0)𝐴(0)𝜙(0)0𝜏01𝜃𝜉=0=𝜓(𝜉𝜃)𝑑𝜂(𝜃)𝐴(0)𝜙(𝜉)𝑑𝜉𝜓(0)0𝜏01𝑑𝜂(𝜃)𝜙(𝜃)0𝜏01𝜃𝜉=0=𝜓(𝜉𝜃)𝑑𝜂(𝜃)𝐴(0)𝜙(𝜉)𝑑𝜉𝜓(0)0𝜏01𝑑𝜂(𝜃)𝜙(𝜃)0𝜏01𝜓(𝜉𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝜃𝜉=0+0𝜏01𝜃𝜉=0𝑑𝜓(𝜉𝜃)=𝑑𝜉𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉0𝜏01𝜓(𝜃)𝑑𝜂(𝜃)𝜙(0)0𝜏01𝜃𝜉=0𝑑𝜓(𝜉𝜃)𝑑𝜉𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉=𝐴𝜓(0)𝜙(0)0𝜏01𝜃𝜉=0𝐴𝜓(𝜉𝜃)𝑑𝜂(𝜃)𝜙(𝜉)𝑑𝜉=𝐴𝜓(𝑠),𝜙(𝜃).(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 ±𝑖𝜔0 are eigenvalues of 𝐴(0) and they are also eigenvalues of 𝐴 corresponding to 𝑖𝜔0 and 𝑖𝜔0, respectively. We have the following result.

Lemma 3.2. The vector 𝑞(𝜃)=(1,𝛾)𝑇𝑒𝑖𝜔0𝜃,𝜃𝜏01,,0(3.12) where 𝛾=𝑖𝜔0𝛼1𝑓+𝑎(0)𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01,(3.13) is the eigenvector of 𝐴(0) corresponding to the eigenvalue 𝑖𝜔0, and 𝑞(𝑠)=𝐷1,𝛾𝑒𝑖𝜔𝑠,𝑠0,𝜏01,(3.14) where 𝛾=𝑖𝜔0+𝛼1𝑓+𝑎(0)𝛼2𝑓𝑏(0)𝑒𝑖𝜔𝜏02,(3.15) is the eigenvector of 𝐴 corresponding to the eigenvalue 𝑖𝜔0, moreover, 𝑞(𝑠),𝑞(𝜃)=1, where 𝐷=1+𝛾𝛾+𝛾𝜏02𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02+𝛾𝜏01𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01.(3.16)

Proof. Let 𝑞(𝜃) be the eigenvector of 𝐴(0) corresponding to the eigenvalue 𝑖𝜔0 and 𝑞(𝑠) be the eigenvector of 𝐴 corresponding to the eigenvalue 𝑖𝜔0, namely, 𝐴(0)𝑞(𝜃)=𝑖𝜔0𝑞(𝜃) and 𝐴𝑞𝑇(𝑠)=𝑖𝜔0𝑞𝑇(𝑠). From the definitions of 𝐴(0) and 𝐴, we have 𝐴(0)𝑞(𝜃)=𝑑𝑞(𝜃)/𝑑𝜃 and 𝐴𝑞𝑇(𝑠)=𝑑𝑞𝑇(𝑠)/𝑑𝑠. Thus, 𝑞(𝜃)=𝑞(0)𝑒𝑖𝜔0𝜃 and 𝑞(𝑠)=𝑞(0)𝑒𝑖𝜔0𝑠. In addition, 0𝜏01𝑑𝜂(𝜃)𝑞(𝜃)=𝐴1𝑞(0)+𝐵𝑞𝜏2+𝐶𝑞𝜏1=𝐴(0)𝑞(0)=𝑖𝜔0𝑞(0).(3.17) That is, 𝑖𝜔0𝛼1𝑓+𝑎𝛼(0)2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02𝑖𝜔0𝛼1𝑓𝑎𝑞00(0)(0)=.(3.18) Therefore, we can easily obtain 𝛾=𝑖𝜔0𝛼1𝑓+𝑎(0)𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01(3.19) and so 𝑞(0)=1,𝑖𝜔0(𝛼1+𝑎)𝑓(0)𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝑇(3.20) hence 𝑞(𝜃)=1,𝑖𝜔0𝛼1𝑓+𝑎(0)𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝑇𝑒𝑖𝜔0𝜃.(3.21) On the other hand, 0𝜏01𝑞(𝑡)𝑑𝜂(𝑡)=𝐴𝑇1𝑞𝑇(0)+𝐵𝑇𝑞𝑇𝜏02+𝐶𝑇𝑞𝑇𝜏01=𝐴𝑞𝑇(0)=𝑖𝜔0𝑞𝑇(0).(3.22) Namely, 𝑖𝜔0+𝛼1𝑓+𝑎𝛼(0)2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝑖𝜔0+𝛼1𝑓𝑎𝑞(0)00(0)=.(3.23) Therefore, we can easily obtain 𝛾=𝑖𝜔0+𝛼1𝑓+𝑎(0)𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02(3.24) and so 𝑞(0)=1,𝑖𝜔0+𝛼1𝑓+𝑎(0)𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02(3.25) hence 𝑞(𝑠)=1,𝑖𝜔0+𝛼1𝑓+𝑎(0)𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02𝑒𝑖𝜔0𝑠.(3.26) In the sequel, we will verify that 𝑞(𝑠),𝑞(𝜃)=1. In fact, from (3.10), we have 𝑞(𝑠),𝑞(𝜃)=𝐷1,𝛾(1,𝛾)𝑇0𝜏01𝜃𝜉=0𝐷1𝛾𝑒𝑖𝜔0(𝜉𝜃)𝑑𝜂(𝜃)(1,𝛾)𝑇𝑒𝑖𝜔0𝜉=𝑑𝜉𝐷1+𝛾𝛾0𝜏011,𝛾𝜃𝑒𝑖𝜔0𝜃𝑑𝜂(𝜃)(1,𝛾)𝑇=𝐷1+𝛾𝛾1,𝛾𝐵𝜏02𝑒𝑖𝜔0𝜏02+𝐶𝜏01𝑒𝑖𝜔0𝜏01(1,𝛾)𝑇=𝐷1+𝛾𝛾+𝛾𝜏02𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02+𝛾𝜏01𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01=1.(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 𝐶0 at 𝜇=0. Let 𝑥𝑡 be the solution of (1.2) when 𝜇=0.

Define𝑧(𝑡)=𝑞,𝑥𝑡,𝑊(𝑡,𝜃)=𝑥𝑡(𝜃)2Re{𝑧(𝑡)𝑞(𝜃)}(3.28) on the center manifold 𝐶0, and we have𝑊𝑧(𝑡,𝜃)=𝑊(𝑡),𝑧(𝑡),𝜃,(3.29) where𝑊𝑧(𝑡),𝑧(𝑡),𝜃=𝑊𝑧,𝑧=𝑊20𝑧22+𝑊11𝑧𝑧+𝑊02𝑧22+(3.30) and 𝑧 and 𝑧 are local coordinates for center manifold 𝐶0 in the direction of 𝑞 and 𝑞. Noting that 𝑊 is also real if 𝑥𝑡 is real, we consider only real solutions. For solutions 𝑥𝑡𝐶0 of (1.2),̇𝑧(𝑡)=𝑞(𝑠),̇𝑥𝑡=𝑞(𝑠),𝐴(0)𝑥𝑡+𝑅(0)𝑥𝑡=𝑞(𝑠),𝐴(0)𝑥𝑡+𝑞(s),𝑅(0)𝑥𝑡=𝐴𝑞(𝑠),𝑥𝑡+𝑞(0)𝑅(0)𝑥𝑡0𝜏01𝜃𝜉=0𝑞(𝜉𝜃)𝑑𝜂(𝜃)𝐴(0)𝑅(0)𝑥𝑡(𝜉)𝑑𝜉=𝑖𝜔0𝑞(𝑠),𝑥𝑡+𝑞(0)𝑓0,𝑥𝑡(𝜃)def=𝑖𝜔0𝑧(𝑡)+𝑞(0)𝑓0𝑧(𝑡),.𝑧(𝑡)(3.31) That is,̇𝑧(𝑡)=𝑖𝜔0𝑧+𝑔𝑧,𝑧,(3.32) where𝑔𝑧,𝑧=𝑔20𝑧22+𝑔11𝑧𝑧+𝑔02𝑧22+𝑔21𝑧2𝑧2+.(3.33) Hence, we have𝑔𝑧,𝑧=𝑞(0)𝑓0𝑧,𝑧=𝑓0,𝑥𝑡=𝐷1,𝛾𝑓10,𝑥𝑡,𝑓20,𝑥𝑡𝑇,(3.34) where 𝑓10,𝑥𝑡=𝛼1𝑓+𝑎(0)2𝑥21𝑡𝑓(0)+(0)𝑥3!31𝑡+𝛼(0)2𝑓+𝑏(0)2𝑥22𝑡𝜏01+𝑓(0)𝑥3!32𝑡𝜏01+𝑐𝑥1𝑡(0)𝑥2𝑡𝑓(0)+h.o.t.,20,𝑥𝑡=𝛼2𝑓𝑏(0)2𝑥21𝑡𝜏02+𝑓(0)𝑥3!31𝑡𝜏02+𝛼1𝑓𝑎(0)2𝑥22𝑡𝑓(0)+(0)𝑥3!32𝑡(0)+𝑑𝑥1𝑡(0)𝑥2𝑡(0)+h.o.t..(3.35) Noticing 𝑥𝑡(𝜃)=(𝑥1𝑡(𝜃),𝑥2𝑡(𝜃))𝑇=𝑊(𝑡,𝜃)+𝑧𝑞(𝜃)+𝑧𝑞(𝜃) and 𝑞(𝜃)=(1,𝛾)𝑇𝑒𝑖𝜔0𝜃, we have 𝑥1𝑡(0)=𝑧+𝑧+𝑊(1)20𝑧(0)22+𝑊(1)11(0)𝑧𝑧+𝑊(1)02(0)𝑧22𝑥+,2𝑡(0)=𝛾𝑧+𝛾𝑧+𝑊(2)20𝑧(0)22+𝑊(2)11(0)𝑧𝑧+𝑊(2)02(0)𝑧22𝑥+,1𝑡𝜏02=𝑒𝑖𝜔0𝜏02𝑧+𝑒𝑖𝜔0𝜏02𝑧+𝑊(1)20𝜏02𝑧22+𝑊(1)11𝜏02𝑧𝑧+𝑊(1)02𝜏02𝑧22𝑥+,2𝑡𝜏01=𝛾𝑒𝑖𝜔0𝜏01𝑧+𝛾𝑒𝑖𝜔0𝜏01𝑧+𝑊(2)20𝜏01𝑧22+𝑊(2)11𝜏01𝑧𝑧+𝑊(2)02𝜏01𝑧22+.(3.36) From (3.33) and (3.34), we have 𝑔𝑧,𝑧=𝑞(0)𝑓0𝑧,𝑧=𝐷𝑓10,𝑥𝑡+𝛾𝑓20,𝑥𝑡=𝐷𝛼1𝑓+𝑎(0)2+𝛼2𝑓+𝑏(0)2𝛾2++𝑐𝛾𝐷𝛾𝛼2𝑓𝑏(0)2𝑒2𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)2𝛾2𝑧+𝑑𝛾2+𝐷𝛼1𝑓+𝑎𝛼(0)+2𝑓+𝑏(0)2𝛾+𝛾+2𝑐Re{𝛾}𝐷𝛾𝛼2𝑓𝑏𝛼(0)+1𝑓𝑎(0)𝛾𝑧𝛾+2𝑑Re{𝛾}𝑧+𝐷𝛼1𝑓+𝑎(0)2+𝛼2𝑓+𝑏(0)2𝛾2+𝑐𝛾+𝐷𝛾𝛼2𝑓𝑏(0)2𝑒2𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)2𝛾2+𝑑𝛾𝑧2+𝐷𝛼1𝑓+𝑎(0)22𝑊(1)11(0)+𝑊(1)20+𝛼(0)1𝑓+𝑎(0)2+𝛼2𝑓+𝑏(0)𝛾𝑒𝑖𝜔0𝜏01𝑊(2)11𝜏01+𝛼2𝑓+𝑏(0)2𝛾2𝛾𝑒𝑖𝜔0𝜏01+12𝑐𝑊(2)20(0)+𝛾𝑊(1)20+(0)𝐷𝛾𝛼2𝑓𝑏(0)2𝑊(1)20𝜏02𝑒𝑖𝜔0𝜏02+2𝑒𝑖𝜔0𝜏02𝑊(1)11𝜏02+𝛼2𝑓𝑏(0)2𝑒𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)22𝛾𝑊(2)11(0)+𝑊(2)20(0)𝛼𝛾+1𝑓𝑎(0)2𝛾21𝛾+2𝑑𝑊(2)20(0)+𝛾𝑊(1)20𝑧(0)2𝑧+h.o.t.(3.37) and we obtain 𝑔20=𝐷𝛼1𝑓+𝑎(𝛼0)+2𝑓+𝑏(0)𝛾2++𝑐𝛾𝐷𝛾𝛼2𝑓𝑏(0)𝑒2𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)𝛾2,𝑔+𝑑𝛾11=𝐷𝛼1𝑓+𝑎𝛼(0)+2𝑓+𝑏(0)𝑎+𝑎+2𝑐Re{𝛾}𝐷𝛾𝛼2𝑓𝑏𝛼(0)+1𝑓𝑎(0)𝛾,𝑔𝛾+2𝑑Re{𝛾}02=𝐷𝛼1𝑓+𝑎𝛼(0)+2𝑓+𝑏(0)𝛾2+2𝑐𝛾+𝐷𝛾𝛼2𝑓𝑏(0)𝑒2𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)𝛾2+2𝑑𝛾,𝑔21=𝐷𝛼1𝑓+𝑎(0)2𝑊(1)11(0)+𝑊(1)20+𝛼(0)1𝑓+𝑎𝛼(0)+22𝑓+𝑏𝛾𝑒𝑖𝜔0𝜏01𝑊(2)11𝜏01+𝛼2𝑓+𝑏(0)𝛾2𝛾𝑒𝑖𝜔0𝜏01𝑊+𝑐(2)20(0)+𝛾𝑊(1)20+(0)𝐷𝛾𝛼2𝑓𝑏𝑊(0)(1)20𝜏02𝑒𝑖𝜔0𝜏02+2𝑒𝑖𝜔0𝜏02𝑊(1)11𝜏02+𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)2𝛾𝑊(2)11(0)+𝑊(2)20(0)𝛼𝛾+1𝑓𝑎(0)𝛾2𝑊𝛾+𝑑(2)20(0)+𝛾𝑊(1)20.(0)(3.38) For unknown 𝑊(1)20(0),𝑊(1)20𝜏02,𝑊(1)11(0),𝑊(2)11(0),𝑊(2)11𝜏01,𝑊(2)11𝜏02(3.39) in 𝑔21, we still need to compute them.

Form (3.8), (3.32), we have 𝑊=𝐴𝑊2Re𝑞(0)𝑓𝑞(𝜃),𝜏01𝜃<0,𝐴𝑊2Re𝑞(0)𝑓𝑞(𝜃)+𝑓,𝜃=0def=𝐴𝑊+𝐻𝑧,,𝑧,𝜃(3.40) where𝐻𝑧,𝑧,𝜃=𝐻20𝑧(𝜃)22+𝐻11(𝜃)𝑧𝑧+𝐻02(𝜃)𝑧22+.(3.41) Comparing the coefficients, we obtain𝐴2𝑖𝜔0𝑊20=𝐻20(𝜃),(3.42)𝐴𝑊11(𝜃)=𝐻11(𝜃),.(3.43) And we know that, for 𝜃[𝜏01,0),𝐻𝑧,𝑧,𝜃=𝑞(0)𝑓0𝑞(𝜃)𝑞(0)𝑓0𝑞(𝜃)=𝑔𝑧,𝑧𝑞(𝜃)𝑔𝑧,𝑧𝑞(𝜃).(3.44) Comparing the coefficients of (3.41) with (3.44) gives that𝐻20(𝜃)=𝑔20𝑞(𝜃)𝑔02𝐻𝑞(𝜃),(3.45)11(𝜃)=𝑔11𝑞(𝜃)𝑔11𝑞(𝜃).(3.46) From (3.42), (3.45), and the definition of 𝐴, we geṫ𝑊20(𝜃)=2𝑖𝜔0𝑊20(𝜃)+𝑔20𝑞(𝜃)+𝑔02𝑞(𝜃).(3.47) Noting that 𝑞(𝜃)=𝑞(0)𝑒𝑖𝜔0𝜃, we have𝑊20(𝜃)=𝑖𝑔20𝜔0𝑞(0)𝑒𝑖𝜔0𝜃+𝑖𝑔023𝜔0𝑞(0)𝑒𝑖𝜔0𝜃+𝐸1𝑒2𝑖𝜔0𝜃,(3.48) where 𝐸1=(𝐸1(1),𝐸1(2))𝑇 is a constant vector.

Similarly, from (3.43), (3.46), and the definition of 𝐴, we havė𝑊11(𝜃)=𝑔11𝑞(𝜃)+𝑔11𝑊𝑞(𝜃),(3.49)11(𝜃)=𝑖𝑔11𝜔0𝑞(0)𝑒𝑖𝜔0𝜃+𝑖𝑔11𝜔0𝑞(0)𝑒𝑖𝜔0𝜃+𝐸2,(3.50) where 𝐸2=(𝐸2(1),𝐸2(2))𝑇 is a constant vector.

In what follows, we will seek appropriate 𝐸1, 𝐸2 in (3.48), (3.50), respectively. It follows from the definition of 𝐴 and (3.45), (3.46) that0𝜏01𝑑𝜂(𝜃)𝑊20(𝜃)=2𝑖𝜔0𝑊20(0)𝐻20(0)(3.51)01𝑑𝜂(𝜃)𝑊11(𝜃)=𝐻11(0),(3.52) where 𝜂(𝜃)=𝜂(0,𝜃).

From (3.42), we have𝐻20(0)=𝑔20𝑞(0)𝑔02𝐻𝑞(0)+1,𝐻2𝑇,(3.53) where 𝐻1=𝛼1𝑓+𝑎(0)2+𝛼2𝑓+𝑏(0)2𝛾2,𝐻2=𝛼2𝑓𝑏(0)2𝑒2𝑖𝜔0𝜏02+𝛼1𝑓𝑎(0)2𝛾2.(3.54) From (3.43), we have𝐻11(0)=𝑔11𝑞(0)𝑔11(0)𝑃𝑞(0)+1,𝑃2𝑇,(3.55) where 𝑃1=𝛼1𝑓+𝑎𝛼(0)+2𝑓+𝑏(0)𝛾𝑃𝛾,2=𝛼2𝑓𝑏(𝛼0)+1𝑓𝑎(0)𝛾𝛾.(3.56) Noting that𝑖𝜔0𝐼0𝜏01𝑒𝑖𝜔0𝜃𝑑𝜂(𝜃)𝑞(0)=0,𝑖𝜔0𝐼0𝜏01𝑒𝑖𝜔0𝜃𝑑𝜂(𝜃)𝑞(0)=0(3.57) and substituting (3.48) and (3.53) into (3.51), we have2𝑖𝜔0𝐼0𝜏01𝑒2𝑖𝜔0𝜃𝐸𝑑𝜂(𝜃)1=𝐻1,𝐻2𝑇.(3.58) That is,2𝑖𝜔0𝐼𝐴1𝐵𝑒2𝑖𝜔0𝜏02𝐶𝑒2𝑖𝜔0𝜏01𝐸1=𝐻1,𝐻2𝑇,(3.59) then2𝑖𝜔0𝛼1𝑓+𝑎𝛼(0)2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏022𝑖𝜔0𝛼1𝑓𝑎𝐸(0)1(1)𝐸1(2)=𝐻1𝐻2.(3.60) Hence, 𝐸1(1)=Δ11Δ1,𝐸1(2)=Δ12Δ1,(3.61) where Δ1=det2𝑖𝜔0𝛼1𝑓+𝑎𝛼(0)2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏022𝑖𝜔0𝛼1𝑓𝑎,Δ(0)11𝐻=det1𝛼2𝑓+𝑏(0)𝑒𝑖𝜔0𝜏01𝐻22𝑖𝜔0𝛼1𝑓𝑎,Δ(0)12=det2𝑖𝜔0𝛼1𝑓+𝑎(0)𝐻1𝛼2𝑓𝑏(0)𝑒𝑖𝜔0𝜏02𝐻2.(3.62) Similarly, substituting (3.49) and (3.55) into (3.52), we have0𝜏01𝐸𝑑𝜂(𝜃)2=𝑃1,𝑃2𝑇.(3.63) Then,𝐴1𝐸+𝐵+𝐶2=𝑃1,𝑃2𝑇.(3.64) That is,𝛼1𝑓+𝑎𝛼(0)2𝑓+𝑏𝛼(0)2𝑓𝑏𝛼(0)1𝑓𝑎𝐸(0)2(1)𝐸2(2)=𝑃1𝑃2.(3.65) Hence, 𝐸2(1)=Δ21Δ2,𝐸2(2)=Δ22Δ2,(3.66) where Δ2𝛼=det1𝑓+𝑎𝛼(0)2𝑓+𝑏𝛼(0)2𝑓𝑏𝛼(0)1𝑓𝑎,Δ(0)21=det𝑃1𝛼2𝑓+𝑏(0)𝑃2𝛼1𝑓𝑎,Δ(0)22𝛼=det1𝑓+𝑎(0)𝑃1𝛼2𝑓𝑏(0)𝑃2.(3.67) From (3.48), (3.50), we can calculate 𝑔21 and derive the following values: 𝑐1𝑖(0)=2𝜔0𝜏0𝑔20𝑔11||𝑔211||2||𝑔02||23+𝑔212,𝜇2𝑐=Re1(0)𝜆Re𝜏0,𝛽2𝑐=2Re1,𝑇(0)2𝑐=Im1(0)+𝜇2𝜆Im𝜏0𝜔0𝜏0.(3.68) These formulae give a description of the Hopf bifurcation periodic solutions of (1.2) at 𝜏=𝜏0 on the center manifold. From the discussion above, we have the following result.

Theorem 3.3. The periodic solution is supercritical (subcritical) if 𝜇2>0(𝜇2<0); The bifurcating periodic solutions are orbitally asymptotically stable with asymptotical phase (unstable) if 𝛽2<0(𝛽2>0); the periods of bifurcating periodic solutions increase (decrease) if T2>0(T2<0).

4. Numerical Examples

In this section, we present some numerical results to verify the analytical predictions obtained in the previous section. As an example, we consider the following special case of the system (1.2) with the parameters 𝛼1=0.03,𝛼2=0.1,𝑎=1.5,𝑏=2,𝑐=0,𝑑=4,and𝑓(𝑥)=tanh(𝑥). Then, the system (1.2) becomeṡ𝑥1𝑥(𝑡)=1.47tanh1𝑥(𝑡)+1.9tanh2𝑡𝜏1,̇𝑥2(𝑥𝑡)=2.1tanh1𝑡𝜏2𝑥1.53tanh2(𝑡)4𝑥1(𝑡)𝑥2(𝑡).(4.1) By some complicated computation by means of Matlab 7.0, we get 𝜔01.3211,𝜏00.01,𝜆(𝜏0)0.01401.4926𝑖. We can easily obtain 𝑔20=0.2501+2.3128𝑖,𝑔11=1.2377+0.3484𝑖,𝑔02=0.45330.5693𝑖,𝑔21=2.3022+4.3015𝑖. Thus, we can calculate the following values:  𝑐1(0)=1.06171.7138𝑖,𝜇2=75.8357,𝛽2=2.1234,𝑇2=86.9780. We obtain that the conditions indicated in Theorem 2.6 are satisfied. Furthermore, it follows that 𝜇2>0 and 𝛽2<0. Thus, the equilibrium 𝐸(0,0) is stable when 𝜏<𝜏0 as illustrated by the computer simulations (see Figures 1, 2, and 3). When 𝜏 passes through the critical value 𝜏0, the equilibrium 𝐸(0,0) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the equilibrium 𝐸(0,0). Since 𝜇2>0 and 𝛽2<0, the direction of the Hopf bifurcation is 𝜏>𝜏0 and these bifurcating periodic solutions from 𝐸(0,0) at 𝜏0 are stable, which are depicted in Figures 4, 5, and 6.

697630.fig.001
Figure 1: Trajectories graphs of the system (4.1) with 𝜏1=0.003,𝜏2=0.004, and 𝜏1+𝜏2=0.007<𝜏00.01. The equilibrium 𝐸(0,0) is asymptotically stable. The initial value is (0.05, 0.01).
697630.fig.002
Figure 2: Trajectories graphs of the system (4.1) with 𝜏1=0.003,𝜏2=0.004, and 𝜏1+𝜏2=0.007<𝜏00.01. The equilibrium 𝐸(0,0) is asymptotically stable. The initial value is (0.05, 0.01).
697630.fig.003
Figure 3: Trajectories graphs of the system (4.1) with 𝜏1=0.003,𝜏2=0.004, and 𝜏1+𝜏2=0.007<𝜏00.01. The equilibrium 𝐸(0,0) is asymptotically stable. The initial value is (0.05, 0.01).
697630.fig.004
Figure 4: Trajectories graphs of thesystem (4.1) with 𝜏1=0.01,𝜏2=0.04, and 𝜏1+𝜏2=0.05>𝜏00.01. Hopf bifurcation occurs from the equilibrium 𝐸(0,0). The initial value is (0.05, 0.01).
697630.fig.005
Figure 5: Trajectories graphs of the system (4.1) with 𝜏1=0.01,𝜏2=0.04, and 𝜏1+𝜏2=0.05>𝜏00.01. Hopf bifurcation occurs from the equilibrium 𝐸(0,0). The initial value is (0.05, 0.01).
697630.fig.006
Figure 6: Trajectories graphs of the system (4.1) with 𝜏1=0.01,𝜏2=0.04, and 𝜏1+𝜏2=0.05>𝜏00.01. Hopf bifurcation occurs from the equilibrium 𝐸(0,0). The initial value is (0.05, 0.01).

5. 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.

Acknowledgments

This work is supported by the National Natural Science Foundation of China (no. 10771215 and no. 10771094), Scientific Research Fund of Hunan Provincial Education Department (no. 10C0560) and Doctoral Foundation of Guizhou College of Finance and Economics (2010).

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