Abstract and Applied Analysis

Abstract and Applied Analysis / 2011 / Article

Research Article | Open Access

Volume 2011 |Article ID 697630 | https://doi.org/10.1155/2011/697630

Changjin Xu, Xiaofei He, "Stability and Bifurcation Analysis in a Class of Two-Neuron Networks with Resonant Bilinear Terms", Abstract and Applied Analysis, vol. 2011, Article ID 697630, 21 pages, 2011. https://doi.org/10.1155/2011/697630

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

Academic Editor: Nobuyuki Kenmochi
Received08 Jan 2011
Revised27 Feb 2011
Accepted27 Apr 2011
Published21 Jun 2011

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 [3–13]).

In 2008, Yang et al. [14] investigated the Bautin bifurcation of the two-neuron networks with resonant bilinear terms and without delay:Μ‡π‘₯1𝛼(𝑑)=1𝑓π‘₯+π‘Ž1ξ€Έ+𝛼2𝑓π‘₯+𝑏2ξ€Έ+𝑐π‘₯1π‘₯2,Μ‡π‘₯2(𝛼𝑑)=2𝑓π‘₯βˆ’π‘1ξ€Έ+𝛼1𝑓π‘₯βˆ’π‘Ž2ξ€Έ+𝑑π‘₯1π‘₯2,(1.1) where π‘₯𝑖(𝑑)(𝑖=1,2) represents the state of the 𝑖th neuron at time 𝑑, 𝑓(π‘₯𝑖)(𝑖=1,2) is the connection function between two neurons, and 𝛼1,𝛼2,π‘Ž,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:Μ‡π‘₯1𝛼(𝑑)=1𝑓π‘₯+π‘Ž1ξ€Έ+𝛼2𝑓π‘₯+𝑏2ξ€·π‘‘βˆ’πœ1ξ€Έξ€Έ+𝑐π‘₯1π‘₯2,Μ‡π‘₯2(𝛼𝑑)=2𝑓π‘₯βˆ’π‘1ξ€·π‘‘βˆ’πœ2+𝛼1𝑓π‘₯βˆ’π‘Ž2ξ€Έ+𝑑π‘₯1π‘₯2,(1.2) where π‘₯𝑖(𝑑)(𝑖=1,2) represents the state of the 𝑖-th neuron at time 𝑑, 𝑓(π‘₯𝑖)(𝑖=1,2) is the connection function between two neurons, 𝛼1,𝛼2,π‘Ž,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) π‘“βˆˆπΆ3(𝑅),𝑓(0)=0, and 𝑒𝑓(𝑒)>0, for 𝑒≠0.

Hypothesis (H1) implies that πΈβˆ—(0,0) is an equilibrium of the system (1.2) and linearized system of (1.2) takes the formΜ‡π‘₯1𝛼(𝑑)=1𝑓+π‘Žξ…ž(0)π‘₯1+𝛼2𝑓+π‘ξ…ž(0)π‘₯2ξ€·π‘‘βˆ’πœ1ξ€Έ,Μ‡π‘₯2(𝛼𝑑)=2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘₯1ξ€·π‘‘βˆ’πœ2ξ€Έ+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…ž(0)π‘₯2.(2.1) The associated characteristic equation of (2.1) isπœ†2βˆ’2𝛼1π‘“ξ…žξ€·π›Ό(0)πœ†+21βˆ’π‘Ž2ξ€Έπ‘“ξ…ž2𝛼(0)βˆ’22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)π‘’βˆ’πœ†πœ=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 π‘ƒξ€·πœ†,π‘’βˆ’πœ†πœ1,…,π‘’βˆ’πœ†πœπ‘šξ€Έ=πœ†π‘›+𝑝1(0)πœ†π‘›βˆ’1+β‹―+𝑝(0)π‘›βˆ’1πœ†+𝑝𝑛(0)+𝑝1(1)πœ†π‘›βˆ’1+β‹―+𝑝(1)π‘›βˆ’1πœ†+𝑝𝑛(1)ξ‚„π‘’βˆ’πœ†πœ1+𝑝+β‹―1(π‘š)πœ†π‘›βˆ’1+β‹―+𝑝(π‘š)π‘›βˆ’1πœ†+𝑝𝑛(π‘š)ξ‚„π‘’βˆ’πœ†πœπ‘š=0,(2.3) as (𝜏1,𝜏2,𝜏3,…,πœπ‘š) vary, the sum of orders of the zeros of 𝑃(πœ†,π‘’βˆ’πœ†πœ1,…,π‘’βˆ’πœ†πœπ‘š) 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) 𝛼1π‘“ξ…ž(0)<0 and 𝛼21βˆ’π›Ό22βˆ’π‘Ž2+𝑏2>0;

(H3) |𝛼21βˆ’π‘Ž2|<|𝛼22βˆ’π‘2|.

Lemma 2.2. If (H1)–(H3) hold, then one has the following.
(i)When 𝜏=πœπ‘—def=1πœ”0arccos2𝛼1πœ”0𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2ξƒ­(0)+2π‘—πœ‹,𝑗=0,1,2,….(2.4) Equation (2.2) has a simple pair of imaginary roots Β±π‘–πœ”0, where πœ”0=ξƒŽξ‚™ξ€·π›Ό21+π‘Ž2ξ€Έ2βˆ’ξ‚ƒξ€·π›Ό21βˆ’π‘Ž2ξ€Έ2βˆ’ξ€·π›Ό22βˆ’π‘2ξ€Έ2ξ‚„π‘“ξ…ž4𝛼(0)βˆ’21+π‘Ž2ξ€Έ.(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 Β±π‘–πœ”0 and all other roots have strictly negative real parts.

Proof. Obviously, by assumption (H2), πœ†=0 is not the root of (2.2). When 𝜏=0, then (2.2) becomes πœ†2βˆ’2𝛼1π‘“ξ…žξ€Ίπ›Ό(0)πœ†+21βˆ’π›Ό22βˆ’π‘Ž2+𝑏2ξ€»π‘“ξ…ž2(0)=0.(2.6) It is easy to see that all roots of (2.6) have negative real parts.
Β±π‘–πœ”(πœ”>0) is a pair of purely imaginary roots of (2.2) if and only if πœ” satisfies βˆ’πœ”2βˆ’2𝛼1π‘“ξ…žξ€·π›Ό(0)πœ”π‘–+21βˆ’π‘Ž2ξ€Έπ‘“ξ…ž2𝛼(0)βˆ’2βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)(cosπœ”πœβˆ’π‘–sinπœ”πœ)=0.(2.7) Separating the real and imaginary parts, we get 𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2𝛼(0)cosπœ”πœ=21βˆ’π‘Ž2ξ€Έπ‘“ξ…ž2(0)βˆ’πœ”2,𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)sinπœ”πœ=2𝛼1π‘“ξ…ž(0)πœ”.(2.8) It follows from (2.8) that πœ”4𝛼+221+π‘Ž2ξ€Έπ‘“ξ…ž2(0)πœ”2+𝛼21βˆ’π‘Ž2ξ€Έ2βˆ’ξ€·π›Ό22βˆ’π‘2ξ€Έ2ξ‚„π‘“ξ…ž4(0)=0.(2.9) Thus, we obtain ξƒŽπœ”=𝛼21+π‘Ž2ξ€Έ2βˆ’ξ‚ƒξ€·π›Ό21βˆ’π‘Ž2ξ€Έ2βˆ’ξ€·π›Ό22βˆ’π‘2ξ€Έ2ξ‚„π‘“ξ…ž4𝛼(0)βˆ’21+π‘Ž2ξ€Έ.(2.10) It is clear that πœ” is well defined if condition (H3) holds.
Denote πœ”0=ξƒŽξ‚™ξ€·π›Ό21+π‘Ž2ξ€Έ2βˆ’ξ‚ƒξ€·π›Ό21βˆ’π‘Ž2ξ€Έ2βˆ’ξ€·π›Ό22βˆ’π‘2ξ€Έ2ξ‚„π‘“ξ…ž4𝛼(0)βˆ’21+π‘Ž2ξ€Έ.(2.11) Let πœπ‘—=1πœ”0arccos2𝛼1πœ”0𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2ξƒ­(0)+2π‘—πœ‹,𝑗=0,1,2,….(2.12) From (2.8), we know that (2.2) with 𝜏=πœπ‘—(𝑗=0,1,2,…) has a pair of imaginary roots Β±π‘–πœ”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 πœ†π‘—(𝜏)=𝛼𝑗(𝜏)+π‘–πœ”π‘—(𝜏) be a root of (2.2) near 𝜏=πœπ‘—, and 𝛼𝑗(πœπ‘—)=0,πœ”π‘—(πœπ‘—)=πœ”0,(𝑗=0,1,2…). Due to functional differential equation theory, for every πœπ‘—,π‘˜=0,1,2…, there exists πœ€>0 such that πœ†π‘—(𝜏) is continuously differentiable in 𝜏 for |πœβˆ’πœπ‘—|<πœ€. Substituting πœ†(𝜏) into the left-hand side of (2.2) and taking derivative with respect to 𝜏, we have ξ‚ƒπ‘‘πœ†ξ‚„π‘‘πœŽβˆ’1=βˆ’2πœ†βˆ’2𝛼1π‘“ξ…ž(0)𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)π‘’βˆ’πœ†πœπœ†βˆ’πœπœ†,(2.13) which leads to Reπ‘‘πœ†ξ‚„π‘‘πœŽβˆ’1𝜏=πœπ‘—=2𝛼1π‘“ξ…žξ€·π›Ό(0)22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)πœ”0sinπœ”0πœπ‘—βˆ’2πœ”20𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)cosπœ”0πœπ‘—π›Όξ€Ίξ€·22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)πœ”0sinπœ”0πœπ‘—ξ€»2+ξ€Ίπœ”0𝛼22βˆ’π‘2ξ€Έπ‘“ξ…ž2(0)cosπœ”0πœπ‘—ξ€»2.(2.14) By (2.8), we getReπ‘‘πœ†ξ‚„π‘‘πœŽβˆ’1𝜏=πœπ‘—=2πœ”20+ξ€·3𝛼21+π‘Ž2ξ€Έπ‘“ξ…ž2(0)ξ€Ί2πœ”0𝛼1π‘“ξ…žξ€»(0)2+𝛼21βˆ’π‘Ž2ξ€Έπ‘“ξ…ž2(0)βˆ’πœ”20ξ€»2>0.(2.15) So we havesignReπ‘‘πœ†ξ‚„π‘‘πœπœ=πœπ‘—ξ‚ƒ=signReπ‘‘πœ†ξ‚„π‘‘πœβˆ’1𝜏=πœπ‘—>0.(2.16)

From the above analysis, we have the following results.

Lemma 2.3. Let 𝜏=πœπ‘—, then the following transversality condition: π‘‘ξ€Ίπœ†π‘‘πœRe𝑗|||(𝜏)𝜏=πœπ‘—>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∈(πœπ‘—,πœπ‘—+1), (2.2) has 2(𝑗+1)(𝑗=0,1,2,…) roots with positive real parts.
From Lemma 2.2–2.4, 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=πœπ‘—(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 πΈβˆ—(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 πΈβˆ—(0,0) for 𝜏=𝜏0 and then Β±π‘–πœ”0 is corresponding purely imaginary roots of the characteristic equation at the equilibrium πΈβˆ—(0,0).

For convenience, let 𝜏=𝜏0+πœ‡,πœ‡βˆˆπ‘…. Then πœ‡=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 πœ‡ close to 0. Without loss of generality, we assume that 𝜏01>𝜏02 and let |πœ‡|β‰€πœ01βˆ’πœ02. Since our analysis is local, where 𝜏0=𝜏01+𝜏02 and 𝜏=𝜏01+(𝜏02+πœ‡). We can consider the fixed phase space 𝐢=𝐢([βˆ’πœ01,0],𝑅2).

For (πœ™1,πœ™2)∈𝐢, defineπΏπœ‡πœ™=𝐴1πœ™ξ€·(0)+π΅πœ™βˆ’πœ2ξ€Έξ€·+πΆπœ™βˆ’πœ1ξ€Έ,(3.1) where 𝐴1=𝛼1𝑓+π‘Žξ…ž00𝛼(0)1ξ€Έπ‘“βˆ’π‘Žξ…žξƒͺ𝛼(0),𝐡=002ξ€Έπ‘“βˆ’π‘ξ…žξƒͺ0𝛼(0)0,𝐢=2𝑓+π‘ξ…žξƒͺ.(0)00(3.2) We expand the nonlinear part of the system (1.2) and derive the following expression:𝑓𝑓(πœ‡,πœ™)=1(π‘“πœ‡,πœ™)2ξƒͺ(πœ‡,πœ™),(3.3) where𝑓1𝛼(πœ‡,πœ™)=1𝑓+π‘Žξ…žξ…ž(0)2πœ™21𝑓(0)+ξ…žξ…žξ…ž(0)πœ™3!31ξ‚Ή+𝛼(0)2𝑓+π‘ξ…žξ…ž(0)2πœ™22ξ€·βˆ’πœ1ξ€Έ+π‘“ξ…žξ…žξ…ž(0)πœ™3!32ξ€·βˆ’πœ1ξ€Έξ‚Ή+π‘πœ™1(0)πœ™2𝑓(0)+h.o.t.,2𝛼(πœ‡,πœ™)=2ξ€Έξ‚Έπ‘“βˆ’π‘ξ…žξ…ž(0)2πœ™21ξ€·βˆ’πœ2ξ€Έ+π‘“ξ…žξ…žξ…ž(0)πœ™3!31ξ€·βˆ’πœ2ξ€Έξ‚Ή+𝛼1ξ€Έξ‚Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2πœ™22𝑓(0)+ξ…žξ…žξ…ž(0)πœ™3!32ξ‚Ή(0)+π‘‘πœ™1(0)πœ™2(0)+h.o.t..(3.4) By the representation theorem, there is a matrix function with bounded variation components πœ‚(πœƒ,πœ‡),πœƒβˆˆ[βˆ’πœ01,0] such thatπΏπœ‡ξ€œπœ™=0βˆ’πœ01π‘‘πœ‚(πœƒ,πœ‡)πœ™(πœƒ),forπœ™βˆˆπΆ.(3.5) In fact, we can choose⎧βŽͺβŽͺ⎨βŽͺβŽͺβŽ©π΄πœ‚(πœƒ,πœ‡)=1ξ€·,πœƒ=0,π΅π›Ώπœƒ+𝜏2ξ€Έξ€Ί,πœƒβˆˆβˆ’πœ2ξ€Έ,ξ€·,0βˆ’πΆπ›Ώπœƒ+𝜏01ξ€Έξ€Ί,πœƒβˆˆβˆ’πœ01,βˆ’πœ2ξ€Έ,(3.6) where 𝛿 is the Dirac delta function.

For πœ™βˆˆπΆ([βˆ’πœ01,0],𝑅2), define⎧βŽͺβŽͺ⎨βŽͺβŽͺ⎩𝐴(πœ‡)πœ™=π‘‘πœ™(πœƒ)π‘‘πœƒ,βˆ’πœ01ξ€œβ‰€πœƒ<0,0βˆ’πœ01π‘…ξƒ―π‘‘πœ‚(𝑠,πœ‡)πœ™(𝑠),πœƒ=0,(πœ‡)πœ™=0,βˆ’πœ01β‰€πœƒ<0,𝑓(πœ‡,πœ™),πœƒ=0.(3.7) Then (1.2) is equivalent to the abstract differential equationΜ‡π‘₯𝑑=𝐴(πœ‡)π‘₯𝑑+𝑅(πœ‡)π‘₯𝑑,(3.8) where π‘₯=(π‘₯1,π‘₯2)𝑇,π‘₯𝑑(πœƒ)=π‘₯(𝑑+πœƒ),πœƒβˆˆ[βˆ’πœ01,0].

For πœ“βˆˆπΆ([0,𝜏01],(𝑅2)βˆ—), defineπ΄βˆ—βŽ§βŽͺβŽͺ⎨βŽͺβŽͺβŽ©βˆ’πœ“(𝑠)=π‘‘πœ“(𝑠)𝑑𝑠,s∈0,𝜏01ξ€»,ξ€œ0βˆ’πœ01π‘‘πœ‚π‘‡(𝑑,0)πœ“(βˆ’π‘‘),s=0.(3.9)

For πœ™βˆˆπΆ([βˆ’πœ01,0],𝑅2) and πœ“βˆˆπΆ([0,𝜏01],(𝑅2)βˆ—), define the bilinear formβŸ¨πœ“,πœ™βŸ©=ξ€œπœ“(0)πœ™(0)βˆ’0βˆ’πœ01ξ€œπœƒπœ‰=0πœ“(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)πœ™(πœ‰)π‘‘πœ‰,(3.10) where πœ‚(πœƒ)=πœ‚(πœƒ,0). We have the following result on the relation between the operators 𝐴=𝐴(0) and π΄βˆ—.

Lemma 3.1. 𝐴=𝐴(0) and π΄βˆ— are adjoint operators.

Proof. Let πœ™βˆˆπΆ1([βˆ’πœ01,0],𝑅2) and πœ“βˆˆπΆ1([0,𝜏01],(𝑅2)βˆ—). It follows from (3.10) and the definitions of 𝐴=𝐴(0) and π΄βˆ— that βŸ¨πœ“(𝑠),𝐴(0)πœ™(πœƒ)⟩=ξ€œπœ“(0)𝐴(0)πœ™(0)βˆ’0βˆ’πœ01ξ€œπœƒπœ‰=0=πœ“(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)𝐴(0)πœ™(πœ‰)π‘‘πœ‰ξ€œπœ“(0)0βˆ’πœ01ξ€œπ‘‘πœ‚(πœƒ)πœ™(πœƒ)βˆ’0βˆ’πœ01ξ€œπœƒπœ‰=0=πœ“(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)𝐴(0)πœ™(πœ‰)π‘‘πœ‰ξ€œπœ“(0)0βˆ’πœ01ξ€œπ‘‘πœ‚(πœƒ)πœ™(πœƒ)βˆ’0βˆ’πœ01ξ€Ίξ€»πœ“(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)πœ™(πœ‰)πœƒπœ‰=0+ξ€œ0βˆ’πœ01ξ€œπœƒπœ‰=0π‘‘πœ“(πœ‰βˆ’πœƒ)=ξ€œπ‘‘πœ‰π‘‘πœ‚(πœƒ)πœ™(πœ‰)π‘‘πœ‰0βˆ’πœ01ξ€œπœ“(βˆ’πœƒ)π‘‘πœ‚(πœƒ)πœ™(0)βˆ’0βˆ’πœ01ξ€œπœƒπœ‰=0ξ‚Έβˆ’π‘‘πœ“(πœ‰βˆ’πœƒ)ξ‚Ήπ‘‘πœ‰π‘‘πœ‚(πœƒ)πœ™(πœ‰)π‘‘πœ‰=π΄βˆ—ξ€œπœ“(0)πœ™(0)βˆ’0βˆ’πœ01ξ€œπœƒπœ‰=0π΄βˆ—πœ“(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)πœ™(πœ‰)π‘‘πœ‰=βŸ¨π΄βˆ—πœ“(𝑠),πœ™(πœƒ)⟩.(3.11) This shows that 𝐴=𝐴(0) and π΄βˆ— are adjoint operators and the proof is complete.

By the discussions in the Section 2, we know that Β±π‘–πœ”0 are eigenvalues of 𝐴(0) and they are also eigenvalues of π΄βˆ— corresponding to π‘–πœ”0 and βˆ’π‘–πœ”0, respectively. We have the following result.

Lemma 3.2. The vector π‘ž(πœƒ)=(1,𝛾)π‘‡π‘’π‘–πœ”0πœƒξ€Ί,πœƒβˆˆβˆ’πœ01ξ€»,,0(3.12) where 𝛾=π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…ž(0)𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01,(3.13) is the eigenvector of 𝐴(0) corresponding to the eigenvalue π‘–πœ”0, and π‘žβˆ—ξ€·(𝑠)=𝐷1,π›Ύβˆ—ξ€Έπ‘’π‘–πœ”π‘ ξ€Ί,π‘ βˆˆ0,𝜏01ξ€»,(3.14) where π›Ύβˆ—=βˆ’π‘–πœ”0+𝛼1𝑓+π‘Žξ…ž(0)𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”πœ02,(3.15) is the eigenvector of π΄βˆ— corresponding to the eigenvalue βˆ’π‘–πœ”0, moreover, βŸ¨π‘žβˆ—(𝑠),π‘ž(πœƒ)⟩=1, where 𝐷=1+π›Ύπ›Ύβˆ—+π›Ύβˆ—πœ02𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02+π›Ύπœ01𝛼2𝑓+π‘ξ…ž(0)π‘’π‘–πœ”0𝜏01.(3.16)

Proof. Let π‘ž(πœƒ) be the eigenvector of 𝐴(0) corresponding to the eigenvalue π‘–πœ”0 and π‘žβˆ—(𝑠) be the eigenvector of π΄βˆ— corresponding to the eigenvalue βˆ’π‘–πœ”0, namely, 𝐴(0)π‘ž(πœƒ)=π‘–πœ”0π‘ž(πœƒ) and π΄βˆ—π‘žβˆ—π‘‡(𝑠)=βˆ’π‘–πœ”0π‘žβˆ—π‘‡(𝑠). From the definitions of 𝐴(0) and π΄βˆ—, we have 𝐴(0)π‘ž(πœƒ)=π‘‘π‘ž(πœƒ)/π‘‘πœƒ and π΄βˆ—π‘žβˆ—π‘‡(𝑠)=βˆ’π‘‘π‘žβˆ—π‘‡(𝑠)/𝑑𝑠. Thus, π‘ž(πœƒ)=π‘ž(0)π‘’π‘–πœ”0πœƒ and π‘žβˆ—(𝑠)=π‘žβˆ—(0)π‘’π‘–πœ”0𝑠. In addition, ξ€œ0βˆ’πœ01π‘‘πœ‚(πœƒ)π‘ž(πœƒ)=𝐴1ξ€·π‘ž(0)+π΅π‘žβˆ’πœ2ξ€Έξ€·+πΆπ‘žβˆ’πœ1ξ€Έ=𝐴(0)π‘ž(0)=π‘–πœ”0π‘ž(0).(3.17) That is, βŽ›βŽœβŽœβŽπ‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…žξ€·π›Ό(0)βˆ’2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01βˆ’ξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02π‘–πœ”0βˆ’ξ€·π›Ό1ξ€Έπ‘“βˆ’π‘Žξ…žβŽžβŽŸβŽŸβŽ π‘žξƒ©00ξƒͺ(0)(0)=.(3.18) Therefore, we can easily obtain 𝛾=π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…ž(0)𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01(3.19) and so ξƒ©π‘ž(0)=1,π‘–πœ”0βˆ’(𝛼1+π‘Ž)π‘“ξ…ž(0)𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01ξƒͺ𝑇(3.20) hence ξƒ©π‘ž(πœƒ)=1,π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…ž(0)𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01ξƒͺπ‘‡π‘’π‘–πœ”0πœƒ.(3.21) On the other hand, ξ€œ0βˆ’πœ01π‘žβˆ—(βˆ’π‘‘)π‘‘πœ‚(𝑑)=𝐴𝑇1π‘žβˆ—π‘‡(0)+π΅π‘‡π‘žβˆ—π‘‡ξ€·βˆ’πœ02ξ€Έ+πΆπ‘‡π‘žβˆ—π‘‡ξ€·βˆ’πœ01ξ€Έ=π΄βˆ—π‘žβˆ—π‘‡(0)=βˆ’π‘–πœ”0π‘žβˆ—π‘‡(0).(3.22) Namely, βŽ›βŽœβŽœβŽπ‘–πœ”0+𝛼1𝑓+π‘Žξ…žξ€·π›Ό(0)2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01π‘–πœ”0+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žβŽžβŽŸβŽŸβŽ π‘ž(0)βˆ—ξƒ©00ξƒͺ(0)=.(3.23) Therefore, we can easily obtain π›Ύβˆ—=βˆ’π‘–πœ”0+𝛼1𝑓+π‘Žξ…ž(0)𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02(3.24) and so π‘žβˆ—ξƒ©(0)=1,βˆ’π‘–πœ”0+𝛼1𝑓+π‘Žξ…ž(0)𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02ξƒͺ(3.25) hence π‘žβˆ—ξƒ©(𝑠)=1,βˆ’π‘–πœ”0+𝛼1𝑓+π‘Žξ…ž(0)𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02ξƒͺπ‘’π‘–πœ”0𝑠.(3.26) In the sequel, we will verify that βŸ¨π‘žβˆ—(𝑠),π‘ž(πœƒ)⟩=1. In fact, from (3.10), we have βŸ¨π‘žβˆ—(𝑠),π‘ž(πœƒ)⟩=𝐷1,π›Ύβˆ—ξ‚(1,𝛾)π‘‡βˆ’ξ€œ0βˆ’πœ01ξ€œπœƒπœ‰=0𝐷1π›Ύβˆ—ξ‚π‘’βˆ’π‘–πœ”0(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)(1,𝛾)π‘‡π‘’π‘–πœ”0πœ‰=π‘‘πœ‰π·ξƒ¬1+π›Ύπ›Ύβˆ—βˆ’ξ€œ0βˆ’πœ01ξ‚€1,π›Ύβˆ—ξ‚πœƒπ‘’π‘–πœ”0πœƒπ‘‘πœ‚(πœƒ)(1,𝛾)𝑇=𝐷1+π›Ύπ›Ύβˆ—βˆ’ξ‚€1,π›Ύβˆ—π΅ξ€·ξ‚ξ‚ƒβˆ’πœ02ξ€Έπ‘’βˆ’π‘–πœ”0𝜏02ξ€·+πΆβˆ’πœ01ξ€Έπ‘’βˆ’π‘–πœ”0𝜏01ξ‚„(1,𝛾)𝑇=𝐷1+π›Ύπ›Ύβˆ—+π›Ύβˆ—πœ02𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’π‘–πœ”0𝜏02+π›Ύπœ01𝛼2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01ξ‚„=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 𝐢0 at πœ‡=0. Let π‘₯𝑑 be the solution of (1.2) when πœ‡=0.

Define𝑧(𝑑)=βŸ¨π‘žβˆ—,π‘₯π‘‘βŸ©,π‘Š(𝑑,πœƒ)=π‘₯𝑑(πœƒ)βˆ’2Re{𝑧(𝑑)π‘ž(πœƒ)}(3.28) on the center manifold 𝐢0, and we haveπ‘Šξ€·π‘§(𝑑,πœƒ)=π‘Š(𝑑),𝑧(𝑑),πœƒ,(3.29) whereπ‘Šξ€·π‘§(𝑑),𝑧(𝑑),πœƒ=π‘Šπ‘§,𝑧=π‘Š20𝑧22+π‘Š11𝑧𝑧+π‘Š02𝑧22+β‹―(3.30) and 𝑧 and 𝑧 are local coordinates for center manifold 𝐢0 in the direction of π‘žβˆ— and π‘žβˆ—. Noting that π‘Š is also real if π‘₯𝑑 is real, we consider only real solutions. For solutions π‘₯π‘‘βˆˆπΆ0 of (1.2),̇𝑧(𝑑)=βŸ¨π‘žβˆ—(𝑠),Μ‡π‘₯π‘‘βŸ©=βŸ¨π‘žβˆ—(𝑠),𝐴(0)π‘₯𝑑+𝑅(0)π‘₯π‘‘βŸ©=βŸ¨π‘žβˆ—(𝑠),𝐴(0)π‘₯π‘‘βŸ©+βŸ¨π‘žβˆ—(s),𝑅(0)π‘₯π‘‘βŸ©=βŸ¨π΄βˆ—π‘žβˆ—(𝑠),π‘₯π‘‘βŸ©+π‘žβˆ—(0)𝑅(0)π‘₯π‘‘βˆ’ξ€œ0βˆ’πœ01ξ€œπœƒπœ‰=0π‘žβˆ—(πœ‰βˆ’πœƒ)π‘‘πœ‚(πœƒ)𝐴(0)𝑅(0)π‘₯𝑑(πœ‰)π‘‘πœ‰=βŸ¨π‘–πœ”0π‘žβˆ—(𝑠),π‘₯π‘‘βŸ©+π‘žβˆ—ξ€·(0)𝑓0,π‘₯𝑑(πœƒ)def=π‘–πœ”0𝑧(𝑑)+π‘žβˆ—(0)𝑓0𝑧(𝑑),ξ€Έ.𝑧(𝑑)(3.31) That is,̇𝑧(𝑑)=π‘–πœ”0𝑧+𝑔𝑧,𝑧,(3.32) where𝑔𝑧,𝑧=𝑔20𝑧22+𝑔11𝑧𝑧+𝑔02𝑧22+𝑔21𝑧2𝑧2+β‹―.(3.33) Hence, we have𝑔𝑧,𝑧=π‘žβˆ—(0)𝑓0𝑧,𝑧=𝑓0,π‘₯𝑑=𝐷1,π›Ύβˆ—π‘“ξ€Έξ€·1ξ€·0,π‘₯𝑑,𝑓2ξ€·0,π‘₯𝑑𝑇,(3.34) where 𝑓1ξ€·0,π‘₯𝑑=𝛼1𝑓+π‘Žξ…žξ…ž(0)2π‘₯21𝑑𝑓(0)+ξ…žξ…žξ…ž(0)π‘₯3!31𝑑+𝛼(0)2𝑓+π‘ξ…žξ…ž(0)2π‘₯22π‘‘ξ€·βˆ’πœ01ξ€Έ+π‘“ξ…žξ…žξ…ž(0)π‘₯3!32π‘‘ξ€·βˆ’πœ01ξ€Έξ‚Ή+𝑐π‘₯1𝑑(0)π‘₯2𝑑𝑓(0)+h.o.t.,2ξ€·0,π‘₯𝑑=𝛼2ξ€Έξ‚Έπ‘“βˆ’π‘ξ…žξ…ž(0)2π‘₯21π‘‘ξ€·βˆ’πœ02ξ€Έ+π‘“ξ…žξ…žξ…ž(0)π‘₯3!31π‘‘ξ€·βˆ’πœ02ξ€Έξ‚Ή+𝛼1ξ€Έξ‚Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2π‘₯22𝑑𝑓(0)+ξ…žξ…žξ…ž(0)π‘₯3!32𝑑(0)+𝑑π‘₯1𝑑(0)π‘₯2𝑑(0)+h.o.t..(3.35) Noticing π‘₯𝑑(πœƒ)=(π‘₯1𝑑(πœƒ),π‘₯2𝑑(πœƒ))𝑇=π‘Š(𝑑,πœƒ)+π‘§π‘ž(πœƒ)+π‘§π‘ž(πœƒ) and π‘ž(πœƒ)=(1,𝛾)π‘‡π‘’π‘–πœ”0πœƒ, we have π‘₯1𝑑(0)=𝑧+𝑧+π‘Š(1)20𝑧(0)22+π‘Š(1)11(0)𝑧𝑧+π‘Š(1)02(0)𝑧22π‘₯+β‹―,2𝑑(0)=𝛾𝑧+𝛾𝑧+π‘Š(2)20𝑧(0)22+π‘Š(2)11(0)𝑧𝑧+π‘Š(2)02(0)𝑧22π‘₯+β‹―,1π‘‘ξ€·βˆ’πœ02ξ€Έ=π‘’βˆ’π‘–πœ”0𝜏02𝑧+π‘’π‘–πœ”0𝜏02𝑧+π‘Š(1)20ξ€·βˆ’πœ02𝑧22+π‘Š(1)11ξ€·βˆ’πœ02𝑧𝑧+π‘Š(1)02ξ€·βˆ’πœ02𝑧22π‘₯+β‹―,2π‘‘ξ€·βˆ’πœ01ξ€Έ=π›Ύπ‘’βˆ’π‘–πœ”0𝜏01𝑧+π›Ύπ‘’π‘–πœ”0𝜏01𝑧+π‘Š(2)20ξ€·βˆ’πœ01𝑧22+π‘Š(2)11ξ€·βˆ’πœ01𝑧𝑧+π‘Š(2)02ξ€·βˆ’πœ01𝑧22+β‹―.(3.36) From (3.33) and (3.34), we have 𝑔𝑧,𝑧=π‘žβˆ—(0)𝑓0𝑧,𝑧=𝐷𝑓1ξ€·0,π‘₯𝑑+π›Ύβˆ—π‘“2ξ€·0,π‘₯𝑑=𝐷𝛼1𝑓+π‘Žξ…žξ…ž(0)2+𝛼2𝑓+π‘ξ…žξ…ž(0)2𝛾2ξ‚Ή++π‘π›Ύπ·π›Ύβˆ—ξ‚Έξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)2π‘’βˆ’2π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2𝛾2𝑧+𝑑𝛾2+𝐷𝛼1𝑓+π‘Žξ…žξ…žξ€·π›Ό(0)+2𝑓+π‘ξ…žξ…ž(0)2𝛾+𝛾+2𝑐Re{𝛾}π·π›Ύβˆ—π›Όξ€Ίξ€·2ξ€Έπ‘“βˆ’π‘ξ…žξ…žξ€·π›Ό(0)+1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾𝑧𝛾+2𝑑Re{𝛾}𝑧+𝐷𝛼1𝑓+π‘Žξ…žξ…ž(0)2+𝛼2𝑓+π‘ξ…žξ…ž(0)2𝛾2+𝑐𝛾+π·π›Ύβˆ—ξ‚Έξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)2𝑒2π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2𝛾2+𝑑𝛾𝑧2+𝐷𝛼1𝑓+π‘Žξ…žξ…ž(0)2ξ‚€2π‘Š(1)11(0)+π‘Š(1)20+𝛼(0)1𝑓+π‘Žξ…žξ…žξ…ž(0)2+𝛼2𝑓+π‘ξ…žξ…ž(0)π›Ύπ‘’βˆ’π‘–πœ”0𝜏01π‘Š(2)11ξ€·βˆ’πœ01ξ€Έ+𝛼2𝑓+π‘ξ…žξ…žξ…ž(0)2𝛾2π›Ύπ‘’βˆ’π‘–πœ”0𝜏01+12π‘ξ‚€π‘Š(2)20(0)+π›Ύπ‘Š(1)20+(0)π·π›Ύβˆ—ξ‚Έξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)2ξ‚€π‘Š(1)20ξ€·βˆ’πœ02ξ€Έξ‚π‘’π‘–πœ”0𝜏02+2π‘’βˆ’π‘–πœ”0𝜏02π‘Š(1)11ξ€·βˆ’πœ02ξ€Έ+𝛼2ξ€Έπ‘“βˆ’π‘ξ…žξ…žξ…ž(0)2π‘’βˆ’π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2ξ‚€2π›Ύπ‘Š(2)11(0)+π‘Š(2)20(0)𝛼𝛾+1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2𝛾21𝛾+2π‘‘ξ‚€π‘Š(2)20(0)+π›Ύπ‘Š(1)20𝑧(0)ξ‚Ήξ‚Ό2𝑧+h.o.t.(3.37) and we obtain 𝑔20=𝐷𝛼1𝑓+π‘Žξ…žξ…ž(𝛼0)+2𝑓+π‘ξ…žξ…ž(0)𝛾2ξ€»++π‘π›Ύπ·π›Ύβˆ—ξ‚ƒξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)π‘’βˆ’2π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾2ξ‚„,𝑔+𝑑𝛾11=𝐷𝛼1𝑓+π‘Žξ…žξ…žξ€·π›Ό(0)+2𝑓+π‘ξ…žξ…ž(0)π‘Žξ€»+π‘Ž+2𝑐Re{𝛾}π·π›Ύβˆ—π›Όξ€Ίξ€·2ξ€Έπ‘“βˆ’π‘ξ…žξ…žξ€·π›Ό(0)+1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾,𝑔𝛾+2𝑑Re{𝛾}02=𝐷𝛼1𝑓+π‘Žξ…žξ…žξ€·π›Ό(0)+2𝑓+π‘ξ…žξ…ž(0)𝛾2+2𝑐𝛾+π·π›Ύβˆ—ξ‚ƒξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)𝑒2π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾2+2𝑑𝛾,𝑔21=𝐷𝛼1𝑓+π‘Žξ…žξ…žξ‚€(0)2π‘Š(1)11(0)+π‘Š(1)20+𝛼(0)1𝑓+π‘Žξ…žξ…žξ…žξ€·π›Ό(0)+22𝑓+π‘ξ…žξ…žπ›Ύπ‘’βˆ’π‘–πœ”0𝜏01π‘Š(2)11ξ€·βˆ’πœ01ξ€Έ+𝛼2𝑓+π‘ξ…žξ…žξ…ž(0)𝛾2π›Ύπ‘’βˆ’π‘–πœ”0𝜏01ξ‚„ξ‚€π‘Š+𝑐(2)20(0)+π›Ύπ‘Š(1)20+(0)π·π›Ύβˆ—ξ‚ƒξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…žξ…žξ‚€π‘Š(0)(1)20ξ€·βˆ’πœ02ξ€Έξ‚π‘’π‘–πœ”0𝜏02+2π‘’βˆ’π‘–πœ”0𝜏02π‘Š(1)11ξ€·βˆ’πœ02ξ€Έ+𝛼2ξ€Έπ‘“βˆ’π‘ξ…žξ…žξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…žξ‚€(0)2π›Ύπ‘Š(2)11(0)+π‘Š(2)20(0)𝛼𝛾+1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾2ξ‚€π‘Šπ›Ύ+𝑑(2)20(0)+π›Ύπ‘Š(1)20.(0)(3.38) For unknown π‘Š(1)20(0),π‘Š(1)20ξ€·βˆ’πœ02ξ€Έ,π‘Š(1)11(0),π‘Š(2)11(0),π‘Š(2)11ξ€·βˆ’πœ01ξ€Έ,π‘Š(2)11ξ€·βˆ’πœ02ξ€Έ(3.39) in 𝑔21, we still need to compute them.

Form (3.8), (3.32), we have π‘Šξ…ž=⎧βŽͺ⎨βŽͺβŽ©ξ€½π΄π‘Šβˆ’2Reπ‘žβˆ—ξ€Ύ(0)π‘“π‘ž(πœƒ),βˆ’πœ01ξ€½β‰€πœƒ<0,π΄π‘Šβˆ’2Reπ‘žβˆ—ξ€Ύ(0)π‘“π‘ž(πœƒ)+𝑓,πœƒ=0defξ€·=π΄π‘Š+𝐻𝑧,ξ€Έ,𝑧,πœƒ(3.40) where𝐻𝑧,𝑧,πœƒ=𝐻20𝑧(πœƒ)22+𝐻11(πœƒ)𝑧𝑧+𝐻02(πœƒ)𝑧22+β‹―.(3.41) Comparing the coefficients, we obtainξ€·π΄βˆ’2π‘–πœ”0ξ€Έπ‘Š20=βˆ’π»20(πœƒ),(3.42)π΄π‘Š11(πœƒ)=βˆ’π»11(πœƒ),β‹―.(3.43) And we know that, for πœƒβˆˆ[βˆ’πœ01,0),𝐻𝑧,𝑧,πœƒ=βˆ’π‘žβˆ—(0)𝑓0π‘ž(πœƒ)βˆ’π‘žβˆ—(0)𝑓0ξ€·π‘ž(πœƒ)=βˆ’π‘”π‘§,π‘§ξ€Έπ‘ž(πœƒ)βˆ’π‘”ξ€·π‘§,π‘§ξ€Έπ‘ž(πœƒ).(3.44) Comparing the coefficients of (3.41) with (3.44) gives that𝐻20(πœƒ)=βˆ’π‘”20π‘ž(πœƒ)βˆ’π‘”02π»π‘ž(πœƒ),(3.45)11(πœƒ)=βˆ’π‘”11π‘ž(πœƒ)βˆ’π‘”11π‘ž(πœƒ).(3.46) From (3.42), (3.45), and the definition of 𝐴, we getΜ‡π‘Š20(πœƒ)=2π‘–πœ”0π‘Š20(πœƒ)+𝑔20π‘ž(πœƒ)+𝑔02π‘ž(πœƒ).(3.47) Noting that π‘ž(πœƒ)=π‘ž(0)π‘’π‘–πœ”0πœƒ, we haveπ‘Š20(πœƒ)=𝑖𝑔20πœ”0π‘ž(0)π‘’π‘–πœ”0πœƒ+𝑖𝑔023πœ”0π‘ž(0)π‘’βˆ’π‘–πœ”0πœƒ+𝐸1𝑒2π‘–πœ”0πœƒ,(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 haveΜ‡π‘Š11(πœƒ)=𝑔11π‘ž(πœƒ)+𝑔11π‘Šπ‘ž(πœƒ),(3.49)11(πœƒ)=βˆ’π‘–π‘”11πœ”0π‘ž(0)π‘’π‘–πœ”0πœƒ+𝑖𝑔11πœ”0π‘ž(0)π‘’βˆ’π‘–πœ”0πœƒ+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) thatξ€œ0βˆ’πœ01π‘‘πœ‚(πœƒ)π‘Š20(πœƒ)=2π‘–πœ”0π‘Š20(0)βˆ’π»20ξ€œ(0)(3.51)0βˆ’1π‘‘πœ‚(πœƒ)π‘Š11(πœƒ)=βˆ’π»11(0),(3.52) where πœ‚(πœƒ)=πœ‚(0,πœƒ).

From (3.42), we have𝐻20(0)=βˆ’π‘”20π‘ž(0)βˆ’π‘”02ξ€·π»π‘ž(0)+1,𝐻2𝑇,(3.53) where 𝐻1=𝛼1𝑓+π‘Žξ…žξ…ž(0)2+𝛼2𝑓+π‘ξ…žξ…ž(0)2𝛾2,𝐻2=𝛼2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(0)2π‘’βˆ’2π‘–πœ”0𝜏02+𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)2𝛾2.(3.54) From (3.43), we have𝐻11(0)=βˆ’π‘”11π‘ž(0)βˆ’π‘”11(0)ξ€·π‘ƒπ‘ž(0)+1,𝑃2𝑇,(3.55) where 𝑃1=𝛼1𝑓+π‘Žξ…žξ…žξ€·π›Ό(0)+2𝑓+π‘ξ…žξ…ž(0)𝛾𝑃𝛾,2=𝛼2ξ€Έπ‘“βˆ’π‘ξ…žξ…ž(𝛼0)+1ξ€Έπ‘“βˆ’π‘Žξ…žξ…ž(0)𝛾𝛾.(3.56) Noting thatξƒ©π‘–πœ”0ξ€œπΌβˆ’0βˆ’πœ01π‘’π‘–πœ”0πœƒξƒͺξƒ©π‘‘πœ‚(πœƒ)π‘ž(0)=0,βˆ’π‘–πœ”0ξ€œπΌβˆ’0βˆ’πœ01π‘’βˆ’π‘–πœ”0πœƒξƒͺπ‘‘πœ‚(πœƒ)π‘ž(0)=0(3.57) and substituting (3.48) and (3.53) into (3.51), we have2π‘–πœ”0ξ€œπΌβˆ’0βˆ’πœ01𝑒2π‘–πœ”0πœƒξƒͺπΈπ‘‘πœ‚(πœƒ)1=𝐻1,𝐻2𝑇.(3.58) That is,ξ‚€2π‘–πœ”0πΌβˆ’π΄1βˆ’π΅π‘’βˆ’2π‘–πœ”0𝜏02βˆ’πΆπ‘’βˆ’2π‘–πœ”0𝜏01𝐸1=𝐻1,𝐻2𝑇,(3.59) thenβŽ›βŽœβŽœβŽ2π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…žξ€·π›Ό(0)βˆ’2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01βˆ’ξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏022π‘–πœ”0βˆ’ξ€·π›Ό1ξ€Έπ‘“βˆ’π‘Žξ…žβŽžβŽŸβŽŸβŽ ξƒ©πΈ(0)1(1)𝐸1(2)ξƒͺ=𝐻1𝐻2ξƒͺ.(3.60) Hence, 𝐸1(1)=Ξ”11Ξ”1,𝐸1(2)=Ξ”12Ξ”1,(3.61) where Ξ”1βŽ›βŽœβŽœβŽ=det2π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…žξ€·π›Ό(0)βˆ’2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01βˆ’ξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏022π‘–πœ”0βˆ’ξ€·π›Ό1ξ€Έπ‘“βˆ’π‘Žξ…žβŽžβŽŸβŽŸβŽ ,Ξ”(0)11𝐻=det1βˆ’ξ€·π›Ό2𝑓+π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏01𝐻22π‘–πœ”0βˆ’ξ€·π›Ό1ξ€Έπ‘“βˆ’π‘Žξ…žξƒͺ,Ξ”(0)12=det2π‘–πœ”0βˆ’ξ€·π›Ό1𝑓+π‘Žξ…ž(0)𝐻1βˆ’ξ€·π›Ό2ξ€Έπ‘“βˆ’π‘ξ…ž(0)π‘’βˆ’π‘–πœ”0𝜏02𝐻2ξƒͺ.(3.62) Similarly, substituting (3.49) and (3.55) into (3.52), we haveξƒ©ξ€œ0βˆ’πœ01ξƒͺπΈπ‘‘πœ‚(πœƒ)2=𝑃1,𝑃2𝑇.(3.63) Then,𝐴1𝐸+𝐡+𝐢2=ξ€·βˆ’π‘ƒ1,βˆ’π‘ƒ2𝑇.(3.64) That is,𝛼1𝑓+π‘Žξ…žξ€·π›Ό(0)2𝑓+π‘ξ…žξ€·π›Ό(0)2ξ€Έπ‘“βˆ’π‘ξ…žξ€·π›Ό(0)1ξ€Έπ‘“βˆ’π‘Žξ…žπΈ(0)ξƒͺ2(1)𝐸2(2)ξƒͺ=ξƒ©βˆ’π‘ƒ1βˆ’π‘ƒ2ξƒͺ.(3.65) Hence, 𝐸2(1)=Ξ”21Ξ”2,𝐸2(2)=Ξ”22Ξ”2,(3.66) where Ξ”2𝛼=det1𝑓+π‘Žξ…žξ€·π›Ό(0)2𝑓+π‘ξ…žξ€·π›Ό(0)2ξ€Έπ‘“βˆ’π‘ξ…žξ€·π›Ό(0)1ξ€Έπ‘“βˆ’π‘Žξ…žξƒͺ,Ξ”(0)21=detβˆ’π‘ƒ1𝛼2𝑓+π‘ξ…ž(0)βˆ’π‘ƒ2𝛼1ξ€Έπ‘“βˆ’π‘Žξ…žξƒͺ,Ξ”(0)22𝛼=det1𝑓+π‘Žξ…ž(0)βˆ’π‘ƒ1𝛼2ξ€Έπ‘“βˆ’π‘ξ…ž(0)βˆ’π‘ƒ2ξƒͺ.(3.67) From (3.48), (3.50), we can calculate 𝑔21 and derive the following values: 𝑐1𝑖(0)=2πœ”0𝜏0𝑔20𝑔11||π‘”βˆ’211||2βˆ’||𝑔02||23ξƒͺ+𝑔212,πœ‡2𝑐=βˆ’Re1ξ€Ύ(0)ξ€½πœ†Reξ…žξ€·πœ0,𝛽2𝑐=2Re1ξ€Έ,𝑇(0)2𝑐=βˆ’Im1ξ€Ύ(0)+πœ‡2ξ€½πœ†Imξ…žξ€·πœ0ξ€Έξ€Ύπœ”0𝜏0.(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 πœ‡2>0(πœ‡2<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=βˆ’0.03,𝛼2=βˆ’0.1,π‘Ž=1.5,𝑏=2,𝑐=0,𝑑=βˆ’4,and𝑓(π‘₯)=tanh(π‘₯). Then, the system (1.2) becomesΜ‡π‘₯1ξ€·π‘₯(𝑑)=1.47tanh1ξ€Έξ€·π‘₯(𝑑)+1.9tanh2ξ€·π‘‘βˆ’πœ1,ξ€Έξ€ΈΜ‡π‘₯2(ξ€·π‘₯𝑑)=βˆ’2.1tanh1ξ€·π‘‘βˆ’πœ2ξ€·π‘₯ξ€Έξ€Έβˆ’1.53tanh2(𝑑)βˆ’4π‘₯1(𝑑)π‘₯2(𝑑).(4.1) By some complicated computation by means of Matlab 7.0, we get πœ”0β‰ˆ1.3211,𝜏0β‰ˆ0.01,πœ†ξ…ž(𝜏0)β‰ˆβ€‰0.0140βˆ’1.4926𝑖. We can easily obtain 𝑔20=βˆ’0.2501+2.3128𝑖,𝑔11=1.2377+0.3484𝑖,𝑔02=0.4533βˆ’0.5693𝑖,𝑔21=βˆ’2.3022+4.3015𝑖. Thus, we can calculate the following values:  𝑐1(0)=βˆ’1.0617βˆ’1.7138𝑖,πœ‡2=75.8357,𝛽2=βˆ’2.1234,𝑇2=86.9780. We obtain that the conditions indicated in Theorem 2.6 are satisfied. Furthermore, it follows that πœ‡2>0 and 𝛽2<0. Thus, the equilibrium πΈβˆ—(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 πΈβˆ—(0,0) loses its stability and a Hopf bifurcation occurs, that is, a family of periodic solutions bifurcations from the equilibrium πΈβˆ—(0,0). Since πœ‡2>0 and 𝛽2<0, the direction of the Hopf bifurcation is 𝜏>𝜏0 and these bifurcating periodic solutions from πΈβˆ—(0,0) at 𝜏0 are stable, which are depicted in Figures 4, 5, and 6.

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 πΈβˆ—(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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Copyright Β© 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.


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