Stability and Hopf bifurcation analysis on a ring of four neurons with delays

Applied Mathematics and Computation 213 (2009) 587–599 Contents lists available at ScienceDirect Applied Mathematics and Computation journal homepag...

Download PDF

546KB Sizes 1 Downloads 90 Views

Report

PDF Reader
Full Text

Applied Mathematics and Computation 213 (2009) 587–599

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

Stability and Hopf bifurcation analysis on a ring of four neurons with delays q Haijun Hu, Lihong Huang * College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China

a r t i c l e

i n f o

Keywords: Ring network Linear stability Hopf bifurcation Characteristic equation

a b s t r a c t We consider a four-neuron ring with self-feedback and delays. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. Numerical simulations are presented to illustrate the results. Ó 2009 Elsevier Inc. All rights reserved.

1. Introduction The investigation of neural networks has been the subject of some recent work (see, for example [1,2]) since Hopﬁeld [3] constructed a simpliﬁed neural network model. Based on the Hopﬁeld neural network model, Marcus and Westervelt [4] argued that time delays always occur in the signal transmission and proposed a neural network model with delay. Afterward, many researchers have been studying on a variety of neural networks with delays (see, for example [5–10]). However, most work focused on researching the stability of neural network. It’s well known that the dynamic behaviors such as periodic phenomenon, bifurcation and chaos are also of great interest. In delay differential equations(DDEs), periodic solutions can arise through the Hopf bifurcation. Thus, there is an extensive literature on bifurcation analysis of some special neural networks (e.g. [11–14,23,24,26,27]). But, most of them deal with only the two-equation models or the models with a single delay. To the best of our knowledge, there are few papers on the bifurcations of the high-dimensional models with multiple delays (see [16,25,26]). In this paper, we study the stability and bifurcation of a ring network with two delays described by the following system of DDEs:

8 x_ 1 ðtÞ ¼ r1 x1 ðtÞ þ g 1 ðx1 ðtÞÞ þ f1 ðx4 ðt s2 ÞÞ þ f1 ðx2 ðt s2 ÞÞ; > > > < x_ 2 ðtÞ ¼ r2 x2 ðtÞ þ g 2 ðx2 ðtÞÞ þ f2 ðx1 ðt s1 ÞÞ þ f2 ðx3 ðt s1 ÞÞ; > > x_ 3 ðtÞ ¼ r3 x3 ðtÞ þ g 3 ðx3 ðtÞÞ þ f3 ðx2 ðt s2 ÞÞ þ f3 ðx4 ðt s2 ÞÞ; > : x_ 4 ðtÞ ¼ r4 x4 ðtÞ þ g 4 ðx4 ðtÞÞ þ f4 ðx3 ðt s1 ÞÞ þ f4 ðx1 ðt s1 ÞÞ;

ð1:1Þ

where x_ ¼ dx=dt, xi ðtÞ represents the state of the ith neuron at time t, r i P 0 is the internal decay rate, fi is the connection function between of neurons, g i represents the nonlinear feedback function, si P 0 is the connection time delay, i ¼ 1; 2; 3; 4. The main difference from the model considered in [16] is that system (1.1) is bidirectional with two loops. This system can model the evolution of a Hopﬁeld–Cohen–Grossberg network consisting of four elements with time delayed nearest-neighbour coupling, as illustrated schematically in Fig. 1.

q Research supported by National Natural Science Foundation of China (10771055, 60835004) and Key Project of Hunan Province Programs for Applied Fundamental Research (2008FJ2008). * Corresponding author. Tel./fax: +86 731 8823056. E-mail address: [email protected] (L. Huang).

0096-3003/$ - see front matter Ó 2009 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2009.03.052

588

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

Fig. 1. A four-neuron ring with delays and self-feedback.

Ring networks have been found in a variety of neural structures such as cerebellum [19], and even in chemistry and electrical engineering. In the ﬁeld of neural networks, rings are studied to gain insight into the mechanisms underlying the behavior of recurrent networks [4,20]. It is well known that neural networks are complex and large-scale nonlinear systems. However, for simplicity, many researchers have directed their attention to the study of simple systems and it is useful since the complexity found may be carried over to large networks in some way. Therefore, we think the discussion on system (1.1) should be helpful in researching large ring networks. We will discuss the linear stability and Hopf bifurcation of (1.1), from which we can ﬁnd that delays do have a great effect on the dynamics of (1.1). Letting s ¼ s1 þ s2 , we shall show that when the delay s passes through a critical value, the zero solution loses its stability and periodic solutions via Hopf bifurcation appears. In the sequel, we’ll determine the Hopf bifurcation direction, that is, make clear whether the bifurcating branch of periodic solution is supercritical or subcritical, and determine the properties of the bifurcating periodic solution, for example, stability on the center manifold and period. The main methods applied to get our results are the normal form method and the center manifold theory introduced in [15]. The rest of the paper is organized as follows. In the next section, we discuss the associated characteristic equation, the linear stability and Hopf bifurcations of (1.1). The direction of the bifurcation, the estimation formula of period and stability of Hopf bifurcating periodic solutions are given in Section 3. Numerical simulations are presented to illustrate the results in Section 4. Finally, conclusions are drawn in Section 5.

2. Stability analysis and Hopf bifurcation To establish the main results for system (1.1), it is necessary to make the following assumption ðH1 Þ f i ; g i 2 C 1 ; f i ð0Þ ¼ g i ð0Þ ¼ 0; for i ¼ 1; 2; 3; 4: It is easily seen that the origin ð0; 0; 0; 0Þ is an equilibrium of (1.1). Linearizing (1.1) about it produces

8 x_ 1 ðtÞ ¼ j1 x1 ðtÞ þ f10 ð0Þx4 ðt s2 Þ þ f10 ð0Þx2 ðt s2 Þ; > > > < x_ ðtÞ ¼ j x ðtÞ þ f 0 ð0Þx ðt s Þ þ f 0 ð0Þx ðt s Þ; 2 2 2 1 1 3 1 2 2 > x_ 3 ðtÞ ¼ j3 x3 ðtÞ þ f30 ð0Þx2 ðt s2 Þ þ f30 ð0Þx4 ðt s2 Þ; > > : x_ 4 ðtÞ ¼ j4 x4 ðtÞ þ f40 ð0Þx3 ðt s1 Þ þ f40 ð0Þx1 ðt s1 Þ; where

ð2:1Þ

ji ¼ ri g 0i ð0Þ; i ¼ 1; 2; 3; 4: The characteristic equation of (2.1) is k4 þ ak3 þ bk2 þ ck þ d þ ðmk2 þ nk þ rÞeks ¼ 0;

ð2:2Þ

where

a ¼ j1 þ j2 þ j3 þ j4 ; b ¼ j1 j2 þ j1 j3 þ j1 j4 þ j2 j3 þ j3 j4 ; c ¼ j1 j2 j3 þ j1 j2 j4 þ j1 j3 j4 þ j2 j3 j4 ; d ¼ j1 j2 j3 j4 ; m ¼ f10 ð0Þ þ f30 ð0Þ f20 ð0Þ þ f40 ð0Þ ; n ¼ j2 f40 ð0Þ þ j4 f20 ð0Þ f10 ð0Þ þ f30 ð0Þ j1 f30 ð0Þ þ j3 f10 ð0Þ f20 ð0Þ þ f40 ð0Þ ; r ¼ j1 f30 ð0Þ þ j3 f10 ð0Þ j2 f40 ð0Þ þ j4 f20 ð0Þ ; s ¼ s1 þ s2 : In this section, we ﬁrst study the distribution of the roots of Eq. (2.2). Obviously, ixðx > 0Þ is a root of Eq. (2.2) if and only if x satisﬁes

x4 iax3 bx2 þ icx þ d þ ðmx2 þ inx þ rÞðcos xs i sin xsÞ ¼ 0:

ð2:3Þ

Separating the real and imaginary parts, we have

x4 bx2 þ d ¼ ðmx2 rÞ cos xs nx sin xs; ax3 cx ¼ ðmx2 rÞ sin xs þ nx cos xs:

ð2:4Þ

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

589

Taking square on the both sides of the equations of (2.4) and summing them up, we obtain

ðx4 bx2 þ dÞ2 þ ðax3 cxÞ2 ¼ ðmx2 rÞ2 þ ðnxÞ2 ;

ð2:5Þ

x8 þ ða2 2bÞx6 þ ðb2 þ 2d 2ac m2 Þx4 þ ðc2 2bd þ 2mr n2 Þx2 þ d2 r2 ¼ 0:

ð2:6Þ

i.e.

2

Let p ¼ a2 2b; q ¼ b þ 2d 2ac m2 ; u ¼ c2 2bd þ 2mr n2 ;

v ¼ d2 r 2 ;

z4 þ pz3 þ qz2 þ uz þ v ¼ 0:

z ¼ x2 . Then Eq. (2.6) becomes

ð2:7Þ

Thus, the fact that Eq. (2.7) has positive roots is a necessary condition for the existence of the pure imaginary roots of Eq. (2.2). If ri ; fi ; g i ði ¼ 1; 2; 3; 4Þ of the system (1.1) are given, it is easy to solve Eq. (2.7) with the aid of a computer. Moreover, since the form of Eq. (2.7) is identical to that of Eq. (2.4) in [16], we can get Lemmas 2.1 and 2.2 analogously. The proofs are omitted. Lemma 2.1 [16]. If

v < 0, then Eq. (2.7) has at least one positive root.

Denote

hðzÞ ¼ z4 þ pz3 þ qz2 þ uz þ v :

ð2:8Þ

0

Then we have h ðzÞ ¼ 4z3 þ 3pz2 þ 2qz þ u: Set

4z3 þ 3pz2 þ 2qz þ u ¼ 0: Let y ¼ z þ

p . 4

ð2:9Þ

Then Eq. (2.9) becomes

3

y þ p1 y þ q1 ¼ 0; q 2

where p1 ¼

3 2 p ; 16

ð2:10Þ

q1 ¼

p3 32

pq 8

þ

u : 4

Deﬁne

pﬃﬃﬃ q 2 p 3 1 þ i 3 1 þ 1 ; r¼ ; 2ﬃ 3 ﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ r2ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ 3 q pﬃﬃﬃﬃ q 3 y1 ¼ 1 þ D þ 1 D; 2 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ q q 3 3 y 2 ¼ 1 þ D r þ 1 D r2 ; 2 2 rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ pﬃﬃﬃﬃ pﬃﬃﬃﬃ q q 3 3 y3 ¼ 1 þ Dr2 þ 1 Dr; 2 2 p zi ¼ yi ; i ¼ 1; 2; 3: 4 D¼

Lemma 2.2 [16]. Suppose that

v P 0, then we have the following:

ðiÞ If D P 0, then Eq. (2.9) has positive roots if and only if z1 > 0 and hðz1 Þ < 0; ðiiÞ If D < 0, then Eq. (2.9) has positive roots if and only if there exists at least one z 2 fz1 ; z2 ; z3 g such that z > 0 and hðz Þ 6 0. Suppose that Eq. (2.7) has positive roots. Without loss of generality, we assume that it has four positive roots, denoted by zk ; k ¼ 1; 2; 3; 4. Then Eq. (2.6) has four positive roots,

x1 ¼

pﬃﬃﬃﬃﬃ z1 ;

x2 ¼

pﬃﬃﬃﬃﬃ z2 ;

x3 ¼

pﬃﬃﬃﬃﬃ pﬃﬃﬃﬃﬃ z3 and x4 ¼ z4 :

By (2.4), we have

sin xk s ¼ cos xk s ¼

ðam nÞx5k þ ðnb cm arÞx3k þ ðcr ndÞxk ðmx2k rÞ2 þ ðnxk Þ2

;

mx6k þ ðan bm rÞx4k þ ðbr þ dm cnÞx2k dr ðmx2k rÞ2 þ ðnxk Þ2

:

590

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

Thus, denoting

a ¼

b ¼

ðam nÞx5k þ ðnb cm arÞx3k þ ðcr ndÞxk ðmx2k rÞ2 þ ðnxk Þ2

;

mx6k þ ðan bm rÞx4k þ ðbr þ dm cnÞx2k dr ðmx2k rÞ2 þ ðnxk Þ2

(

1

xk ðarccos b

sðjÞ k ¼

þ 2jpÞ;

xk ð2p arccos b 1

;

a P 0; ð2:11Þ

þ 2jpÞ; a < 0;

where k ¼ 1; 2; 3; 4 and j ¼ 0; 1; . . . ; then ixk is a pair of purely imaginary roots of Eq. (2.2) with ðjÞ quence fsk gþ1 j¼0 is increasing, and

lim

j!þ1

s ¼ sðjÞ k : Clearly, the se-

sðjÞ k ¼ 1; 2; 3; 4: k ¼ þ1;

Thus we can deﬁne

n

o

ð0Þ ; x0 ¼ xk0 ; z0 ¼ zk0 : s0 ¼ sð0Þ k0 ¼ min sk

ð2:12Þ

16k64

To discuss the distribution of the roots of the exponential polynomial Eq. (2.2), we need the following result from Ruan and Wei [17]. Lemma 2.3 [17]. Consider the exponential polynomial

h i ð0Þ ð0Þ ð1Þ n1 ð1Þ eks1 þ Pðk; eks1 ; . . . ; eksm Þ ¼ kn þ p1 kn1 þ þ pn1 k þ pð0Þ þ þ pn1 k þ pð1Þ n þ p1 k n h i ðmÞ ðmÞ eksm ; þ p1 kn1 þ þ pn1 k þ pðmÞ n ðiÞ

where si P 0 ði ¼ 1; 2; . . . ; mÞ and pj ði ¼ 0; 1; 2; . . . ; m; j ¼ 1; 2; . . . ; nÞ are constants. As ðs1 ; s2 ; . . . ; sm Þ vary, the sum of the orders of Pðk; eks1 ; . . . ; eksm Þ on the open right half plane can change only if a zero appears on or crosses the imaginary axis. Using Lemmas 2.1–2.3, we can easily obtain the following results on the distribution of roots of the Eq. (2.2). Lemma 2.4. Assume that ðH2 Þ a > 0; aðb þ mÞ > c þ n; d þ r > 0; aðb þ mÞðc þ nÞ > ðc þ nÞ2 þ a2 ðd þ rÞ: ðiÞ If one of the following holds: ðaÞ v < 0; ðbÞ v P 0; D P 0; z1 > 0 and hðz1 Þ 6 0; ðcÞ v P 0; D < 0, and there exists a z 2 fz1 ; z2 ; z3 g such that z > 0 and hðz Þ 6 0, then all roots of Eq. (2.2) have negative real parts when s 2 ½0; s0 Þ; ðiiÞ If the conditions ðaÞ–ðcÞ of ðiÞ are not satisﬁed, then all roots of Eq. (2.2) have negative real parts for all s P 0. Proof. When 4

s ¼ 0, Eq. (2.2) becomes 3

k þ ak þ ðb þ mÞk2 þ ðc þ nÞk þ d þ r ¼ 0:

ð2:13Þ

By the Routh–Hurwitz criterion, all roots of Eq. (2.13) have negative real parts if and only if

a > 0; aðb þ mÞ > c þ n; d þ r > 0; aðb þ mÞðc þ nÞ > ðc þ nÞ2 þ a2 ðd þ rÞ: From Lemmas 2.1 and 2.2, we know that if (a)–(c) are not satisﬁed, then Eq. (2.2) has no roots with zero real part for all s P 0; ðjÞ if one of ðaÞ; ðbÞ and ðcÞ holds, when s – sk ; k ¼ 1; 2; 3; 4; j ¼ 0; 1; . . ., Eq. (2.2) has no roots with zero real part and s0 is the minimum value of s so that Eq. (2.2) has purely imaginary roots. Applying Lemma 2.3, we obtain the conclusion of the lemma. Let

kðsÞ ¼ aðsÞ þ ixðsÞ

ð2:14Þ

be the root of Eq. (2.2) near

s ¼ sðjÞ k satisfying

¼ xk : aðsðjÞ x sðjÞ k Þ ¼ 0; k Then, the following lemma holds. 0

ðjÞ

Lemma 2.5. Suppose h ðzk Þ – 0, where hðzÞ is deﬁned by (2.8). If s ¼ sk , then ixk is a pair of simple purely imaginary roots of Eq. (2.2). Moreover,

dðRekðsÞÞ ðjÞ – 0 ds s¼s k

and the sign of

dðRekðsÞÞ js¼sðjÞ ds k

0

is consistent with that of h ðzk Þ.

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

591

Proof. Denote

FðkÞ ¼ k4 þ ak3 þ bk2 þ ck þ d;

ð2:15Þ

GðkÞ ¼ mk2 þ nk þ r: Then Eq. (2.2) can be written as

FðkÞ þ GðkÞeks ¼ 0;

ð2:16Þ

and (2.6) can be transformed into the following form

FðixÞFðixÞ GðixÞGðixÞ ¼ 0:

ð2:17Þ

Thus, together with (2.7) and (2.8), we have

hðx2 Þ ¼ FðixÞFðixÞ GðixÞGðixÞ:

ð2:18Þ

Differentiating both sides of Eq. (2.18) with respect to x, we obtain

h i 0 2xh ðx2 Þ ¼ i F 0 ðixÞFðixÞ F 0 ðixÞFðixÞ G0 ðixÞGðixÞ þ G0 ðixÞGðixÞ :

ð2:19Þ

If ixk is not simple, then xk must satisfy

i ðjÞ d h ¼ 0; FðkÞ þ GðkÞeksk dk k¼ixk

that is xk must satisfy ðjÞ

F 0 ðixk Þ þ G0 ðixk Þeisk

xk

ðjÞ

sk Gðixk Þeisk ðjÞ

xk

¼ 0:

With Eq. (2.16), we have

sðjÞ k ¼

G0 ðixk Þ F 0 ðixk Þ : Gðixk Þ Fðixk Þ

Thus, by (2.17) and (2.19) we obtain

Ims

Since

ðjÞ k

( ) ( ) 0 G ðixk Þ F 0 ðixk Þ G0 ðixk ÞGðixk Þ F 0 ðixk ÞFðixk Þ G0 ðixk ÞGðixk Þ F 0 ðixk ÞFðixk Þ ¼ Im ¼ Im ¼ Im Gðixk Þ Fðixk Þ Gðixk ÞGðixk Þ Fðixk ÞFðixk Þ Fðixk ÞFðixk Þ h i i G0 ðixk ÞGðixk Þ F 0 ðixk ÞFðixk Þ G0 ðixk ÞGðixk Þ þ F 0 ðixk ÞFðixk Þ xk h0 ðx2k Þ ¼ ¼ : 2Fðixk ÞFðixk Þ jFðixk Þj2

ðjÞ sðjÞ k is real, i.e. Imsk ¼ 0, we have

0

0

h ðzk Þ ¼ h ðx2k Þ ¼ 0: 0

We get a contradiction to the condition h ðzk Þ – 0: This proves the ﬁrst conclusion. Differentiating both sides of Eq. (2.16) with respect to s, we obtain

0 dk F ðkÞ þ G0 ðkÞeks sGðkÞeks kGðkÞeks ¼ 0; ds which implies

h i h i kGðkÞ F 0 ðkÞeks þ G0 ðkÞ sGðkÞ k F 0 ðkÞFðkÞ þ G0 ðkÞGðkÞ sjGðkÞj2 dkðsÞ kGðkÞ ¼ 0 ¼ : ¼ 0 0 ds F ðkÞeks þ G0 ðkÞ sGðkÞ2 F ðkÞeks þ G0 ðkÞ sGðkÞ2 F ðkÞeks þ G0 ðkÞ sGðkÞ

It follows together with (2.19) that

n h io Re k F 0 ðkÞFðkÞ þ G0 ðkÞGðkÞ sjGðkÞj2 s¼sðjÞ dðRekðsÞÞ k ¼ 0 2 0 ds F ðkÞeks þ G ðkÞ sGðkÞ s¼sðjÞ k h i ixk F 0 ðixk ÞFðixk Þ þ G0 ðixk ÞGðixk Þ þ F 0 ðixk ÞFðixk Þ G0 ðixk ÞGðixk Þ ¼ 2 ðjÞ ðjÞ 2F 0 ðixk Þeisk xk þ G0 ðixk Þ sk Gðixk Þ 0

0

x2k h ðx2k Þ x2k h ðzk Þ ¼ 2 ¼ 2 – 0: ðjÞ ðjÞ 0 0 ðjÞ ðjÞ 0 0 F ðixk Þeisk xk þ G ðixk Þ sk Gðixk Þ F ðixk Þeisk xk þ G ðixk Þ sk Gðixk Þ

592

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

Clearly, the sign of

dðRekðsÞÞ js¼sðjÞ ds k

0

is determined by that of h ðzk Þ. This completes the proof.

Remark 2.1. In the proof of Lemma 5, we not only obtain that the transversality condition is satisﬁed, but also get the conclusion that ixk is a pair of simple purely imaginary roots of the associated characteristic equation under the same condition as that in [21,22]. Moreover, our proof method avoids prolix calculations and can be applied to a general characteristic equation FðkÞ þ GðkÞeks ¼ 0 where FðkÞ and GðkÞ are polynomials of arbitrary order. Applying Lemmas 2.4 and 2.5, we obtain the following theorem.

sðjÞ and s0 be deﬁned by (2.11) and (2.12), respectively. Suppose that ðH1 Þ and ðH2 Þ hold. k

Theorem 2.1. Let

ðiÞ If the conditions ðaÞ v < 0; ðbÞ v P 0; D P 0; z1 > 0 and hðz1 Þ 6 0; ðcÞ v P 0; D < 0, and there exists a z 2 fz1 ; z2 ; z3 g such that z > 0 and hðz Þ 6 0 are not satisﬁed, then the zero solution of system (1.1) is asymptotically stable for all s P 0; ðiiÞ If one of the conditions ðaÞ, ðbÞ and ðcÞ of ðiÞ is satisﬁed, then the zero solution of system (1.1) is asymptotically stable when s 2 ½0; s0 Þ. 0 ðiiiÞ If the condition of ðiiÞ is satisﬁed, and h ðzk Þ–0, then system (1.1) undergoes a Hopf bifurcation at ð0; 0; 0; 0Þ when ðjÞ s ¼ sk ðk ¼ 1; 2; 3; 4; j ¼ 0; 1; 2; . . .Þ.

3. Direction and stability of bifurcation In the previous section, we have shown that the Hopf bifurcation occurs at some value s0 ¼ s1 þ s2 for system (1.1). In this section, by using the normal form theory and center manifold reduction due to Hassard et al. [15], we will work out an algorithm for determining the direction, stability and the period of the bifurcating periodic solutions. In what follows, it’s necessary to assume that fi ; g i 2 C 3 ; i ¼ 1; 2; 3; 4. Without loss generality, we assume that s0 ¼ s1 þ s2 with s2 > s1 and s ¼ s0 þ l ¼ ðs1 þ lÞ þ s2 , where s0 is deﬁned by for system (1.1). We choose the phase space as (2.12) and jl j 64 s2 s1 . Then l ¼ 0 is the Hopf bifurcation value C ¼ C ½s2 ; 0 ; C Þ, where for convenience in computation we use C4 instead of R4 . Letting XðtÞ ¼ ðx1 ðtÞ; x2 ðtÞ; x3 ðtÞ; x4 ðtÞÞT and X t ðhÞ ¼ Xðt þ hÞ 2 C, we can transform system (1.1) into an operator equation of the form:

_ XðtÞ ¼ Ll ðX t Þ þ Gðl; X t Þ

ð3:1Þ

Ll ð/Þ ¼ B/ð0Þ þ B1 /ðs1 lÞ þ B2 /ðs2 Þ

ð3:2Þ

with

and

0 Gðl; /Þ ¼

g 001 ð0Þ 2 f 00 ð0Þ /1 ð0Þ þ 1 2 /24 ð 2 B g002 ð0Þ 2 f200 ð0Þ 2 B 1 ð 1 B 2 /002 ð0Þ þ 2 / g 3 ð0Þ 2 f300 ð0Þ 2 B /3 ð0Þ þ 2 /2 ð @ 2 g 004 ð0Þ 2 f 00 ð0Þ / ð0Þ þ 4 2 /23 ð 1 4 2

s s

0

þ

1

s2 Þ þ /22 ðs2 Þ C lÞ þ /23 ðs1 lÞ C C C s2 Þ þ /24 ðs2 Þ A lÞ þ /21 ðs1 lÞ

g 000 ð0Þ 3 f 000 ð0Þ 1 /1 ð0Þ þ 1 3! /34 ð 3! 000 000 B g2 ð0Þ 3 f2 ð0Þ 3 B 1 B 3! /0002 ð0Þ þ 3! 000/1 ð g 3 ð0Þ 3 f3 ð0Þ 3 B /3 ð0Þ þ 3! /2 ð @ 3! g 000 ð0Þ 3 f 000 ð0Þ 4 / ð0Þ þ 4 3! /33 ð 1 4 3!

1

s2 Þ þ /32 ðs2 Þ C lÞ þ /33 ðs1 lÞ C C þ Oðk/k4 Þ; C s2 Þ þ /34 ðs2 Þ A lÞ þ /31 ðs1 lÞ

s s

ð3:3Þ

where /ðhÞ ¼ ð/1 ðhÞ; /2 ðhÞ; /3 ðhÞ; /4 ðhÞÞT 2 C, and

0

j1

0

B 0 B B¼B @ 0 0

0

1

0

0

j2

0

0

j3

0 C C C; 0 A

0

0

j4

0

0

0

0

1

B f 0 ð0Þ 0 f 0 ð0Þ 0 C B C 2 B1 ¼ B 2 C; @ 0 0 0 0A f40 ð0Þ 0 f40 ð0Þ 0

0

0 f10 ð0Þ 0 f10 ð0Þ

1

B0 0 0 0 C B C B2 ¼ B C: @ 0 f30 ð0Þ 0 f30 ð0Þ A 0

0

0

0

By the Riesz representation theorem, there exists a function gðh; lÞ of bounded variation for h 2 ½s2 ; 0, such that

Ll / ¼

Z

0

s2

dgðh; lÞ/ðhÞ;

/ 2 C:

ð3:4Þ

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

593

In fact, we can choose

8 h ¼ 0; > < B; gðh; lÞ ¼ B1 dðh þ s1 þ lÞ; h 2 ½s l; 0Þ; > : h 2 ½s2 ; s1 lÞ; B2 dðh þ s2 Þ;

ð3:5Þ

where dðhÞ is the Dirac function. For / 2 C, deﬁne

8 < d/ðhÞ ; h 2 ½s2 ; 0Þ; dh Al /ðhÞ ¼ R 0 : s dgðs; lÞ/ðsÞ; h ¼ 0;

ð3:6Þ

2

and

Rl / ¼

h 2 ½s2 ; 0Þ;

0;

ð3:7Þ

Gðl; /Þ; h ¼ 0:

Then, since dX t =dt ¼ dX t =dh, system (3.1) can be further rewritten as

X_ t ¼ Al X t þ Rl X t :

ð3:8Þ

Recall that when l ¼ 0, the associated characteristic equation of system (3.1) has a pair of simple imaginary roots ix0 . Then there is a neighborhood of l ¼ 0 such that for any l in it there exists a two-dimensional local center manifold C 0 of (3.1) in C, which contains the zero element of C and the orbits of Hopf periodic solutions are also located in C 0 . In order to construct the coordinates to describe the center manifold near l ¼ 0, we need an inner product and the adjoint operator A0 of A0 . Letting 4 C ¼ C ½0; s2 ; C , for w 2 C , A0 is deﬁned by

8 < dwðsÞ ; s 2 ð0; s2 ; ds A0 wðsÞ ¼ R 0 : s dgT ðn; 0ÞwðnÞ; s ¼ 0;

ð3:9Þ

2

where gT is the transpose of g. For / 2 C and w 2 C , we deﬁne the following bilinear form as an inner product

hwðsÞ; /ðhÞi ¼ wð0Þ /ð0Þ

Z

Z

0

h¼s2

h

T ðn hÞdgðh; 0Þ/ðnÞdn; w

ð3:10Þ

n¼0

P where a b ¼ 4i¼1 ai bi for a ¼ ða1 ; a2 ; a3 ; a4 ÞT and b ¼ ðb1 ; b2 ; b3 ; b4 ÞT . Then as usual, hw; A0 /i ¼ hA0 w; /i. It’s easy to see that ix0 is the eigenvalue of A0 , then ix0 is that of A0 . Suppose that qðhÞ is the eigenvector of A0 corresponding to ix0 , namely,

A0 qðhÞ ¼ ix0 qðhÞ: We compute here with qðhÞ ¼ ð1; a; b; cÞT eix0 h . It follows from the deﬁnition of A0 and (3.2) that

0

ix0 þ j1 B f 0 ð0Þeix0 s1 B 2 B @ 0 f40 ð0Þe

ix0 1

s

0

ix0 þ j2

f20 ð0Þeix0 s1

10 1 0 1 1 0 CB a C B 0 C 0 CB C B C C ¼ B C: CB f30 ð0Þeix0 s2 A@ b A @ 0 A c 0 ix0 þ j4

f10 ð0Þeix0 s2

f10 ð0Þeix0 s2

f30 ð0Þeix0 s2

ix0 þ j3

0

f40 ð0Þeix0 s1

By direct computation, we obtain

f20 ð0Þeix0 s1 f10 ð0Þðix0 þ j3 Þ þ f30 ð0Þðix0 þ j1 Þ a¼ ; f10 ð0Þðix0 þ j2 Þðix0 þ j3 Þ

i x s f 0 ð0Þe 0 1 f10 ð0Þðix0 þ j3 Þ þ f30 ð0Þðix0 þ j1 Þ c¼ 4 : f10 ð0Þðix0 þ j3 Þðix0 þ j4 Þ

b¼

f30 ð0Þðix0 þ j1 Þ ; f10 ð0Þðix0 þ j3 Þ

Similarly, it can be veriﬁed that q ðsÞ ¼ Mð1; a ; b ; c ÞT eix0 s is the eigenvector of A0 corresponding to ix0 where

eix0 s2 f10 ð0Þðix0 þ j3 Þ þ f30 ð0Þðix0 þ j1 Þ ; ðix0 þ j2 Þðix0 þ j3 Þ

eix0 s2 f10 ð0Þðix0 þ j3 Þ þ f30 ð0Þðix0 þ j1 Þ c ¼ : ðix0 þ j3 Þðix0 þ j4 Þ

a ¼

b ¼

ix0 þ j1 ; ix0 þ j3

594

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

By (3.10), we get

hq ðsÞ; qðhÞi ¼ q ð0Þ qð0Þ

Z

Z

0 s

2

h

q T ðn hÞdgðh; 0ÞqðnÞdn

n¼0

¼ Mð1; a ; b ; c Þð1; a; b; cÞT

Z

s

" ¼ M 1 þ aa þ bb þ cc h

Z

0

Z

0 2

h

Mð1; a ; b ; c Þeix0 ðnhÞ dgðh; 0Þð1; a; b; cÞT eix0 n dn

n¼0

#

ð1; a ; b ; c Þheix0 h dgðh; 0Þð1; a; b; cÞT

s2

i ¼ M 1 þ aa þ bb þ cc þ a f20 ð0Þ þ c f40 ð0Þ ð1 þ bÞs1 eix0 s1 þ f10 ð0Þ þ b f30 ð0Þ ða þ cÞs2 eix0 s2 :

Then, we can choose

h i 1 M ¼ 1 þ aa þ bb þ cc þ a f20 ð0Þ þ c f40 ð0Þ ð1 þ bÞs1 eix0 s1 þ ðf10 ð0Þ þ b f30 ð0ÞÞða þ cÞs2 eix0 s2 such that hq ðsÞ; qðhÞi ¼ 1. Using the same method it is easy to proof hq ðsÞ; qðhÞi ¼ 0, we omit it. Let

zðtÞ ¼ hq ; X t i

ð3:11Þ

Wðt; hÞ ¼ X t ðhÞ 2RefzðtÞqðhÞg:

ð3:12Þ

and

On the center manifold C 0 , Wðt; hÞ ¼ WðzðtÞ; zðtÞ; hÞ, where

Wðz; z; hÞ ¼ W 20 ðhÞ

z2 z2 z3 þ W 11 ðhÞzz þ W 02 ðhÞ þ W 30 ðhÞ þ ; 2 2 6

ð3:13Þ

z and z are local coordinates for center manifold C 0 in the direction of q and q . Note that W is real if X t is real. We shall deal with real solutions only. It’s easy to see that hq ; Wi ¼ 0. For the solution X t 2 C 0 of (3.8), since l ¼ 0, we have

z_ ðtÞ ¼ hq ; X_ t i ¼ hq ; A0 X t i þ hq ; R0 X t i ¼ hA0 q ; X t i þ q ð0ÞGð0; X t Þ ¼ ix0 zðtÞ þ q ð0ÞGð0; Wðz; z; hÞ þ 2RefzqðhÞgÞ ¼ ix0 zðtÞ þ q ð0Þf0 ðz; zÞ;

ð3:14Þ

which can be written in abbreviated form as

z_ ðtÞ ¼ ix0 zðtÞ þ gðz; zÞ;

ð3:15Þ

where

gðz; zÞ ¼ q ð0Þf0 ðz; zÞ ¼ g 20

z2 z2 z2 z þ : þ g 11 zz þ g 02 þ g 21 2 2 2

By (3.11) and (3.12) we have X t ðhÞ ¼ ðx1t ðhÞ; x2t ðhÞ; x3t ðhÞ; x4t ðhÞÞT ¼ Wðt; hÞ þ zqðhÞ þ zqðhÞ, and then

z2 z2 ð1Þ ð1Þ þ W 11 ð0Þzz þ W 02 ð0Þ þ Oðjðz; zÞj3 Þ; 2 2 z2 z2 ð2Þ ð2Þ ð2Þ x2t ð0Þ ¼ az þ az þ W 20 ð0Þ þ W 11 ð0Þzz þ W 02 ð0Þ þ Oðjðz; zÞj3 Þ; 2 2 z2 z2 ð3Þ ð3Þ ð3Þ x3t ð0Þ ¼ bz þ bz þ W 20 ð0Þ þ W 11 ð0Þzz þ W 02 ð0Þ þ Oðjðz; zÞj3 Þ; 2 2 z2 z2 ð4Þ ð4Þ ð4Þ x4t ð0Þ ¼ cz þ cz þ W 20 ð0Þ þ W 11 ð0Þzz þ W 02 ð0Þ þ Oðjðz; zÞj3 Þ; 2 2 z2 z2 ð1Þ ð1Þ ð1Þ ix0 s1 i x s 0 x1t ðs1 Þ ¼ ze þ ze 1 þ W 20 ðs1 Þ þ W 11 ðs1 Þzz þ W 02 ðs1 Þ þ Oðjðz; zÞj3 Þ; 2 2 2 z2 z ð2Þ ð2Þ ð2Þ x2t ðs2 Þ ¼ azeix0 s2 þ azeix0 s2 þ W 20 ðs2 Þ þ W 11 ðs2 Þzz þ W 02 ðs2 Þ þ Oðjðz; zÞj3 Þ; 2 2 2 2 ð3Þ ð3Þ ð3Þ ix0 s1 ix0 s1 z z x3t ðs1 Þ ¼ bze þ bze þ W 20 ðs1 Þ þ W 11 ðs1 Þzz þ W 02 ðs1 Þ þ Oðjðz; zÞj3 Þ; 3 2 2 2 z z ð4Þ ð4Þ ð4Þ x4t ðs2 Þ ¼ czeix0 s2 þ czeix0 s2 þ W 20 ðs2 Þ þ W 11 ðs2 Þzz þ W 02 ðs2 Þ þ Oðjðz; zÞj3 Þ: 2 2 ð1Þ

x1t ð0Þ ¼ z þ z þ W 20 ð0Þ

ð3:16Þ

595

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

It follows together with (3.3) that

h gðz; zÞ ¼ q ð0Þf0 ðz; zÞ ¼ M a2 þ c2 f100 ð0Þ þ b f300 ð0Þ e2ix0 s2 þ 1 þ b2 a f200 ð0Þ þ c f400 ð0Þ e2ix0 s1 þ g 001 ð0Þ i z2 h þa2 a g 002 ð0Þ þ b2 b g 003 ð0Þ þ c2 c g 004 ð0Þ þ M g 001 ð0Þ þ jaj2 a g 002 ð0Þ þ jbj2 b g 003 ð0Þ þ jcj2 c g 004 ð0Þ 2 h i 2 þ c 2 f100 ð0Þ þ b f300 ð0Þ e2ix0 s2 þ jaj2 þ jcj2 f100 ð0Þ þ b f300 ð0Þ þ 1 þ jbj2 a f200 ð0Þ þ c f400 ð0Þ zz þ M a i z2 2 b g 00 ð0Þ þ c 2 a f 00 ð0Þ þ c f 00 ð0Þ e2ix0 s1 þ g 00 ð0Þ þ a 2 a g 002 ð0Þ þ b 2 c g 004 ð0Þ þ 1þb 2 4 1 3 2 nh i ð1Þ ð3Þ þ M 2 W 11 ðs1 Þ þ bW 11 ðs1 Þ a f200 ð0Þ þ c f400 ð0Þ þ 1 þ jbj2 b a f2000 ð0Þ þ c f4000 ð0Þ eix0 s1 h i ð2Þ ð4Þ þ 2 aW 11 ðs2 Þ þ cW 11 ðs2 Þ f100 ð0Þ þ b f300 ð0Þ þ jaj2 a þ jcj2 c f1000 ð0Þ þ b f3000 ð0Þ eix0 s2 ð1Þ ð3Þ ðs Þ a f 00 ð0Þ þ c f 00 ð0Þ eix0 s1 þ a W ð2Þ ð4Þ þ W 20 ðs1 Þ þ bW f100 ð0Þ þ b f300 ð0Þ eix0 s2 1 2 4 20 20 ðs2 Þ þ cW 20 ðs2 Þ ð1Þ ð1Þ 2 2 2 000 00 000 000 þ g 000 1 ð0Þ þ jaj aa g 2 ð0Þ þ jbj bb g 3 ð0Þ þ jcj cc g 4 ð0Þ þ W 20 ð0Þ þ 2W 11 ð0Þ g 1 ð0Þ ð2Þ ð2Þ ð3Þ ð0Þ þ 2bW ð3Þ ð0Þ b g 00 ð0Þ W 20 ð0Þ þ 2aW 11 ð0Þ a g 002 ð0Þþ bW þ a 3 20 11 o z2 z ð4Þ 00 W ð4Þ þ c þ : 20 ð0Þ þ 2cW 11 ð0Þ c g 4 ð0Þ 2 Comparing the coefﬁcients with (3.16), we have

h g 20 ¼ M a2 þ c2 f100 ð0Þ þ b f300 ð0Þ e2ix0 s2 þ 1 þ b2 a f200 ð0Þ þ c f400 ð0Þ e2ix0 s1 þ g 001 ð0Þ þ a2 a g 002 ð0Þ i þb2 b g 003 ð0Þ þ c2 c g 004 ð0Þ ;

ð3:17Þ

h i g 11 ¼ M g 001 ð0Þ þ jaj2 a g 002 ð0Þ þ jbj2 b g 003 ð0Þ þ jcj2 c g 004 ð0Þ þ jaj2 þ jcj2 f100 ð0Þ þ b f300 ð0Þ þ 1 þ jbj2 a f200 ð0Þ þ c f400 ð0Þ ; ð3:18Þ g 02

h

2

2 a f 00 ð0Þ þ c f 00 ð0Þ e2ix0 s1 þ g 00 ð0Þ þ a 2 þ c f100 ð0Þ þ b f300 ð0Þ e2ix0 s2 þ 1 þ b 2 a g 002 ð0Þ ¼M a 2 4 1 i 2 b g 00 ð0Þ þ c 2 c g 004 ð0Þ ; þb 3 nh i ð1Þ ð3Þ 2 W 11 ðs1 Þ þ bW 11 ðs1 Þ a f200 ð0Þ þ c f400 ð0Þ þ 1 þ jbj2 b a f2000 ð0Þ þ c f4000 ð0Þ eix0 s1 h i ð2Þ ð4Þ þ 2 aW 11 ðs2 Þ þ cW 11 ðs2 Þ f100 ð0Þ þ b f300 ð0Þ þ jaj2 a þ jcj2 c f1000 ð0Þ þ b f3000 ð0Þ eix0 s2 ð1Þ ð3Þ ðs Þ a f 00 ð0Þ þ c f 00 ð0Þ eix0 s1 þ W 20 ðs1 Þ þ bW 1 2 4 20 ð2Þ ð4Þ 2 2 000 000 W 20 W 20 þ a ðs2 Þ þ c ðs2 Þ f100 ð0Þ þ b f300 ð0Þ eix0 s2 þ g 000 1 ð0Þ þ jaj aa g 2 ð0Þ þ jbj bb g 3 ð0Þ ð1Þ ð1Þ 00 þ jcj2 cc g 000 4 ð0Þ þ W 20 ð0Þ þ 2W 11 ð0Þ g 1 ð0Þ o ð2Þ ð2Þ ð4Þ ð4Þ ð3Þ ð0Þ þ 2bW ð3Þ ð0Þ b g 00 ð0Þ þ c W 20 W 20 ð0Þ þ 2aW 11 ð0Þ a g 002 ð0Þþ bW ð0Þ þ 2cW 11 ð0Þ c g 004 ð0Þ : þ a 3 20 11

ð3:19Þ

g 21 ¼ M

ð3:20Þ

In order to determine g 21 , we still need to compute W 20 ðhÞ and W 11 ðhÞ. From (3.8) and (3.12), we have

_ ¼ X_ t z_ q z_ q ¼ W

A0 W 2Refq ð0Þf0 ðz; zÞqg; h 2 ½s2 ; 0Þ; A0 W 2Refq ð0Þf0 ðz; zÞqg þ f0 ðz; zÞ; h ¼ 0:

ð3:21Þ

We rewrite this as

_ ¼ A0 W þ Hðz; z; hÞ; W

ð3:22Þ

where

Hðz; z; hÞ ¼ H20

z2 z2 þ H11 zz þ H02 þ : 2 2

Taking the derivative of W with respect to t in (3.13), we have

_ ¼ W z z_ þ W z z_ : W

ð3:23Þ

596

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

Thus, together with (3.21)–(3.23), we obtain

ðA0 2ix0 ÞW 20 ðhÞ ¼ H20 ðhÞ; A0 W 11 ðhÞ ¼ H11 ðhÞ;

ð3:24Þ

ðA0 þ 2ix0 ÞW 02 ðhÞ ¼ H02 ðhÞ; . . . : By (3.21) and (3.22), we know that for h 2 ½s2 ; 0Þ,

ðhÞ Hðz; z; hÞ ¼ 2Refq ð0Þf0 ðz; zÞqðhÞg ¼ gðz; zÞqðhÞ gðz; zÞq

1 1 1 1 ðhÞ: ¼ g 20 z2 þ g 11 zz þ g 02 z2 þ qðhÞ g20 z2 þ g11 zz þ g02 z2 þ q 2 2 2 2 Comparing the coefﬁcients with (3.23), we can obtain that

ðhÞ; H20 ðhÞ ¼ g 20 qðhÞ g02 q ðhÞ: H11 ðhÞ ¼ g 11 qðhÞ g11 q It follows from (3.24) that

(

_ 20 ðhÞ ¼ 2ix0 W 20 ðhÞ þ g qðhÞ þ g02 q ðhÞ; W 20 _ ðhÞ: W 11 ðhÞ ¼ g qðhÞ þ g11 q

ð3:25Þ

11

It’s easy to obtain the solutions of (3.25):

8 < W 20 ðhÞ ¼ g20 qð0Þ eix0 h g02 qð0Þ eix0 h þ E1 e2ix0 h ; ix0

3ix0

ð3:26Þ

: W 11 ðhÞ ¼ g11 qð0Þ eix0 h g11 qð0Þ eix0 h þ E2 ; ix0

ix 0

where E1 and E2 are constant vectors and can be determined by setting h ¼ 0 in Hðz; z; hÞ. From

Hðz; z; 0Þ ¼ 2Refq ð0Þf0 ðz; zÞqð0Þg þ f0 ðz; zÞ; we have

0

ða2 þ c2 Þf100 ð0Þe2ix0 s2 þ g 001 ð0Þ

1

C B B ð1 þ b2 Þf200 ð0Þe2ix0 s1 þ a2 g 002 ð0Þ C C ð0Þ þ B H20 ð0Þ ¼ g 20 qð0Þ g02 q C B 2 2 @ ða þ c2 Þf300 ð0Þe2ix0 s2 þ b g 003 ð0Þ A

ð3:27Þ

ð1 þ b2 Þf400 ð0Þe2ix0 s1 þ c2 g 004 ð0Þ

and

1 ðjaj2 þ jcj2 Þf100 ð0Þ þ g 001 ð0Þ C B B ð1 þ jbj2 Þf200 ð0Þ þ jaj2 g 002 ð0Þ C C: ð0Þ þ B H11 ð0Þ ¼ g 11 qð0Þ g11 q C B 2 2 2 @ ðjaj þ jcj Þf300 ð0Þ þ jbj g 003 ð0Þ A 0

ð3:28Þ

ð1 þ jbj2 Þf400 ð0Þ þ jcj2 g 004 ð0Þ

From (3.24) and the deﬁnition of A0 , we have

BW 20 ð0Þ þ B1 W 20 ðs1 Þ þ B2 W 20 ðs2 Þ ¼ 2ix0 W 20 ð0Þ H20 ð0Þ;

ð3:29Þ

BW 11 ð0Þ þ B1 W 11 ðs1 Þ þ B2 W 11 ðs2 Þ ¼ H11 ð0Þ: Substituting (3.26) into (3.29) and noticing that

i x0 I

Z

!

0

ix 0 h

e

dgðhÞ qð0Þ ¼ 0

s2

and

ix0 I

Z

0

! ð0Þ ¼ 0; eix0 h dgðhÞ q

s2

we can obtain

0

ða2 þ c2 Þf100 ð0Þe2ix0 s2 þ g 001 ð0Þ

1

C B B ð1 þ b2 Þf200 ð0Þe2ix0 s1 þ a2 g 002 ð0Þ C C; E1 ¼ ðB B1 e2ix0 s1 B2 e2ix0 s2 þ 2ix0 IÞ1 B C B 2 2 @ ða þ c2 Þf300 ð0Þe2ix0 s2 þ b g 003 ð0Þ A

ð1 þ b2 Þf400 ð0Þe2ix0 s1 þ c2 g 004 ð0Þ

597

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

0

ðjaj2 þ jcj2 Þf100 ð0Þ þ g 001 ð0Þ

1

C B B ð1 þ jbj2 Þf200 ð0Þ þ jaj2 g 002 ð0Þ C C: E2 ¼ ðB B1 B2 Þ1 B C B 2 2 2 @ ðjaj þ jcj Þf300 ð0Þ þ jbj g 003 ð0Þ A ð1 þ jbj2 Þf400 ð0Þ þ jcj2 g 004 ð0Þ According to the above analysis, we can see that all g ij in (3.17)–(3.20) have been expressed in terms of the parameters. Thus we can compute the following quantities

c1 ð0Þ ¼ 2xi 0 g 20 g 11 2jg 11 j2 13 jg 02 j2 þ 12 g 21 ; Refc1 ð0Þg l2 ¼ Refk ; 0 ðs0 Þg m2 ¼ 2Refc1 ð0Þg;

T 2 ¼ x10 Imfc1 ð0Þg þ l2 Imfk0 ðs0 Þg : Using the method of [15], we obtain the following result. Theorem 3.1. Assume that fi ; g i 2 C 3 ; i ¼ 1; 2; 3; 4. Then, the direction of the Hopf bifurcation is determined by the sign of l2 : if l2 > 0ðl2 < 0Þ, then the Hopf bifurcation is supercritical(subcritical) and the bifurcating periodic solutions exist for s > s0 ðs < s0 Þ. m2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (unstable) if m2 < 0ðm2 > 0Þ. T 2 determines the period of the bifurcating periodic solutions: the period increases (decreases) if T 2 > 0ðT 2 < 0Þ. Moreover, the period of the bifurcating periodic solutions of (1.1) can be estimated by

2p ð1 þ T 2 l2 þ Þ: x0

0.4

0.4

0.2

0.2

0

0

x4

x3

TðlÞ ¼

−0.2

−0.2

−0.4 0.5

−0.4 0.5 0 −0.5

x

−0.4

0.2

0

−0.2

0.4 0

x2

x1

2

−0.5

−0.4

0

−0.2

0.2

0.4

x

1

0.4 0.4 0.2

x4

x4

0.2 0

−0.2

−0.2 0.5 −0.4 0.4

0

−0.4 0.4 0.2

0.2

0 0

−0.2

x3

0.5 0

0

−0.2 −0.4

−0.5

x1

x3

−0.4

Fig. 2. s1 = 0.6, s2 = 0.9, s = s1 + s2 = 1.5 < s0, the zero steady state is stable.

−0.5

x2

598

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

4. An example and numerical simulation In this section, we will study the speciﬁc example of (1.1) along with numerical simulation. Consider the following system:

8 x_ 1 ðtÞ ¼ 0:5x1 ðtÞ 0:5 tanhðx1 ðtÞÞ þ 0:4 tanhðx4 ðt s2 ÞÞ þ 0:4 tanhðx2 ðt s2 ÞÞ; > > > < x_ ðtÞ ¼ 1:5x ðtÞ þ 0:5 tanhðx ðtÞÞ 1:2 tanhðx ðt s ÞÞ 1:2 tanhðx ðt s ÞÞ; 2 2 2 1 1 3 1 > x_ 3 ðtÞ ¼ 0:6x3 ðtÞ 0:4 tanhðx3 ðtÞÞ þ 0:6 tanhðx2 ðt s2 ÞÞ þ 0:6 tanhðx4 ðt s2 ÞÞ; > > : x_ 4 ðtÞ ¼ 1:8x4 ðtÞ þ 0:8 tanhðx4 ðtÞÞ 0:8 tanhðx3 ðt s1 ÞÞ 0:8 tanhðx1 ðt s1 ÞÞ;

ð4:1Þ

which has a steady state ð0; 0; 0; 0Þ. It is easy to verify that the hypotheses ðH1 Þ and ðH2 Þ hold and compute

p ¼ 4;

q ¼ 2;

v ¼ 3:

u ¼ 4;

In this case hðzÞ ¼ z þ pz þ qz þ uz þ v ¼ 0 has only one positive real root z0 ¼ 1, and then x0 ¼ 1. According to (2.11) and (2.12), we obtain 4

3

2

p s0 ¼ : 2

From Theorem 2.1, the zero solution of system (4.1) is asymptotically stable if s ¼ s1 þ s2 < s0 , and the Hopf bifurcation occurs at s0 ¼ p2 . 00 Furthermore, noticing that tanh ð0Þ ¼ 0, from (3.17)–(3.20), we can easily obtain

g 20 ¼ g 11 ¼ g 02 ¼ 0;

0.4

0.2

0.2

4

0.4

0

x

x3

g 21 ¼ 4:1342 i1:1219;

0

−0.2

−0.2

−0.4 0.5

−0.4 0.5 0 −0.5

x2

−0.4

0.2

0

−0.2

0.4 0

x

x1

−0.5

−0.4

2

0

−0.2

x

0.2

0.4

1

0.4

0.4 0.3

0.2

0.2

x

4

x4

0.1

0

0 −0.2

−0.1 −0.2 −0.3 −0.4 0.4

0.5 0.2

0 0

−0.2

x3

−0.4

−0.4 0.4 0.2

0.5 0 −0.2

−0.5

x1

0

x3

−0.4

−0.5

Fig. 3. s1 = 0.7, s2 = 0.9, s = s1 + s2 = 1.6 > s0, the bifurcating periodic solution is stable.

x

2

H. Hu, L. Huang / Applied Mathematics and Computation 213 (2009) 587–599

599

and then

c1 ð0Þ ¼ 2:0671 i0:5610;

l2 ¼ 15:7284; m2 ¼ 4:1342; T 2 ¼ 5:875:

Therefore, from Theorem 3.1, we know that the Hopf bifurcation is supercritical, the bifurcating periodic solution is asymptotically stable, the period of the bifurcating periodic solution increases and the period of the bifurcating periodic solution is approximately

TðlÞ ¼ 2pð1 þ 5:875l2 þ Þ: The numerical simulations, which are performed by the DDEs solver developed by Shampine and Thompson [18] are given in Figs. 2 and 3. 5. Conclusions In this paper, a four-neuron ring with self-feedback and delays has been investigated. By choosing the sum s of two delays as a bifurcation parameter and analyzing the corresponding characteristic equation, we have shown that the equilibrium of the model loses its stability and periodic solutions via Hopf bifurcation occurs when s passes through a critical value. This implies that the connection time delays are able to alter the dynamics of system (1.1) signiﬁcantly. The direction of Hopf bifurcation and the stability of the bifurcating periodic orbits have also been discussed by applying the normal form theory and the center manifold theorem. Finally, simulation results have veriﬁed and demonstrated the correctness of the theoretical results. At the end of this paper, We would like to point out that in order to understand fully the periodic phenomenon, it is signiﬁcant as well as challenging to study whether the periodic solutions bifurcating from local Hopf bifurcation globally exist. In addition, it is worth searching for an effective bifurcation control method to obtain some desirable system behaviors. We leave them as our future work. Acknowledgement We wish to thank one reviewer for her/his valuable comments that lead to an improvement of the manuscript. References [1] T. Faria, On a planar system modelling a neuron network with memory, Journal of Differential Equation 168 (2000) 129–149. [2] Z. Yuan, L. Huang, Y. Chen, Convergence and periodicity of solutions for a discrete-time network model of two neurons, Mathematical and Computer Modelling 35 (2002) 941–950. [3] J.J. Hopﬁeld, Neurons with graded response have collective computational properties like two-state neurons, Proceedings of the National Academy of Sciences of the USA 81 (1984) 3088–3092. [4] C.M. Marcus, R.M. Westervelt, Stability of analog neural networks with delay, Physical Review A 39 (1989) 347–359. [5] K. Gopalsamy, I. Leung, Delay induced periodicity in a neural network of excitation and inhibition, Physica D 89 (1) (1996) 395–426. [6] X. Liao, J. Yu, Qualitative analysis of bi-directional associative memory with time delays, International Journal of Circuit Theory and Applications 26 (3) (1998) 219–229. [7] P.V.D. Driessche, X. Zou, Global attractivity in delayed Hopﬁeld neural network models, SIAM Journal on Applied Mathematics 58 (6) (1998) 1878– 1890. [8] J. Cao, M. Dong, Exponential stability of delayed bidirectional associative memory networks, Applied Mathematics and Computation 135 (2003) 105–112. [9] M. Joy, Results concerning the absolute stability of delayed neural networks, Neural Networks 13 (2000) 613–616. [10] S. Arik, V. Tavsanoglu, Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays, Neurocomputing 68 (2005) 161–176. [11] J. Belair, S.A. Campbell, P. van den Driessche, Frustration, stability, and delay-induced oscillations in a neural network model, SIAM Journal of Applied Mathematics 56 (1) (1996) 245–255. [12] G. Röst, Neimark–Sacker bifurcation for periodic delay differential equations, Nonlinear Analysis 60 (2005) 1025–1044. [13] Y. Chen, J. Wu, Minimal instability and unstable set of a phaselocked periodic orbit in a delayed neural network, Physica D 134 (1999) 185–199. [14] S. Guo, L. Huang, Hopf bifurcating periodic orbits in a ring of neurons with delays, Physica D 183 (2003) 19–44. [15] B.D. Hassard, N.D. Kazarinoff, Y.H. Wan, Theory and Applications of Hopf Bifurcation, Cambridge University Press, Cambridge, 1981. [16] X. Li, J. Wei, On the zeros of a fourth degree exponential polynomial with applications to a neural network model with delays, Chaos, Solitons and Fractals 26 (2005) 519–526. [17] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dynamics of Continuous Discrete and Impulsive Systems Series A: Mathematics and Analytical 10 (2003) 863–874. [18] L.F. Shampine, S. Thompson, Solving DDEs in MATLAB, Applied Numerical Mathematics 37 (2001) 441–458. [19] J.C. Eccles, M. Ito, J. Szenfagothai, The Cerebellum as Neuronal Machine, Springer, New York, 1967. [20] M.W. Hirsch, Convergent activation dynamics in continuous-time networks, Neural Networks 2 (1989) 331–349. [21] Y. Song, M. Han, J. Wei, Stability and Hopf bifurcation analysis on a simpliﬁed BAM neural network with delays, Physica D 200 (2005) 185–204. [22] M. Xiao, J. Cao, Stability and Hopf bifurcation in a delayed competitive web sites model, Physics Letters A 353 (2006) 138–150. [23] L.P. Shayer, S.A. Campbell, Stability, bifurcation and multistability in a system of two coupled neurons with multiple time delays, SIAM Journal of Applied Mathematics 61 (2) (2000) 673–700. [24] J. Wu, Symmetric functional-differential equations and neural networks with memory, Transactions of the American Mathematical Society 350 (1998) 4799–4838. [25] J. Wei, Y. Yuan, Synchronized Hopf bifurcation analysis in a neural network model with delays, Journal of Mathematical Analysis and Applications 312 (2005) 205–229. [26] J. Cao, M. Xiao, Stability and Hopf bifurcation in a simpliﬁed BAM neural network with two time delays, IEEE Transactions of Neural Networks 18 (2) (2007) 416–430. [27] Z. Cheng, J. Cao, Bifurcation and stability analysis of a neural network model with distributed delays, Nonlinear Dynamics 46 (2006) 363–373.

Stability and Hopf bifurcation analysis on a ring of four neurons with delays

Stability and Hopf bifurcation analysis on a ring of four neurons with delays

Recommend Documents