Stability and Hopf bifurcation analysis in bidirectional ring network model

Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

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Stability and Hopf bifurcation analysis in bidirectional ring network model Linjun Wang a,b,⇑, Xu Han a a State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Automotive Engineering, Hunan University, Changsha City 410082, PR China b College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China

a r t i c l e

i n f o

Article history: Received 15 September 2010 Received in revised form 20 December 2010 Accepted 20 December 2010 Available online 28 December 2010 Keywords: Delay Hopf bifurcation Neural network Normal form Center manifold

a b s t r a c t Using the system parameter instead of the delay as the bifurcation parameter, linear stability and Hopf bifurcation analysis including its direction and stability of bidirectional ring network model are investigated in this paper. The main tools to obtain our results are the normal form method and center manifold theory. Numerical simulations show that the theoretically predicted values are in excellent agreement with the numerically observed behavior. Ó 2010 Elsevier B.V. All rights reserved.

1. Introduction Bifurcation behavior of artificial neural networks is a very interesting phenomenon which has been intensively studied during the last two decades. Moreover, they have many other applications. For example, content-addressable memories, in which information is stored as stable equilibrium points of the system. In 1984, Hopfield considered a simplified neural network model, where each neuron is represented by a linear circuit consisting of a resistor and a capacitor, and connected to the other neurons via nonlinear sigmoidal activation functions [1]. Since then, dynamical characteristics of neural networks had become a subject of intensive research activity. Later, it developed that delays were arising in neural networks due to the processing of information. Most of the models of neural networks are described by the systems of delay differential equations [2–7]. In addition, for the general theory of delay differential equations (DDEs), we refer to [8–15]. Due to the complexity of the analysis, most of the works only have focused on the situations where most of them deal with the two-equation models or traditionally regarding the delay of signal transmission as the parameter. However, very few papers about bifurcation analysis regarding the eigenvalues of the connection matrix as the bifurcation parameters can be found and very limited, especially in highdimensional bidirectional ring network model with delay. In this paper, we consider bidirectional ring network model as the following system (1.1) of DDEs consisting of delay, the internal decay rate and the connection strength. More precisely, we will give some conditions on the linear stability of the trivial solution of the system and investigate the Hopf bifurcation including its direction and stability of system (1.1), which is given as the following model (see Fig. 1)

8 > < x_ ¼ x þ af ðyðt  sÞÞ þ bf ðzðt  sÞÞ; y_ ¼ y þ af ðzðt  sÞÞ þ bf ðxðt  sÞÞ; > :_ z ¼ z þ af ðxðt  sÞÞ þ bf ðyðt  sÞÞ;

ð1:1Þ

⇑ Corresponding author at: State Key Laboratory of Advanced Design and Manufacturing for Vehicle Body, College of Mechanical and Automotive Engineering, Hunan University, Changsha City 410082, PR China. E-mail address: [email protected] (L. Wang). 1007-5704/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cnsns.2010.12.022

L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

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Fig. 1. Architecture of the model described by (1.1).

where s is nonnegative and denotes the synaptic transmission delay. The strengths of the self and nearest-neighbour coupling are denoted by a and b, respectively. They are the nonzero connection weighters. f : R ? R is the activation function. Throughout this paper, we always assume that f : R ? R is adequately smooth, and satisfies the following conditions: (C1) f(0) = 0, f0 (0) = 1; (C2) x f(x) – 0 for when x – 0. The rest of the paper is organized as follows: In Section 2, we discuss the linear stability of the trivial solution. In Section 3, we study the Hopf bifurcation of equilibria. Section 4 is devoted to numerical simulations and we conclude this paper in Section 5. 2. Linear stability of the trivial solution Linearizing (1.1) at trivial solution produces the following linear system:

8 > < x_ ¼ x þ ayðt  sÞ þ bzðt  sÞ; y_ ¼ y þ azðt  sÞ þ bxðt  sÞ; > :_ z ¼ z þ axðt  sÞ þ byðt  sÞ; whose characteristic matrix is

0

kþ1

B Dðl; kÞ ¼ @ beks aeks

aeks

beks

C aeks A:

kþ1 be

1

ks

kþ1

So the characteristic equation is

0 ¼ det Dðl; kÞ ¼ ðk þ 1Þ3  3abe2ks ðk þ 1Þ  ða3 þ b3 Þe3ks :

ð2:1Þ

Then it can be decomposed as

" ks

det Dðl; kÞ ¼ ½k þ 1  ða þ bÞe

 kþ1þ

aþb 2

#" # ! ! pffiffiffi pffiffiffi aþb 3 3 ks ks kþ1þ : þ  ða  bÞi e ða  bÞi e 2 2 2

ð2:2Þ

We first consider

pz ðkÞ ¼ k þ 1  zeks ; z 2 C:

ð2:3Þ

We define a curve R as the following parametric equations



uðtÞ ¼ cosðstÞ  t sinðstÞ;

v ðtÞ ¼ t cosðstÞ þ sinðstÞ;

ð2:4Þ

where t 2 R. ðtÞ Similar to the analysis in [13], it is not difficult to check that the curve R is symmetric about the u-axis. Let hðtÞ ¼ vuðtÞ . Then 2 2 0 h (t) = u (t)(1 + s + st ) > 0 for all t 2 R such that u(t) – 0. This implies that, as t increases, the corresponding point (u(t), v(t))

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on the curve R moves anticlockwise around the origin. Moreover, it follows from u2 + v2 = 1 + t2 that R+ = {(u(t), v(t)) : t 2 R+} is simple, i.e., it cannot intersect with itself. Let ftn gþ1 non-negative zeros of n¼0 be the monotonic increasing sequence of theS v(t), and cn = u(tn) for all n 2 N0 :¼ {0, 1, 2, . . .}. For each n 2 N0, let Rn = {(u(t), v(t)):t 2 [  tn+1,  tn] [tn, tn+1]}, which is a closed curve with (0, 0) inside, and it is shown in Fig. 2. The following lemma plays a profound role in analyzing the distribution of zeros of (2.1). Lemma 2.1. Consider pz(k) defined in (2.3), then the following statements are true: (i) pz(k) has a purely imaginary zero if and only if z 2 R. Moreover, if z = u(h) + iv(h) then the purely imaginary zero is ih except that there is a pair of conjugate purely imaginary zeros ±itn if z = cn for n 2 N. (ii) For each fixed z0 = u(h0) + iv(h0) 2 R, there exists an open d-neighborhood of z0 in the complex plane, denoted by B(z0, d), and an analytical function k : B(z0, d) ? C such that k(z0) = ih0 and k(z) is a zero of pz(k) for all z 2 B(z0, d). In addition, along the vector v(n) = (v0 (h0),  u0 (h0))M(n), the directional derivative of Re{k(z)} at z0 is positive, where n 2 (p/2,p/2) and

MðnÞ ¼



cos n

sin n

 sin n cos n

 :

(iii) pz(k) has only zeros with strictly negative real parts if and only if z is inside the curve R0; exactly j 2 N zeros with negative real parts if z is between Rj1 and Rj. Especially, if z 2 R0, pz(k) has either a simple real zero 0 (if z = 1) or a simple purely imaginary zero (if Im (z) – 0), or a pair of simple purely imaginary zeros (if z = c1), and all other zeros have strictly negative real parts. For the sake of convenience, we introduce the following notations:

8  X ¼ fðb; aÞ 2 R2þ : b  a ¼ cj ; b  a–cj ; b  a–  ck g; > > > 1j < > > > :

k 2 N0 ; j 2 N0 :

X2j ¼ fðb; aÞ 2 R2þ : b  a ¼ cj ; b  a–cj ; b  a–  ck g;

k 2 N0 ; j 2 N0 :

R2þ ¼ fðb; aÞ 2 R2 : a > 0g:

 We call X 1j ; X2j ; j 2 N 0 ; critical lines.

10 for t ≥0 for t<0

8 Σ

6

2

Σ1

4 2

Σ

v(t)

0

0 −2 −4 −6 −8

−10 −10

−8

−6

−4

−2 u(t)

0

Fig. 2. The parametric curve R.

2

4

6

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Corollary 2.2 (i) All the zeros of detD(l, k) have negative real parts if and only if jb + aj < 1 and a2 + b2  ab < 1. S  S þ S  (ii) If and only if l ¼ ðb; aÞ 2 Xþ X1n X2n X2n for some n 2 N, detD(l,k) has a pair of simple conjugate purely imaginary 1n zeros denoted by ±itn. S  S þ S  (iii) If and only if l ¼ ðb; aÞ 2 Xþ X10 X20 X20 det Dðl; kÞ has a simple zero k = 0. Moreover, if l ¼ ðb; aÞ 2 Xþ10 ; all the 10 zeros but k = 0 of detD(l, k) have strictly negative real parts. pffiffiffiffiffiffiffiffiffiffiffiffiffi (iv) If 1 < a2 þ b2  ab < 1 þ f2 ; then detD(l, k) has exactly a pair of simple purely imaginary zeros and all the other zeros pffiffiffiffiffiffiffiffiffiffiffiffiffi have strictly negative real parts. If a2 þ b2  ab > 1 þ f2 ; then detD(l, k) has exactly a pair of simple purely imaginary zeros and at least one zero with positive real part, where f is the minimal positive solution to the equation f sin (sf) = cos (sf).     pffiffi pffiffi (v) detD(l,k) has a purely imaginary zero ih if and only if aþb ; 23 ða  bÞ or  aþb ; 23 ða  bÞ lies on the curve R, where h 2 2 pffiffi pffiffi and tðhÞ ¼ 23 ða  bÞ or uðhÞ ¼  aþb and tðhÞ ¼ 23 ðb  aÞ: satisfies uðhÞ ¼ aþb 2 2

Theorem 2.3 (i) The trivial solution of system (1.1) is asymptotically stable if and only if jb + aj < 1 and a2 + b2  ab < 1.     pffiffi pffiffi ; 23 ða  bÞ or  aþb ; 23 ða  bÞ lies on the curve, then near (b, a) a Hopf bifurcation occurs in system (1.1), (ii) If either aþb 2 2 i.e., a unique branch of periodic solutions bifurcates from the origin. 3. Hopf bifurcation Set C ¼ Cð½s; 0; R3 Þ and k/ks = sups6h60k/(h)k. We define one solution x(t) of the system (1.1) by xt(h) = x(t + h), s 6 h 6 0. If x(t) is continuous, then xt ðhÞ 2 C. Let X(t) = (x(t), y(t),z(t))T. Let BC be the set of all the functions from [  s, 0] to R3. Thus we can rewrite the system (1.1) as

X_ t ¼ AðlÞX t þ RðlÞX t ;

ð3:1Þ

where AðlÞ : C ! BC is given by

8 duðhÞ > ; h 2 ½s; 0; > > > 2dh 3 < u1 ð0Þ þ au2 ðsÞ þ bu3 ðsÞ AðlÞuðhÞ ¼ 6 7 > 4 u2 ð0Þ þ au3 ðsÞ þ bu1 ðsÞ 5; h ¼ 0; > > > : u3 ð0Þ þ au1 ðsÞ þ bu2 ðsÞ and the nonlinear operator RðlÞ : C ! BC is

RðlÞu ¼

1 00 1 f ð0ÞBðlÞðu; uÞ þ f t0 ð0ÞCðlÞðu; u; uÞ þ    ; 2 6

with BðlÞ : C  C ! BC; CðlÞ : C  C  C ! BC being defined as follows:

BðlÞðu; uÞðhÞ ¼ 0;

h 2 ½s; 0

2

3

au2 ðsÞw2 ðsÞ þ bu3 ðsÞw3 ðsÞ 6 7 BðlÞðu; uÞð0Þ ¼ 4 bu3 ðsÞw3 ðsÞ þ au1 ðsÞw1 ðsÞ 5; au1 ðsÞw1 ðsÞ þ bu2 ðsÞw2 ðsÞ CðlÞðu; w; /ÞðhÞ ¼ 0;

h 2 ½s; 0

2

3

au2 ðsÞw2 ðsÞ/2 ðsÞ þ bu3 ðsÞw3 ðsÞ/3 ðsÞ 6 7 CðlÞðu; w; /Þð0Þ ¼ 4 bu3 ðsÞw3 ðsÞ/3 ðsÞ þ au1 ðsÞw1 ðsÞ/1 ðsÞ 5; au1 ðsÞw1 ðsÞ/1 ðsÞ þ bu2 ðsÞw2 ðsÞ/2 ðsÞ for u; w; / 2 C. By the Riesz representation theorem, there exists a matrix N(h, l) whose components are functions of bounded variation in h 2 [s, 0] such that

ðAðlÞuÞð0Þ ¼

Z

0

s

dNðh; lÞuðhÞ; u 2 C:

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It is easy to see that Hopf bifurcation may occur when satisfying l0 ¼ ðb0 ; a0 Þ 2 R2þ and the condition that A(l) has a pair of conjugate purely imaginary zeros. In fact, this is true if l0 2 R. In this section, we only discuss codimension one Hopf bifurcation, i.e., A(l) has a pair of conjugate purely imaginary zeros. Throughout this section, we assume that one of the following two cases is satisfied: 



pffiffi 3 ð 0 2

(1)

0 l0 ¼  a0 þb ; 2

(2)

0 l0 ¼ a0 þb ; 3 ða0  b0 Þ 2 R:In Case (1), a2 þ b20  a0 b0 ¼ 1 þ x20 , where x0 ¼ a2 þ b2  ab  1. Let  2 p2ffiffi    0 pffiffi 0 l ¼  aþb ; 23 ða  bÞ ; l0 ¼  a0 þb ; 23 ða0  b0 Þ ; kðlÞ be the analytical function satisfying k(l0) = x0 and 2 2 k(l) + 1 + l ek(l)s = 0 for all sufficiently small jl j, where x0 = tj . From Lemma 2.1, we know that ddv Refkðl0 Þg < 0; where v is the outward- pointing normal vector to the curve R at l0. So A(l) has only a pair of conjugate imaginary zeros ±ix0 at l0 = (b0, a0), where x0 = tj. In order to study the Hopf bifurcation, we need to compute the reduced sys-



a  b 0 Þ 2 R; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi



pffiffi

tem on the center manifold with the pair of conjugate complex, purely imaginary eigenvalues ±ix0. By this reduction we can determine the Hopf bifurcation direction and the stability of the bifurcating periodic solutions. For this purpose, we further assume that x f(x) – 0 when x – 0. It is easy to check that

qðhÞ ¼ ð1; d1 ; d2 ÞT eix0 h ;

h 2 ½s; 0;

where

d1 ¼ d2 ¼

ðix0 þ 1Þa þ b2 eix0 s j ; a2 eix0 s þ ðix0 þ 1Þb ðix0 þ 1Þb þ a2 eix0 s b2 eix0 s þ ðix0 þ 1Þa

;

is an eigenvector of A(l0) associated with the eigenvalue ix0. The adjoint operator A⁄(l0) is given by

( 

ðA ðl0 ÞwÞðnÞ ¼

 dw ; n 2 ð0; s; dn R0 wðsÞdNðt; l0 Þ; n ¼ 0: s

For the convenience in computation, allow the functions with the range in C3 instead of in R3. Thus the domains of we 1shall  1 3 ⁄ A(l0) and A (l0) are C ½s; 0; C and C ½0; s; C3 ; respectively, where C3 is the space of three-dimensional complex row vectors. It follows that ix0 is an eigenvalue of A⁄(l0) and

A ðl0 ÞpðnÞ ¼ ix0 pðnÞ for some nonzero row-vector function p(n), n 2 [0,s]. In order to construct the coordinates to describe the center manifold Cl near the origin, we use the bilinear form [6,16]

 uð0Þ  hw; ui ¼ wð0Þ

Z

Z

0

h¼s

h

  hÞdNðh; lÞuðnÞdn wðn

n¼0

for w 2 Cð½0; s; C3 Þ and u 2 Cð½s; 0; C3 Þ. In particular, let h; i ¼ h; il0 . Then

hw; Aðl0 Þui ¼ hA ðl0 Þw; ui; hu; wi 2 DomðAðl0 ÞÞ  DomðA ðl0 ÞÞ: We normalize q and p so that

hp; qi ¼ 1;

i ¼ 0: hp; q

By direct computation, we obtain that

2 ; d 1 ; 1Þeix0 n ; pðnÞ ¼ Dðd

n 2 ½0; s;

where 2

D ¼ fð2d2 þ d1 Þ½1 þ sð1 þ ix0 Þg1 : For each X 2 Dom (A(l)) with sufficiently small kl  l0k, we associate it with the pair (z, x), where z = hp, Xi and x ¼ X  zq  zq ¼ X  2Refzqg. For a solution Xt of (3.1) at l with sufficiently small kl  l0k, we define z(t) = hp, Xti and xðz; z; lÞ ¼ X t  2RefzðtÞqg. In fact, z and z are the local coordinates for Cl in the directions of p and p. It is easy to see that hp, xi = 0. D E Now, for the solutions Xt 2 Cl of (3.1), p; X_ t i ¼ hp; AðlÞX t þ RðlÞX t . Then, on the center manifold Cl with sufficiently small kl  l0k, we have

z_ ðtÞ ¼ ix0 z þ gðz; zÞ;

ð3:2Þ

L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

3689

where

wðt; hÞ ¼ w20 ðhÞ

z2 z2 z z3 þ w30 ðhÞ þ    ; þ w11 ðhÞzz þ w02 ðhÞ 2 2 6

ð3:3Þ

and

gðz; zÞ ¼ Pð0Þf0 ðz; zÞ ¼ g 20 ðhÞ

z2 z2 z2 z þ : þ g 11 ðhÞzz þ g 02 ðhÞ þ g 21 ðhÞ 6 2 2

ð3:4Þ

It is easy to obtain

_ ¼ w



h 2 ½s; 0Þ; A0 w  2RefPð0Þf0 ðz; zÞqg; A0 w  2RefPð0Þf0 ðz; zÞqg þ f0 ðz; zÞ; h ¼ 0:

We rewrite this as

_ ¼ A0 w þ Hðz; z; hÞ; w

ð3:5Þ

where

Hðz; z; hÞ ¼ H20 ðhÞ

z2 z2 z3 þ H11 ðhÞzz þ H02 ðhÞ þ H30 ðhÞ þ    ; 2 2 6

and

_ ¼ wz z_ þ wzz_ : w

ð3:6Þ

Expanding the above series and comparing the coefficients, we obtain

ðA0  2iw0 Þw20 ðhÞ ¼ H20 ðhÞ; A0 w11 ðhÞ ¼ H11 ðhÞ;

ð3:7Þ

ðA0 þ 2iw0 Þw02 ðhÞ ¼ H02 ðhÞ; . . . Noticing that

xðt  sÞ ¼ wð1Þ ðt; sÞ þ zðtÞeiw0 s þ zðtÞeiw0 s ; 1 eiw0 s ; yðt  sÞ ¼ wð2Þ ðt; sÞ þ zðtÞd1 eiw0 s þ zðtÞd  eiw0 s ; zðt  sÞ ¼ wð3Þ ðt; sÞ þ zðtÞd2 eiw0 s þ zðtÞd 2 where ðjÞ

wðjÞ ðt; sÞ ¼ w20 ðsÞ

z2 z2 ðjÞ ðjÞ þ w11 ðsÞzz þ w02 ðsÞ þ    ðj ¼ 1; 2; 3Þ; 2 2

then it follows that

0

1

ay2 ðt  sÞ þ bz2 ðt  sÞ f ð0Þ B C Dðd2 ; d1 ; 1Þ@ bz2 ðt  sÞ þ ax2 ðt  sÞ A gðz; zÞ ¼ Pð0ÞGðwðz; z; hÞ þ 2RefzqðhÞg; 0Þ ¼ 2 ax2 ðt  sÞ þ by2 ðt  sÞ 0 3 1 3 ay ðt  sÞ þ bz ðt  sÞ 1 B C þ f 000 ð0ÞDðd2 ; d1 ; 1Þ@ bz3 ðt  sÞ þ ax3 ðt  sÞ A þ    6 ax3 ðt  sÞ þ by3 ðt  sÞ 00



 1 1 1 1 Daðd1 þ 1Þ f 00 ð0Þx2 ðt  sÞ þ f 000 ð0Þx3 ðt  sÞ þ Dðd2 a þ bÞ f 00 ð0Þy2 ðt  sÞ þ f 000 ð0Þy3 ðt  sÞ 2 3 2 3

 1 1 þ Dðd1 þ d2 Þb f 00 ð0Þz2 ðt  sÞ þ f 000 ð0Þz3 ðt  sÞ 2 3 h i 1 ð1Þ ð1Þ 00 2 2iw0 s 2 2iw þ z e 0 s þ 2zz þ w20 ðt; sÞz2 zeiw0 s þ 2w11 ðt; sÞz2zeiw0 s ¼ Daðd1 þ 1Þf ð0Þ z e 2 h i 1 2 2 eiw0 s þ 2zzd1 d 1 þ wð2Þ ðt; sÞz2zd 1 eiw0 s þ 2wð2Þ ðt; sÞz2 zd1 eiw0 s þ Dðd2 a þ bÞf 00 ð0Þ z2 d1 e2iw0 s þ z2 d 1 20 11 2 h i 1 2 2 e2iw0 s þ 2zzd2 d 2 þ wð3Þ ðt; sÞz2zd 2 eiw0 s þ 2wð3Þ ðt; sÞz2zd2 eiw0 s þ Dðd1 þ d2 Þbf 00 ð0Þ z2 d2 e2iw0 s þ z2 d 2 20 11 2 h i 1 h i

1 1 2  iw0 s 2  iw0 s þ Daðd1 þ 1Þf 000 ð0Þ z2zeiw0 s þ Dðd2 a þ bÞf 000 ð0Þ z2zd1 d þ Dðd1 þ d2 Þbf 000 ð0Þ z2zd2 d þ : 1e 2e 2 2 2 ð3:8Þ ¼

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Comparing the coefficients with (3.4), we have

h i 00 2 2 g 20 ¼ Df ð0Þ aðd1 þ 1Þ þ ðd2 a þ bÞd1 þ ðd1 þ d2 Þbd2 e2iw0 s ; 00 1 þ ðd1 þ d2 Þbd2 d  2 ; g 11 ¼ Df ð0Þ aðd1 þ 1Þ þ ðd2 a þ bÞd1 d

00 2 þ ðd1 þ d2 Þbd 2 e2iw0 s ; g 02 ¼ Df ð0Þ aðd1 þ 1Þ þ ðd2 a þ bÞd 1 2 h i 00 ð1Þ ð2Þ iw0 s 1 þ ðd1 þ d2 Þbwð3Þ ðt; sÞd 2 aðd1 þ 1Þw20 ðt; sÞ þ ðd2 a þ bÞw20 ðt; sÞd g 21 ¼ Df ð0Þe 20 h i 000 2 2 iw0 s þ Df ð0Þ aðd1 þ 1Þ þ ðd2 a þ bÞd1 d 1 þ ðd1 þ d2 Þbd2 d2 e h i 00 ð1Þ ð2Þ ð3Þ þ 2Df ð0Þ aðd1 þ 1Þw11 ðt; sÞ þ ðd2 a þ bÞw11 ðt; sÞd1 þ ðd1 þ d2 Þbw11 ðt; sÞd2 eiw0 s :

ð3:9Þ

We still need to compute w11(t, h) and w20(t, h) for h 2 [  s,0). Then we have

ðhÞ Hðz; z; hÞ ¼ 2RefPð0Þf0 ðz; zÞqðhÞg ¼ gðz; zÞqðhÞ  gðz; zÞq   z2 z2 z2 z þ    qðhÞ ¼  g 20 ðhÞ þ g 11 ðhÞzz þ g 02 ðhÞ þ g 21 ðhÞ 6 2 2   2 2  z z z2 z ðhÞ: þ  q  g20 ðhÞ þ g11 ðhÞzz þ g02 ðhÞ þ g 21 ðhÞ 6 2 2

ð3:10Þ

Comparing the coefficients with (3.6) gives that

ðhÞ; H20 ðhÞ ¼ g 20 qðhÞ  g02 ðhÞq ðhÞ: H11 ðhÞ ¼ g 11 qðhÞ  g11 ðhÞq

ð3:11Þ

It follows from (3.7) that

ð0Þeiw0 h : _ 20 ðhÞ ¼ 2iw0 w20 ðhÞ þ g 20 qð0Þeiw0 h þ g02 q w Solving for w20(h), we obtain

w20 ðhÞ ¼

ig 20 ig02 ð0Þeiw0 h þ E1 e2iw0 h ; q qð0Þeiw0 h þ w0 3w0

ð3:12Þ

and similarly, we have

w11 ðhÞ ¼ 

ig 11 ig11 ð0Þeiw0 h þ E2 ; q qð0Þeiw0 h þ w0 w0

where E1 and E2 are both three-dimensional vectors, and can be determined by setting h = 0 in Hðz; z; hÞ. In fact,

Hðz; z; 0Þ ¼ 2RefPð0Þf0 ðz; zÞqð0Þg þ f0 ðz; zÞ:

ð3:13Þ

Noticing that

f0 ðz; zÞ ¼ GðX t ðhÞ; 0Þ 0 2 1 ay ðsÞ þ bz2 ðsÞ f 00 ð0Þ @ 2 ¼ bz ðsÞ þ ax2 ðsÞ A 2 ax2 ðsÞ þ by2 ðsÞ 0 3 1 ay ðsÞ þ bz3 ðsÞ 1 000 @ 3 þ f ð0Þ bz ðsÞ þ ax3 ðsÞ A þ Oðjuj4 Þ 6 ax3 ðsÞ þ by3 ðsÞ ð1Þ

2 ! f 00 ð0Þ 0a w ðt; sÞ þ zðtÞeiw0 s þ zðtÞeiw0 s ¼

2 2 a wð1Þ ðt; sÞ þ zðtÞeiw0 s þ zðtÞeiw0 s ! ð2Þ iw s  iw s f 00 ð0Þ a w ðt; sÞ þ zðtÞd1 e 0 þ zðtÞd1 e 0 0 þ 1 eiw0 s 2 2 b wð2Þ ðt; sÞ þ zðtÞd1 eiw0 s þ zðtÞd ! ð3Þ 2 eiw0 s f 00 ð0Þ b w ðt; sÞ þ zðtÞd2 eiw0 s þ zðtÞd þ ð3Þ

2 eiw0 s 0 þ    ; 2 b w ðt; sÞ þ zðtÞd2 eiw0 s þ zðtÞd

ð3:14Þ

and comparing with (3.5), we have

0

1 ad21 þ bd22 B C 2 C 2iw0 s ð0Þ þ f 00 ð0ÞB H20 ð0Þ ¼ g 20 qð0Þ  g02 0q ; @ a þ bd2 Ae 2 a þ bd1

ð3:15Þ

L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

0

3691

1

ajd1 j2 þ bjd2 j2 B 00 ð0Þ þ f ð0Þ@ a þ bjd2 j2 C H11 ðhÞ ¼ g 11 qð0Þ  g11 0q Ae2iw0 s : a þ bjd1 j2

ð3:16Þ

From (3.7) and the definition of A0, we have

w20 ð0Þ þ Bw20 ðsÞ ¼ 2iw0 w20 ð0Þ  H20 ð0Þ

ð3:17Þ

w11 ð0Þ þ Bw11 ðsÞ ¼ H11 ð0Þ;

ð3:18Þ

and

where

0

0 B ¼ @b

a b1 0 a A:

a b 0 Substituting (3.12) into (3.17) and noticing that D(s, iw0)q(0) = 0, we have

0

1

ad21 þ bd22 B C 1 00 E1 ¼ f ð0ÞD ðs; 2iw0 Þ@ a þ bd22 A; 2 a þ bd1 where

Dðs; 2iw0 Þ ¼ ð1 þ 2iw0 ÞI  Be2iw0 s : Similarly, we have

0

1

ajd1 j2 þ bjd2 j2 B C 1 00 E2 ¼ f ð0ÞD ðs; 0Þ@ a þ bjd2 j2 A: 2 a þ bjd1 j Based on the above analysis, we can see that each gijin (3.9) is determined by the parameters in system (1.1). Thus we can compute the following values:

  i 1 g g 20 g 11  2jg 11 j2  jg 02 j2 þ 21 ; 2b0 3 2 RefC 1 ðlð0ÞÞg ; U2 ¼  Refk0ðl0 Þg ImfC 1 ðlð0ÞÞg þ l2 Imfk0ðl0 Þg ; T2 ¼  w0 B2 ¼ 2RefC 1 ð0Þg;

C 1 ð0Þ ¼

ð3:19Þ

which determine the quantities of bifurcating periodic solutions in the center manifold at the critical value l0, i.e., U2 determines the directions of the Hopf bifurcation: if U2 > 0 (respectively, <0), then the Hopf bifurcation is supercritical (respectively, subcritical); B2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are stable (respectively, unstable) if B2 < 0 (respectively, >0); and T2 determines the period of the bifurcating periodic solutions: the period increase (respectively, decrease) if T2 > 0 (respectively, < 0). Then we have the following results. Theorem 3.1. Let C1(0) be given in (3.19). Then (i) the bifurcating periodic solutions exist for l = l0 and if

RefC 1 ð0Þg > 0 ðrespectiv ely; < 0Þ; then the Hopf bifurcation is supercritical (respectively, subcritical); (ii) the bifurcating periodic solutions are stable (respectively, unstable) if

RefC 1 ð0Þg < 0 ðrespectiv ely; > 0Þ; (iii) T2 determines the period of the bifurcating periodic solutions: the period increases (respectively, decreases) if T2 > 0 (respectively, <0). Then we have, Theorem 3.2. Suppose that f00 (0) = 0, f000 (0) – 0. Let

h i h i pffiffiffi h ¼ 2k þ 4ðl þ mÞða  bÞ2 ða2 þ b2 þ abÞ  ð1 þ s þ w20 sÞða þ bÞ þ 3w0 ða  bÞ h i pffiffiffi pffiffiffi þ 4 3ðm  lÞða  bÞ2 ða2 þ b2 Þ  w0 ða þ bÞ  3ða  bÞð1 þ s þ w20 sÞ ;

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L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

then at l = l0, the system (1.1) undergoes a Hopf bifurcation. The direction of Hopf bifurcation and the stability of bifurcation periodic solutions are determined by the sign of hf000 (0). More precisely, if hf000 (0) < 0 (respectively, >0), then the bifurcating periodic solutions are orbitally asymptotically stable (respectively, unstable), and the Hopf bifurcation is subcritical (respectively, supercritical), i.e., the bifurcated periodic solutions exist for jlj2 > a2 + b2  ab(respectively,
h i 000 2 2 iw0 s g 21 ¼ Df ð0Þ aðd1 þ 1Þ þ ðd2 a þ bÞd1 d : 1 þ ðd1 þ d2 Þbd2 d2 e

ð3:20Þ

Therefore,

(

)

aðd1 þ 1Þ þ ðd2 a þ bÞd21 d1 þ ðd1 þ d2 Þbd22 d2 iw0 s : e 2RefC 1 ð0Þg ¼ Refg 21 g ¼ Re f ð0Þ 2 ð2d2 þ d1 Þ½1 þ sð1 þ iw0 Þ 000

From (2.1) and the definition of d1, d2, it is easy to find that

d1 ¼ Nd2 ;

jd1 j ¼ N;

jd2 j ¼

1 ; N

where

N¼

ð2b2  a2  abÞ2 þ 3a2 ða  bÞ2 2a2  b2  abÞ2 þ 3b2 ða  bÞ2

:

In fact, from (2.1), we can obtain

d1 ¼

z1 ; z2

d2 ¼

z2 ; z1

jz1 j2 ¼ N1 ;

jz2 j2 ¼ N2 ;

N¼

N1 ; N2

where

z1 ¼ 2b2  a2  ab 

pffiffiffi 3aða  bÞi;

z2 ¼ 2a2  b2  ab 

pffiffiffi 3bða  bÞi:

Then

9 8  b b 2= < ð a þ bN Þd þ a bN þ d þ d d 1 1 2 N N 2   Refg 21 g ¼ Re f 000 ð0Þeiw0 s 2 ; : 2d þ d ½1 þ sð1 þ iw Þ 2

1

0

8 9 bN31 N 42 z1 = 3 b < ð a þ bN ÞN Z þ ð a bN þ ÞN z þ 1 2 1 2 N N ¼ Re f 000 ð0Þeiw0 s : ; ð2N2 z2 þ N1 z1 Þ½1 þ sð1 þ iw0 Þ ( ) ½k þ lz1 z2 þ mz1 z2 ð1 þ s  iw0 sÞ ; ¼ Re f 000 ð0Þeiw0 s ½ð1 þ sÞ2 þ w20 s2 c where

  b bN 5 N4 N21 N32 þ 1 2 ; k ¼ 2ða þ bNÞN1 N 22 þ abN þ N N   3 5 b bN N l ¼ 2 abN þ N4 þ 2 1 2 ; N 2 N m ¼ ða þ bNÞN21 ;

c ¼ j2N2 z2 þ N1 z1 j2 : Then

Refg 21 g ¼ f 000 ð0Þ

1 ½ða þ bÞ2 þ 3ða  bÞ2 ½ð1 þ sÞ2 þ w20 s2 c

h:

So we can see that Theorem 3.2 is true. This completes the proof.

h

4. Numerical simulation example In this section, some numerical results of simulating the system (1.1) are presented at different data of a, b, s. Using the method of numerical simulations, we will find that the theoretically predicted results are in excellent agreement with the

L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

3693

numerically observed behavior. For this purpose, we must specify the activation function. Herein we take f(x) = tanh(x), i.e., we consider the system as follows:

8 > < x_ ¼ x þ a tanhðyðt  sÞÞ þ b tanhðzðt  sÞÞ; y_ ¼ y þ a tanhðzðt  sÞÞ þ b tanhðxðt  sÞÞ; > :_ z ¼ z þ a tanhðxðt  sÞÞ þ b tanhðyðt  sÞÞ;

ð4:1Þ

It is easy to check that f000 (0) = tanh000 (0) = 2 < 0. Obviously, the activation function f(x) satisfies the concavity and bound39 < 1; edness conditions. When a = 0.25, b = 0.35, s = 0.5, by a simple calculation, we have ja þ bj ¼ 0:6 < 1; a2 þ b2  ab ¼ 400 then from Theorem 2.3, we know that the origin is delay-independently asymptotically stable, which can be shown in Fig. 3. 3 < 1; which together with Theorem 2.3, Taking a = 0.5, b = 0.25, s = 0.25, by checking that ja þ bj ¼ 0:75 < 1; a2 þ b2  ab ¼ 16 pffiffi pffiffi implies that the origin is delay-independently asymptotically stable (see Fig. 4). Choosing b ¼ 1; a ¼  36 (or  26), by Theorem 3.2, we know that the corresponding waveform and phase plot for s ¼ p4 are shown in Figs. 5 and 6, respectively. 0.8

x(t) y(t) z(t)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6

0

20

40

t

60

80

100

Fig. 4. The trivial solution of (4.1) is asymptotically stable, where a = 0.5, b = 0.25, s = 0.25.

0.8

x(t) y(t) z(t)

0.6 0.4 0.2 0 −0.2 −0.4 −0.6

0

20

40

t

60

80

100

Fig. 3. The trivial solution of (4.1) is asymptotically stable, where a = 0.25, b = 0.35, s = 0.5.

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L. Wang, X. Han / Commun Nonlinear Sci Numer Simulat 16 (2011) 3684–3695

1 0.5 z

0 −0.5 −1 1 0.5 0 y

−0.5 −1

−1

−0.5

0

0.5

1

x

Fig. 5. The bifurcating periodic solutions occur when a ¼ 

pffiffi 6 ;b 3

¼ 1; s ¼ p4.

2

z

1 0 −1 −2 2 1 0 −1 y

−2

−2

−1

1

0

2

x

Fig. 6. The bifurcating periodic solutions occur when a ¼ 

pffiffi 6 ;b 2

¼ 1; s ¼ p4.

5. Conclusion In this paper, bidirectional ring network model has been investigated. Linear stability of the system is investigated by analyzing the associated characteristic transcendental equation. By means of space decomposition, we subtly discuss the distribution of zeros of the characteristic equation, and then derive some sufficient conditions ensuring that all the characteristic roots have negative real parts. By regarding the eigenvalues of the connection matrix of the system as bifurcation parameters, we discuss Hopf bifurcation of the equilibria. Meanwhile, with the help of center manifold reduction and normal form theory, we study Hopf bifurcation of the equilibria, and obtain the detailed information about the bifurcation direction and stability of various bifurcated periodic solutions. Finally, numerical simulations have demonstrated the correctness of the theoretical results. Acknowledgement This work is supported by the National Science Foundation of China for Distinguished Young Scholars (10725208). References [1] Huang L, Wu J. Dynamics of inhibitory artificial neural networks with threshold nonlinearity. Fields Ins Commun 2001;29:235–43. [2] Béair J, Campbell SA, Van den Driessche P. Frustration, stability, and delay-induced oscillations in a neural network model. SIAM J Appl Math 1996;56:245–55.

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[3] Driessche PV, Zou X. Global attractivity in delayed Hopfield neural network models. SIAM J Appl Math 1998;58:1878–90. [4] Gopalsamy K, Leung I. Delay induced periocicity in a neural network of exctitation and inhibition. Physica D 1996;89:395–426. [5] Huang L, Wu J. The role of threshold in preventing delay-induced oscillations of frustrated neural networks with McCulloch–Pitts nonlinearity. Int J Math Game Theory Algebra 2001;11(6):71–100. [6] Hale J, Lunel SV. Introduction to functional differential equations. New York: Springer; 1993. [7] Wu J. Introduction to neural dynamics and signal transmission delay. Berlin: Walter de Gruyter; 2001. [8] Hale J. Theory of functional differential equations. New York: Springer; 1997. [9] Hale J, Kocak H. Dynamics and bifurcations. New York: Springer; 1991. [10] Chen Y, Huang Y, Wu J. Desynchronization of large scale delayed neural networks. Proc Amer Math Soc 2000;128:2365–71. [11] Faria T. On a planar system modelling a neuron network with memory. J Diff Equat 2000;168:129–49. [12] Wu J. Symmetric functional differential equations and neural networks with memory. Trans Amer Math Soc 1998;350:4799–838. [13] Guo S, Chen Y, Wu J. Two-parameter bifurcations in a network of two neurons with multiple delays. J Diff Equat 2008;244:444–86. [14] Guo S, Huang L, Wang L. Linear stability and Hopf bifurcation in a two-neuron network with three delays. Int J Bifur Chaos Appl Sci Eng 2004;14:2799–810. [15] Peng J, Guo S. Synchronous dynamics of two coupled oscillators with inhibitory-to-inhibitory connection. Commun Nonlinear Sci Numer Simulat 2010;15:4131–48. [16] Belair J. Stability in a model of a delayed neural network. J Dynam Diff Equat 1993;5:603–23.