Hopf–pitchfork bifurcation in an inertial two-neuron system with time delay

Neurocomputing 97 (2012) 223–232

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Hopf–pitchfork bifurcation in an inertial two-neuron system with time delay Tao Dong a,b,n, Xiaofeng Liao a,nn, Tingwen Huang c, Huaqing Li a a b c

State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 400044, China College of Software and Engineering, Chongqing University of Posts and Telecommunications, Chongqing 400065, China Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatar

a r t i c l e i n f o

abstract

Article history: Received 16 January 2012 Received in revised form 12 April 2012 Accepted 12 June 2012 Communicated by: H. Jiang Available online 5 July 2012

In this paper, we have considered a delayed differential equation modeling two-neuron system with both inertial terms and time delay. By studying the distribution of the eigenvalues of the corresponding transcendental characteristic equation of the linearization of this equation, we derive the critical values where Hopf–pitchfork bifurcation occurs. Then, by computing the normal forms for the system, the bifurcation diagrams are obtained. Furthermore, we find some interesting phenomena, such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasiperiodic motions, which are verified both theoretically and numerically. & 2012 Elsevier B.V. All rights reserved.

Keywords: Hopf–pitchfork bifurcation Inertial Time delay Two-neuron system

1. Introduction In the past two decades, a large number of delayed neural networks have been proposed to solve various engineering problems [1–3]. The design for such neural-network-based computational systems entails a profound understanding of the dynamics [4–6,27]. In general, the study of existence and stability of periodic solutions in delayed neural networks is not only of theoretical significance but also has important applications such as dynamically associative memories and so on. Consequently, the appearance of a cycle bifurcating from equilibrium with a single parameter, which is known as a Hopf bifurcation, has received considerable attention [7–10]. So far, the existing bifurcation analysis for the delayed neural networks with fixed connection topology has been carried out by regarding the time delay as bifurcation parameter. On the other hand, realistic modeling of networks inevitably requires variation of the connection topology, and furthermore in this case, a wide range of different behaviors can be produced by varying the coupling strength. At the same time, network structure has important implications for neural networks, since synaptic coupling can change through learning. In this paper, the motivation of our

n Corresponding author at: State Key Laboratory of Power Transmission Equipment and System Security, College of Computer Science, Chongqing University, Chongqing 400044, China. Tel.: þ 86 23 65103199. nn Corresponding author. E-mail addresses: [email protected] (T. Dong), xfl[email protected] (X. Liao).

0925-2312/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.neucom.2012.06.008

study is to regard both connection topology and time delay as the bifurcation parameters. In reality, a system may have two independent parameters, and the occurrence of bifurcation in such a network may be a codimension (codim) two bifurcation. So far, few articles deal with the codim two bifurcations such as Hopf–pitchfork bifurcation, double Hopf bifurcation and Bogdanov–Takens singularity etc. [11–14,16]. Particularly, to the best of the authors’ knowledge, there are seldom works on Hopf–pitchfork bifurcation of delayed neural networks. Moreover, there are only a few articles on Hopf–pitchfork bifurcation in delay differential equations [11,15]. Since networks with one or two neurons are prototypes to understand the dynamics of large-scale networks, in this paper, similar to that in [7,9], we consider a class of inertial two-neuron system with time delay, which are of the form ( n€ 1 ðtÞ ¼ n_ 1 ðtÞbn1 ðtÞ þ a1 f 1 ½n2 ðttÞ , ð1:1Þ n€ 2 ðtÞ ¼ n_ 2 ðtÞbn2 ðtÞ þ a2 f 2 ½n1 ðttÞ where b, ai(i¼1,2) are the connection topology and t is time delay. f i : R-R ði ¼ 1,2Þ are continuous activation function. For this model, Liu and Liao studied the dynamical behavior and chaos of the system (1.1) when f i ðnÞ ¼ tanhðnÞ [7,9]. However, in this paper, we will derive a sufficient condition for a Hopf–pitchfork bifurcation of system (1.1) based on normal form theory [19–22]. In the Hopf– pitchfork bifurcation, the characteristic equation of the linearization of system (1.1) at the equilibrium point has a zero root and a pair of purely imaginary root under certain condition. By setting the proper parameters, system (1.1) exhibits codimension two singularity when

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T. Dong et al. / Neurocomputing 97 (2012) 223–232

two parameters vary in a neighborhood of the critical values. Using the normal form method and the center manifold theorem [21], we obtain the normal form for system (1.1) and study its dynamical behaviors. It is shown that the system (1.1) can display very interesting bifurcation phenomena such as the coexistence of two asymptotically stable states, two stable periodic orbits, and two attractive quasi-periodic motions. The remainder of this paper is organized as follows. In Section 2, we analyze the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, and obtain the critical values for Hopf–pitchfork bifurcation. In Section 3, we perform the center manifold reduction and normal form computation, and the normal forms with the Hopf–pitchfork for the inertial two-neuron system with time delay are derived. In Section 4, we carry out the bifurcation analysis and get our main results. In Section 5, the numerical simulation results are shown to demonstrate the theoretical predictions. Finally, conclusions are stated in Section 6.

2. The singularity and the analysis of eigenvalues In order to investigate the dynamic behavior of the inertial twoneuron system, it is more convenient to work with the equations in an abstract form. This can be done by introducing x¼ [x1,x2,x3,x4]T and x(t t)¼ [x1(t t), x2(t t), x3(t t), x4(t t)]T into (1.1). Then the differential equations (1.1) can be rewritten as follows: 8 x_ 1 ðtÞ ¼ x2 ðtÞ, > > > > < x_ 2 ðtÞ ¼ bx1 ðtÞx2 ðtÞ þa1 f 1 ½x3 ðttÞ, ð2:1Þ x_ 3 ðtÞ ¼ x4 ðtÞ, > > > > : x_ 4 ðtÞ ¼ bx3 ðtÞx4 ðtÞ þa2 f ½x1 ðttÞ: 2

(iii) if b o0, l ¼0 is a double root to Eq. (2.4)when t ¼  1/b. Proof. Clearly, l ¼0 is a root to Eq. (2.4) if and only if b2 ¼ Z2. Substituting b2 ¼ Z2 into D(l,t) and taking the derivative with respect to l gives dDðl, tÞ 3 2 ¼ 4l þ 6l þ 2ð1 þ2bÞl þ 2b þ 2tZ2 e2lt , dl

ð2:5Þ

then dDð0, tÞ=dl ¼ 2b þ 2tZ2 :

ð2:6Þ

For any t 40, if b 40, it is easy to see that dD(0,t)/dl 40. If b o0, this implies that dD(0,t)/dl ¼0 with b2 ¼ Z2 if and only if t ¼  1/b, and hence the conclusion of (i) and (ii) follows. From Eq. (2.6), it follows that 2

d Dðl, tÞ 2

dl

2

¼ 12l þ 12l þ2ð1 þ 2bÞ4t2 Z2 e2lt ,

ð2:7Þ

which implies that if b o0, d2D(0,t)/dl2 a0, and the conclusion of (iii) follows. This completes the proof. & From (i) and (ii) of Lemma 1, we can see that system (2.1) undergoes a stationary bifurcation at the origin when (H4) holds. Considering that the origin is always an equilibrium of system (2.1), therefore, the stationary bifurcation occurs in system (2.1) when b2 ¼ Z2 and t a  1/b, which is either pitchfork bifurcation or transcritical bifurcation. In the following, we will show that this stationary bifurcation is pitchfork bifurcation. Theorem 1. Assume that hold, then system (2.1) undergoes a pitchfork bifurcation at the origin when (H4) and t a 1/b are satisfied. More precisely, system (2.1) has a unique zero equilibrium as b2 Z Z2, and three equilibriums among which one is zero and the others are nontrivial as b2 o Z2. 00

To establish our main results, it is necessary to make the following assumptions:

Proof. It follows easily from xf k ðxÞ o 0 for x a0 that we take the 0 maximum value at f k ðxÞ at x ¼0, that is

(H1). f k ðxÞ A C 2 ðR,RÞ, f k ðxÞ Z 0 and there exists a constant L40 such that 9fk(x)9rL, k¼1,2.

f k ð0Þ ¼ maxf k ðxÞ

0

0

00

00

(H2). xfk(x)40, xf k ðxÞ o0 for xa0, and fk(0) ¼0, f k ð0Þ ¼ 0, k¼ 1,2. qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 0 0 0 (H3). Z ¼ a1 a2 f 1 ðxÞf 2 ðxÞ, a1 a2 f 1 ðxÞf 2 ðxÞ 40. (H4). b ¼ 7 Z. From Assumption (H2), we can easily see that fk(0) ¼0(k¼1,2), therefore origin is always the equilibrium of system (2.1). Linearizing system (2.1) at the origin yields the following linear system: 8 x_ 1 ðtÞ ¼ x2 ðtÞ > > > > < x_ 2 ðtÞ ¼ bx1 ðtÞx2 ðtÞ þa1 k1 x3 ðttÞ ð2:2Þ , x_ 3 ðtÞ ¼ x4 ðtÞ > > > > : x_ 4 ðtÞ ¼ bx3 ðtÞx4 ðtÞ þa2 k2 x1 ðttÞ 0

0

where k1 ¼ f 1 ð0Þ,k2 ¼ f 2 ð0Þ. The system (2.2) is 0 l 1 0 B b l þ1 a1 k1 elt B detB B0 l 0 @ b a2 k2 elt 0

characteristic equation for 0

1

C C C¼0 1 C A lþ1 0

Lemma 1. Suppose that (H2) is satisfied. Then (i) if b 40, l ¼0 is a single root to Eq. (2.4). (ii) if b o0, l ¼0 is a single root to Eq. (2.4) when t a  1/b.

ð2:8Þ

Considering that the equilibrium of system (2.1) are the roots of the following equations: 8 x ðtÞ ¼ 0, > > > 2 > < bx1 ðtÞx2 ðtÞ þa1 f 1 ½x3 ðttÞ ¼ 0, ð2:9Þ x ðtÞ ¼ 0, > > > 4 > : bx3 ðtÞx4 ðtÞ þa2 f ½x1 ðttÞ ¼ 0: 2 From (2.9), we observe that x1 satisfies   a1 a2 f 2 ðx1 Þ ¼ 0 x1  f 1

ð2:10Þ

Let h(x)¼x  (a1/b)f1((a2/b)f2(x)), then   a1 a2 0 a2 0 0 f 2 ðxÞ f 2 ðxÞ h ðxÞ ¼ 1 2 f 1

ð2:11Þ

b

b

b

b

Obviously, h0 (0)¼1  Z2/b2, suppose b2 Z Z2, we can get h0 (0) Z0. (i) If a1a2 Z0, we can get

ð2:3Þ

Hence, the following four-order exponential polynomial equation is obtained:

Dðl, tÞ :¼ ðl2 þ l þ bZelt Þðl2 þ l þ b þ Zelt Þ ¼ 0

0

ð2:4Þ

0

h ðxÞ Z1

Z2 b2

Z0

ð2:12Þ

(ii) If a1a2 o0, it is easy to see that h0 (x)Z0 Therefore, h0 (x)Z0 means h(x) is strictly monotonically increasing with R, we can find that (2.8) has only a zero equilibrium. In the following, we consider the case b2 o Z2. In this case, we observe that h0 (0) o0 and therefore there exists a neighborhood N(0) of x¼0 such that h0 (x)o0 for any xAN(0). Noting that h(0)¼0 we have h(x) o0 when xAN(0) and x4 0.

T. Dong et al. / Neurocomputing 97 (2012) 223–232

In addition, from (H1), we can see lim hðxÞ ¼ þ1. Thus, there x-1 exists x0 40 such that h(x0)¼0 for xA(0,x0). Let H(x) be defined by   a1 a2 HðxÞ ¼ x þ x0  f 1 f 2 ðx þx0 Þ ð2:13Þ

b

b

then H(0) ¼0 H0 ðxÞ ¼ 1

a1 a2

b2

0

f1



a2

b

 0 f 2 ðx þx0 Þ f 2 ðx þ x0 Þ

ð2:14Þ

Pðl,elt ,. . .,eltm Þ on the open right half plane can change only if a zero appears on or crosses the imaginary axis. Lemma 4. All the roots of Eq. (2.4), except the zero root, have negative real parts when b ¼ Z and 0 o t o t0. Proof. Let b ¼ Z, for t ¼0, the Eq. (2.4) can be rewritten in the following form: 2

2

ðl þ lÞðl þ l þ 2ZÞ ¼ 0

From (H1) and (H2), we know that for xZ0,     0 a2 0 a2 f 2 ðx þ x0 rf 1 f 2 ðx0 Þ f1

b

ð2:15Þ

b

Considering that H0 (0) ¼h0 (x0) Z0 and using the same arguments as (i) and (ii), we can easily show that H0 (x)Z0 for xZ0, that is, h0 (x)Z0 for x Zx0. Thus we show that the equation h(x) ¼0 has only one solution for (0, þN). From (H1) and (H2), we see that (2.6) has only one solution. Similarly, one can prove that (2.9) also has only one solution such that the first component is less than zero. This completes the proof. & In the following, we consider the case that Eq. (2.4) not only has a zero root, but also has a pair of purely imaginary root 7io, when (H4) holds. If io(o 40) is a root of Eq. (2.4) with b2 ¼ Z2, rewriting the characteristic Eq. (2.4) according to its real and imaginary parts gives ( o40 ð2b þ 1Þo20 þ b2 ¼ Z2 cosð2t0 o0 Þ, ð2:16Þ 2o30 þ2bo0 ¼ Z2 sinð2t0 o0 Þ: By squaring and adding the above equations, it follows that

o80 þ b1 o60 þ b2 o40 þ b3 o20 þ b4 ¼ 0,

ð2:17Þ 2

2

3

2

where b1 ¼2(1  2b), b2 ¼ (1 2b) þ2b , b3 ¼  4b þ2b , b4 ¼ b4  Z4. Let z ¼ o20 , then (2.17) becomes ð2:18Þ z3 þ b1 z2 þ b2 z þ b3 ¼ 0: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 * * 3 * 2 * n Let z ¼ 1=3ðb1 þ b1 3b2 Þ, h(z ) ¼(z ) þb1(z ) þb2z þ b3, then we have the following results (see [23] for details) about the distributions of the positive roots of (2.18) Lemma 2. [23]. (i) If b3 o0, then (2.18) has at least one positive root; 2 (ii) If b3 Z0 and b1 3b2 r 0 , then (2.18) has no positive root; 2 (iii) If b3 Z0 and b1 3b2 4 0 , then (2.18) has positive roots if and only if z* 40 and h(z*) r0.

ð2:20Þ pffiffiffiffiffiffiffiffiffiffiffiffiffi The roots of (2.20) are l1 ¼ 0, l2 ¼ 1, l3,4 ¼ 1 7 18Z=2. It is easy to see that all the roots of Eq. (2.20), except the zero root, have negative real parts. Using Lemma 3 and similar to the approach in [24,25], we complete the proof. & Summarizing the discussions above, we have the following theorem. Theorem 2. Suppose hold and from Lemmas 2–4, we have the following results: (1) For any t 40, if t a 1/b is satisfied, then system (2.1) undergoes a pitchfork bifurcation at origin. (2) For any t 40, if t ¼ t0 a  1/b and either b3 o0 or b3 Z0 and 2 b1 3b2 4 0 , z* 40, h(z*)r0 are satisfied, then system (2.1) undergoes a Hopf–Pitchfork bifurcation at origin.

3. Hopf–pitchfork bifurcation In this section, we give the normal form of Hopf–Pitchfork bifurcation. We only consider the case b 40, and the case b o0 is similar. Rescaling the time by t/t=t to normalize the delay, and expanding the function fk(x)(k¼1,2) in Eq. (2.1), we get 8 x_ 1 ðtÞ ¼ tx2 ðtÞ, > > > > 3 < x_ 2 ðtÞ ¼ tðbx1 ðtÞx2 ðtÞ þa1 ðk1 x3 ðt1Þ þ 1 f 001 ð0Þx2 ðt1Þ þ 1 f 000 3 2 6 1 ð0Þx3 ðt1Þ þ h:o:tÞÞ, _ ðtÞ ¼ t x ðtÞ, x > 3 4 > > > : x_ 4 ðtÞ ¼ tðbx3 ðtÞx4 ðtÞ þa2 ðk2 x1 ðt1Þ þ 1 f 000 ð0Þx2 ðt1Þ þ 1 f 000 ð0Þx3 ðt1Þ þh:o:tÞÞ: 2 2

1 2o0

arccos

o40 ð2b þ 1Þo20 þ b2 , Z2

where t0 ¼ t0k0 ¼ min

k ¼ 1,2,3

n

ð2:19Þ

o

t0k0 :

Lemma 3. Consider the exponential polynomial [26] lt

Pðl,e

ltm

,. . .,e

n

n1 ð0Þ Þ ¼ l þ pð0Þ þ    þpn1 l þpnð0Þ 1 l ð1Þ n1 ð1Þ lt þ ½p1 l þ    þ pn1 l þ pð1Þ n e n1 ðmÞ ltm þ    þ ½pðmÞ þ    þ pn1 l þ pðmÞ , n e 1 l

(j ¼1,2,y,m) are constants. As where ti Z0 (i¼ 1,2,....,m) and pðiÞ j (t1,t2,y,tm) vary, the sum of the order of the zeros of

6 2

1

1

ð3:1Þ Near (Z,t0), we rewrite b and t as b ¼ Z þ m1 and t ¼ t0 þ m2, make Taylor expansion of the fi(x), then Eq. (3.1) becomes 8 x_ 1 ðtÞ ¼ ðt0 þ m2 Þx2 ðtÞ, > > > > > _ 2 ðtÞ ¼ ðt0 þ m2 ÞððZ þ m1 Þx1 ðtÞx2 ðtÞ þa1 ðk1 x3 ðt1Þ x > > > 1 00 > 3 < þ f 1 ð0Þx2 ðt1Þ þ 1 f 000 3 2 6 1 ð0Þx3 ðt1Þ þh:o:tÞÞ, ð3:2Þ : _ ðtÞ ¼ ð t þ m Þx ðtÞ, x > 3 0 2 4 > > > > x_ 4 ðtÞ ¼ ðt0 þ m2 ÞððZ þ m1 Þx3 ðtÞx4 ðtÞ þa2 ðk2 x1 ðt1Þ > > > > : þ 1 f 00 ð0Þx2 ðt1Þ þ 1 f 000 ð0Þx3 ðt1Þ þh:o:tÞÞ: 2 2

Suppose (2.18) has positive roots, without loss of generality, assume that (2.18) has three positive roots z1,z2,z3, and it is easy to pffiffiffiffiffi see that ok ¼ zk , k¼1, 2, 3 are the three positive roots. From (2.16), we get

t0 ¼

225

6 2

1

1

The linearization of (3.2) at origin is 8 x_ 1 ðtÞ ¼ t0 x2 ðtÞ, > > > > < x_ 2 ðtÞ ¼ t0 ðZx1 ðtÞx2 ðtÞ þ a1 k1 x3 ðt1ÞÞ, x_ 3 ðtÞ ¼ t0 x4 ðtÞ, > > > > : x_ 4 ðtÞ ¼ t0 ðZx3 ðtÞx4 ðtÞ þ a2 k2 x1 ðt1ÞÞ:

ð3:3Þ

Let

ZðyÞ ¼ AdðyÞ þBdðy þ 1Þ, where 2

0

6 6 Z A ¼ t0 6 60 4 0

1

0

1

0

0

0

0

Z

ð3:4Þ

0

3

7 0 7 7, 1 7 5

1

2

0

6 60 B ¼ t0 6 60 4 a2 k2

0

0

0

a1 k1

0

0

0

0

0

3

7 07 7: 07 5 0

226

T. Dong et al. / Neurocomputing 97 (2012) 223–232

Let C ¼C([ 1,0], C2) and define a linear operator L on C as follows: Z 0 Lm j ¼ dZðyÞjðyÞ, j A C: ð3:5Þ 1 T

1

2

3

As a result, we obtain

b1 ¼

Let X¼(x1,x2,x3,x4) and let F(xt,m)¼(F ,F ,F ,F ) where

,

b2 ¼ b3 io0 b3 , b3 ¼

4 T

1 1io0

ð3:15Þ

Zb1 io0 it0 o0 e : a2 k2

Suppose c2(y) is an eigenvector of A* corresponding to 0. According to the definition of A*, we have that c2(y) is a constant vector (1,u1,u2,u3) such that

1

F ¼ m2 x2 , F 2 ¼ ðt0 m1 þ Zm2 Þx1 ðtÞm1 m2 x1 ðtÞm2 x2 ðtÞ þa1 k1 m2 x3 ðt1Þ   1 00 1 000 f 1 ð0Þx23 ðt1Þ þ f 1 ð0Þx33 ðt1Þ þ h:o:t , þ a1 ðt0 þ m2 Þ 2 6

ð1,u1 ,u2 ,u3 ÞðA þ BÞ ¼ 0:

ð3:16Þ

As a result, we obtain

Z

Z

F 3 ¼ m2 x4 ,

u1 ¼ 1, u2 ¼

F 4 ¼ ðt0 m1 þ Zm2 Þx3 ðtÞm1 m2 x3 ðtÞm2 x4 ðtÞ þa2 k2 m2 x1 ðt1Þ   1 00 1 000 f 2 ð0Þx21 ðt1Þ þ f 2 ð0Þx31 ðt1Þ þ h:o:t þ a2 ðt0 þ m2 Þ 2 6

It is easy toD checkE /c2,f1S¼0, /c1,f2S¼ 0. In order to guarantee that c1 , j1 ¼ 1 and /c2,f2S¼1, we have to determine factors D1 and D. Since D E h c1 , j1 ¼ D ð1, b1 , b2 , b3 Þð1, a1 , a2 , a3 ÞT # Z

Then Eq. (3.2) can be transformed into X_ ðtÞ ¼ Lm ðX t Þ þFðX t , mÞ:

ð3:6Þ

0

þ



ð3:9Þ

Z

0

# ð1, s1 , s2 , s3 ÞBð1,u1 ,u2 ,u3 ÞT de

1

Hence, we obtain D¼

2*

s A ½1,0Þ, ð3:10Þ

s ¼ 0:

1 ð1 þ a1 b1 þ a2 b2 þ a3 b3 Þt0 eit0 o0 ða1 k1 a2 b1 þa2 k2 b3 Þ

D1 ¼

1 : ð1 þ s1 u1 þ s2 u2 þ s3 u3 Þt0 ða2 k2 s3 þa1 k1 s1 u2 Þ

,

ð3:20Þ

ð3:21Þ

By the continuous projection p, we can decompose the enlarged phase space by L ¼ {0,io0t0,  io0t0} as BC¼PKerp. Let u ¼ fxþy, then Eq. (3.6) can be decomposed as x_ ¼ Jx þ cð0ÞFðjx þ y, mÞ, y_ ¼ AQ 1 y þ ðIpÞX 0 Fðjx þy, mÞ,

ð3:22Þ

where J ¼ diagðio0 t0 ,io0 t0 ,0Þ y A Q 1 :¼ Q \ C 1  Kerp. AQ 1 is the restriction of an operator A from Q1 to the Banach space Kerp. Neglecting higher order terms with respect to parameters m1 and m2, Eq. (3.22) can be written as

As a result, we obtain

a1 ¼ io0 , o20 io0 ðZo20 Þ it0 o0 e :

c2 , j2 ¼ D1 ð1, s1 , s2 , s3 Þð1,u1 ,u2 ,u3 ÞT þ

ð3:19Þ

Aj1 ð0Þ þBj1 ð1Þ ¼ io0 t0 j1 ð0Þ:

a3 ¼



"

ð3:8Þ

As we know that A and A* have eigenvalues 0 and 7io0t0. Now we need to compute their corresponding eigenvectors. Let j ¼ ðj1 , j1 , j2 Þ, suppose j1 ðyÞ ¼ ð1, a1 , a2 , a3 ÞT eio0 t0 y is an eigenvector of A corresponding to io0t0. Then Af ¼io0t0f, it follows from the definition of A that

ðZo20 Þ it0 o0 e , a1 k1

ð1, b1 , b2 , b3 Þeio0 t0 ðe þ 1Þ Bð1, a1 , a2 , a3 ÞT eio0 t0 e de

¼ D1 ðð1 þ s1 u1 þ s2 u2 þ s3 u3 Þt0 ða2 k2 s3 þ a1 k1 s1 u2 ÞÞ:

and its adjoint A :C ([0,1],C )-BC 8 _,
a2 ¼

ð3:17Þ

ð3:18Þ

0

The infinitesimal generator A:C  1-BC is ( j_ , y A ½1,0Þ, _ þX 0 ½Ljj _ ð0Þ ¼ R 0 Aj ¼ j d Z ðtÞ j ðtÞ, y ¼ 0, 1 1

:

ð3:7Þ

It is easy to show that the bilinear form can be written as Z 0Z y   c, j ¼ cð0ÞUjð0Þ cðxyÞdZðyÞjðxÞdx:

*

a2 k2

¼ Dð1 þ a1 b1 þ a2 b2 þ a3 b3 t0 eit0 o0 ða1 k1 a2 b1 þ a2 k2 b3 ÞÞ,

1 1 F 2 ðj, mÞ þ F 3 ðj, mÞ þ h:o:t: 2 3!

1

, u3 ¼

1

Write the Taylor expansion of F as Fðj, mÞ ¼

a2 k2

ð3:11Þ

a1 k1

Suppose f2(y) is an eigenvector of A corresponding to 0. According to the definition of A, we have that f2(y) is a constant vector (1,s1,s2,s3)T such that

1 1 1 1 f ðx,y, mÞ þ f 3 ðx,y, mÞ þ h:o:t, 2 2 3! 1 2 1 2 y_ ¼ AQ 1 y þ f 2 ðx,y, mÞ þ f 3 ðx,y, mÞ þh:o:t:, 2 3! x_ ¼ Jx þ

ð3:23Þ

where 1

f j ðx,y, mÞ ¼ cð0ÞFðjx þ y, mÞ, 2

ðAþ BÞð1, s1 , s2 , s3 ÞT ¼ 0:

ð3:12Þ

f j ðx,y, mÞ ¼ ðIpÞX 0 Fðjx þ y, mÞ: On the center manifold, Eq. (3.23) can be transformed as the following normal form:

As a result, we obtain

s1 ¼ Z, s2 ¼ 0, s3 ¼ a2 k2 :

ð3:13Þ io0 t0 s

x_ ¼ Jx þ

1 1 1 g ðx,0, mÞ þ g 13 ðx,0, mÞ þ h:o:t: 2 2 3!

ð3:24Þ

Let c ¼ ðc1 , c1 , c2 Þ, suppose c1 ðyÞ ¼ ð1, b1 , b2 , b3 Þe is an eigenvector of A* corresponding to io0t0. Then A*c ¼  io0t0c, it follows from the definition of A* that

First we compute g 12 ðx,0, mÞ. Let M2 denote the operator defined 4 in V 52 ðcomplexes;  ker pÞ, with

c1 Að0Þ þ c1 Bð1Þ ¼ io0 t0 j1 ð0Þ:

M12 : V 52 ðcomplexes; Þ/V 52 ðcomplexes; Þ,

ð3:14Þ

4

4

T. Dong et al. / Neurocomputing 97 (2012) 223–232

and ðM12 pÞðx,

mÞ ¼ Dz pðx, mÞBzBpðx, mÞ, 4

where V 52 ðcomplexes; Þ denotes the linear space of the second order homogeneous polynomials in five variables (x1,x2,x3,m1,m2), and with coefficients in C4. If we do not consider the strong resonant cases, it is easy to check that one may choose the decomposition: 4

V 52 ðcomplexes; Þ ¼ ImðM 12 Þ  KerðM 12 ÞC , with complementary space 80 1 10 10 10 10 0 0 x1 x3 0 > < mi x1 C CB CB B CB CB KerðM 12 ÞC ¼ span @ 0 A, @ 0 A, @ mi x2 A, @ x2 x3 A, @ 0 A, > : m x 0 0 0 0 i 3 0 10 10 10 10 19 0 0 0 0 0 > = B0CB 0 CB 0 CB 0 CB 0 C @ A, @ A, @ A, @ A, @ A : > x23 m21 m22 m1 m2 ; x1 x2 g 12

is the function giving the quadratic terms in Then, we can get (x,m) for y¼0, and is determined by g 12 ðx,0, mÞ ¼ ProjðkerðM1 ÞC Þ 2 1 1 f 2 ðx,0, mÞ, where f 2 ðx,0, mÞ is the function giving the quadratic terms in (x,m) for y¼0. It follows that 1 1 1 g ðx,0, mÞ ¼ ProjðkerðM1 ÞC Þ f 2 ðx,0, mÞ 2 2 2 0 1 ða11 m1 þa12 m2 Þx1 þ a13 x1 x3 B C ða11 m1 þa12 m2 Þx2 þ a13 x2 x3 ¼@ A, ða21 m1 þ a22 m2 Þx3 þ a23 x1 x2 þa24 x23

is the third order terms of the equation which is obtained after computing the second order terms of the normal form, and 1 U 12 ðx,0Þ is the solution of equation M 12 U 12 ðx,0Þ ¼ f 2 ðx,0,0Þ and 1 2 3 h¼(h ,h ,h ) is a second order homogeneous polynomial in (x1,x2,x3,m1,m2) with coefficients in Q1. 00 From (H2), we knowf k ðxÞ ¼ 0, ðk ¼ 1,2Þ, we can easily com1 pute that U 12 ðx,0Þ ¼ 0. Therefore, we have ðDx f 2 ÞU 12 ðDx U 12 Þg 12 þ 1 1 ðDx f 2 Þh ¼ 0 and thus we need to only compute f 3 ðx,0,0Þ. It follows that 1 1 f ðx,0,0Þ 6 3 2

3 000 000 a1 Db1 f 1 ð0Þða2 eio0 t0 x1 þ a2 eio0 t0 x2 þ s2 x3 Þ3 þ a2 Db3 f 1 ð0Þðeio0 t0 x1 þ eio0 t0 x2 þ x3 Þ3 1 6 000 000 3 3 7 io0 t0 io0 t0 io0 t0 io0 t0 x1 þ a2 e x2 þ s2 x3 Þ þ a2 Db3 f 1 ð0Þðe x1 þ e x2 þ x3 Þ 5, ¼ t0 4 a1 Db1 f 1 ð0Þða2 e 6 000 000 a1 D1 u1 f 1 ð0Þða2 eio0 t0 x1 þ a2 eio0 t0 x2 þ s2 x3 Þ3 þ a2 D1 u3 f 1 ð0Þðeio0 t0 x1 þ eio0 t0 x2 þx3 Þ3

Thus, one can obtain that 2

b11 x21 x2 þ b12 x1 x23

3

6 7 1 1 1 1 2 27 g ðx,0,0Þ ¼ projs2 f 3 ðx,0,0Þ ¼ 6 4 b11 x1 x2 þ b12 x2 x3 5, 6 3 6 b21 x1 x2 x3 þb22 x33

ð3:26Þ

where 1 t0 Deio0 t0 ða1 b1 a22 a2 f 0001 ð0Þ þ a2 b3 f 0002 ð0ÞÞ, 2 1 000 000 b12 ¼ t0 Deio0 t0 ða1 b1 a2 s22 f 2 ð0Þ þa2 b3 f 2 ð0ÞÞ, 2 000 000 b21 ¼ t0 D1 ða1 a2 a2 s2 u1 f 1 ð0Þ þ a2 u3 f 2 ð0ÞÞ,

b11 ¼

ð3:25Þ

where

227

b22 ¼

1 t0 D1 ða1 s32 u1 f 0001 ð0Þ þ a2 u3 f 0002 ð0ÞÞ: 6

Then, we can write Eq. (3.24) as

a11 ¼ t0 Dðb1 þ b3 a2 Þ, a12 ¼ Dða1 þ b2 a3 b1 ðZ þ a1 a1 k1 a2 eio0 t0 Þ b3 ðZa2 þ a3 a2 k2 eio0 t0 ÞÞ, a13 ¼ 0,a21 ¼ t0 D1 ðu1 þ s2 u3 Þ, a22 ¼ D1 ðs1 þ u2 s3 u1 ðZ þ s1 a1 k1 s2 Þu3 ðZs2 þ s3 a2 k2 ÞÞ, a23 ¼ a24 ¼ 0: To find the third-order normal form, let M3 denote the 4 operator defined in V 33 ðcomplexes;  kerpÞ, with M 13 : V 33 ðC 4 Þ/ 4 3 V 3 ðcomplexes; Þ, and ðM13 pÞðx, mÞ ¼ Dz pðx, mÞBzBpðx, mÞ: 4

where V 33 ðcomplexes; Þ denotes the linear space of the second order homogeneous polynomials in three variables (x1,x2,x3), and with coefficients in C3. Then, it is easy to check that one may choose the decomposition:

x_ 1 ¼ io0 t0 x1 þ ða11 m1 þ a12 m2 Þx1 þ b11 x21 x2 þb12 x1 x23 þh:o:t, x_ 2 ¼ io0 t0 x2 þ ða11 m1 þ a12 m2 Þx2 þ b11 x1 x22 þb12 x2 x23 þh:o:t, x_ 3 ¼ ða21 m1 þ a22 m2 Þx3 þ a24 x23 þ b21 x1 x2 x3 þ b22 x33 þ h:o:t:

Table 1 The twelve types of unfolding [18]. Case

Ia

Ib

II

III

IVa

IVb

V

VIa

VIb

VIIa

VIIb

VIII

d b c d  bc

þ1 þ þ þ

þ1 þ þ 

þ1 þ  

þ1   þ

þ1   þ

þ1   

1 þ þ 

1 þ  þ

1 þ  

1  þ 

1  þ 

1   

4

V 33 ðcomplexes; Þ ¼ ImðM 13 Þ  kerðM 13 ÞC , with complementary space 80 1 10 10 2 0 x1 x23 > < x1 x2 C B CB CB s2 :¼ span @ 0 A, @ 0 A, @ x1 x22 A, > : 0 0 0 19 0 10 10 0 0 0 > = C B CB CB @ x2 x23 A, @ 0 A, @ 0 A , > x33 x1 x2 x3 ; 0

L2 L3

L0

D6

D4

H0 L1

µ2

D5

D3

D7

D2 D1

µ1 D9

then we have

ð3:27Þ

D8

1 1 1 1 1 1 g ðx,0,0Þ ¼ projðkerðM1 ÞC Þ f 3 ðx,0, mÞ ¼ projs2 f 3 ðx,0, mÞ þ h:o:t, 3 6 3 6 6

where 1

1

f 3 ðx,0, mÞ ¼ f 3 ðx,0,0Þ þ

i h i 3h 1 1 ðDx f 2 ÞU 12 Dx U 12 g 12 þ ðDx f 2 Þh , ðx,0,0Þ ðx,0,0Þ 2

Fig. 1. The bifurcation set and phase portraits in the case VIa.

228

T. Dong et al. / Neurocomputing 97 (2012) 223–232

4. Bifurcation analysis

where

r1 ðmÞ ¼ Re½a11 m1 þRe½a12 m2 , b30 ¼ Re½b11 , b12 ¼ Re½b12 , r2 ðmÞ ¼ a21 m1 , g21 ¼ b21 , g03 ¼ b22 :

Make changes to variables x1 ¼ o1  io2, x2 ¼ o1 þio2, x3 ¼ o3, and then a change to cylindrical coordinates according to o1 ¼ g cosx, o2 ¼ g sinx, o3 ¼ z, Eq. (3.27) becomes

Since the third equation describes a rotation around the z-axis, it is irrelevant to our discussion and we will omit it. Hence we obtain a system in the plane (r,z), up to the third order

2

3

r_ ¼ r1 ðmÞr þ b30 r þ b12 r z þ h:o:t,

z_ ¼ r2 ðmÞz þ g21 r 2 z þ g03 z3 þ h:o:t,

2 r_ ¼ r1 ðmÞr þ b30 r 3 þ b12 r z þh:o:t,

x_ ¼ o0 þ ðImða11 Þm1 þImða12 Þm2 Þz þ h:o:t,

z_ ¼ r2 ðmÞz þ g21 r2 z þ g03 z3 þ h:o:t:

ð4:1Þ

ð4:2Þ

0.1

0.05

0.1 0.05 x1(t)

x2(t)

0

−0.05

0 −0.05 −0.1 0.1

−0.1

0.05 0

−0.15 0

100

200

300

400

500

−0.05

600

−0.1

x3(t)

t

−0.15

−0.05

−0.1

0.1

0.05

0

x2(t)

Fig. 2. The stable trivial equilibrium: (a) waveform diagram for variable x3(t) and (b) phase diagram for variable x1(t), x2(t), x3(t).

0.4

0.2

0.2

0.1 x1(t)

x1(t)

0.3 0.4

0

0 −0.1

−0.2

−0.2

−0.4 0.01

−0.3

0.4 0.005

0.2

x2(t)

0

0 −0.005

−0.4 0.4

x3(t)

−0.2 −0.01

0.2

0.4 0

x2(t)

−0.2

−0.4

−0.4

200

250

−0.2

0.2

0 x3(t)

−0.4

0.4

0.4

0.3 0.2

0

0.1 x2(t)

x1(t)

0.2

−0.2

0 −0.1

−0.4 0.4

−0.2 0.2

0.4

x3(t)

0.2

0

0

−0.2

−0.2 −0.4

−0.4

−0.3

x4(t) −0.4 0

50

100

150

300

350

400

t

Fig. 3. (a) Two asymptotically stable nontrivial equilibria coexist. (b), (c) and (d) the bifurcation occurs from the trivial equilibrium.

450

500

T. Dong et al. / Neurocomputing 97 (2012) 223–232

qffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi Let r 1 ¼ b30 r and r 2 ¼ g03 z, we obtain the following: 2 2 1 þ r 1 þ br 2 Þ, 2 2 2 þ cr 1 þ dr 2 Þ,

r 1 ¼ r 1 ðe r 2 ¼ r 2 ðe

ð4:3Þ

where e1 ¼ r1, b¼ b12/9g039, e2 ¼ r2, c¼ g21/9b309, d ¼ 71. We note that (r1,r2)¼(0,0) is always an equilibrium and that up to three other equilibriums can appear, as follows: pffiffiffiffiffiffiffiffiffi ðr 1 ,r 2 Þ ¼ ð e1 ,0Þ, for e1 o0 ðr 1 ,r 2 Þ ¼ ð0,

ðr 1 ,r 2 Þ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffi e2 =dÞ, for e2 d o0

ð4:4Þ

rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! be2 de1 ce2 e1 be2 de1 ce2 e1 , for , 40 A A A A

where A¼d  bc. Based on [12], we know that pitchfork bifurcations occur from an equilibrium (0,0) on the line L0 : m2 ¼ Re½a12 =Re½a11 m1 , and also that pitchfork bifurcations occur from an equilibrium pffiffiffiffiffiffiffiffiffi ð e1 ,0Þ, on the line L1 : m2 ¼ 1=Re½a12 ðða21 =cÞRe½a11 Þm1 , and pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi from an equilibrium ð be2 de1 =A, ce2 e1 =AÞ on the line L2 : m2 ¼ 1=Re½a12 ðb=dÞa21 Re½a11 Þm1 . Using the conditions for existence of this fixed point from (4.4), we find that Hopf bifurcations can occur only on the line L3 : m2 ¼ 1=Re½a12 ðbd=dð1cÞa21 Re½a11 Þm1 when A 40. On the curve H0 : m2 ¼ 1=Re½a12 ððbd=dð1cÞÞa21 Re½a11 Þm1 þ Oðm21 Þ

(see [12]), the system undergoes a saddle connection bifurcation, i.e., there is a pair of symmetric heteroclinic orbits connecting the two nontrivial saddle points. From [18], we know by the different signs of b,c,d in Table 1, Eq. (4.2) has twelve different types of unfolding, which mean twelve essentially distinct types of phase portraits and bifurcation diagrams. It is easy to see that Hopf bifurcation can occur in the case VIa and VIIa, since for such bifurcations, the line L3 must lie between the slopes of the pitchfork lines L1 and L2. From (4.1), we know that sign(c)¼sign(d), so the Hopf bifurcation can only occur in the case VIa. Now we will focus on singularities in the case VIa. The phase portraits and bifurcation diagrams of the VIa case can be given out and are shown in Fig. 1. From [11,17,18], we know that an equilibrium outside the vertical axis in Fig. 1 is corresponding to a periodic solution of the original system (2.1). So, we shall call the source (saddle, sink) periodic solution of (2.1) when the equilibrium is a source (saddle, sink) in Fig. 1. Furthermore, we have the following results. In region D1, the trivial equilibrium is a source, and two nontrivial equilibriums are saddle points which are stable; when the parameters vary across the line L0 from region D1 to D2, the trivial equilibrium becomes a saddle point, and an unstable limit cycle (source) is bifurcated; in region D3, the above unstable periodic solution becomes a saddle from a source, and two unstable periodic solutions (source) are bifurcated; on H0, there

0.5

0.45

0.4 x1(t)

0.4 0.35 x1(t)

229

0.3 0.2 −0.02

0.3 0.25

−0.01

0.2 0.5 0.4 x3(t)

0.3 0.2

−0.02

−0.01

0

0.01

x2(t)

0.02

0 0.01

x4(t)

0.02

0.2

0.25

0.35

0.3

0.4

0.45

x3(t)

−0.2

−0.25 −0.2

−0.3 x1(t)

−0.25

x1(t)

−0.3

−0.35

−0.35

−0.4

−0.4 −0.45 −0.1 −0.2 −0.3 x3(t)

−0.4 −0.5

−0.02

−0.01

0 x4(t)

0.01

0.02

−0.45 0.02 x2(t)

0 −0.02

−0.45

−0.4

−0.3

−0.35

−0.25

−0.2

x3(t)

Fig. 4. Two stable nontrivial periodic orbits. (a, b) The stable periodic orbit at E1(0.3996,0,0.3996,0) and (c, d) the stable periodic orbit at E2( 0.3996,0,  0.3996,0).

T. Dong et al. / Neurocomputing 97 (2012) 223–232

is a pair of symmetric heteroclinic orbits, each of which connects a nontrivial saddle point and the saddle periodic solution; in region D4, there are two quasi-periodic motions, attractors, which are bifurcated from the two source periodic solutions; in region D5, two stable periodic solutions appear, while the above quasiperiodic motions disappear; from region D5 to D6, two stable periodic solutions disappear; in region D7, two stable nontrivial equilibriums overlap and become the origin which is a sink; next, in region D8, the unstable periodic solution disappears and the trivial equilibrium becomes a stable saddle point. In region D9, there are two nontrivial equilibriums exist which are stable saddle points. Summarizing the above analysis, we obtain the following conclusions. Theorem 3. System (2.1) undergoes a Hopf–pitchfork bifurcation at the origin when b ¼ Z and t ¼ t0. And for (H3), we have

E1(0.3996,0,0.3996,0) and the diagram (c) and (d) in Fig. 4 shows the stable periodic orbit at E2(  0.3996,0, 0.3996,0). (4) If m1 ¼  0.045 and m2 ¼0.6, then (m1,m2)AD5. System (2.1) simultaneously undergoes a subcritical Hopf bifurcation at E1(0.3996,0,0.3996,0) and E2( 0.3996,0,  0.3996,0). In Fig. 5, two unstable limit cycles emerge around the above two nontrivial equilibriums and these equilibria themselves are stable. The phase portrait (a) and its corresponding waveform graph (b) and (c) show that trajectories, starting from an

2 1 x1(t)

230

0 −1

(1) The trivial equilibrium is asymptotically stable when m2 o0, that is (m1,m2)AD8[D9. (2) There are two asymptotically stable nontrivial equilibria which coexist when (m1,m2)AD1[D9. (3) The two nontrivial periodic orbits undergo a secondary Hopf bifurcation, giving rise to the appearance of quasi-periodic motions when (m1,m2)AL3. (4) There are two stable nontrivial periodic orbits which coexist when (m1,m2)AD4. (5) There are two attractive quasi-periodic motions which coexist when (m1,m2)AD5. (6) There is a heteroclinic orbit which connects two nontrivial saddle points when (m1,m2)AH0.

−2 −2

2 1

−1

0

0 x2(t)

−1

1 2

x3(t)

−2

1.5

1

5. Numerical simulation

x1(t)

0.5

0

In this section, to demonstrate the main results obtained in the previous sections, we choose f 1 ðxÞ ¼ f 2 ðxÞ ¼ tanhðxÞ. It is easy to 0 00 000 compute tanh ð0Þ ¼ 1, tanh ð0Þ ¼ 0 and tanh ð0Þ ¼ 2. If b ¼0.9 and a1 ¼a2 ¼0.9, then we can obtain Z ¼0.9 and t0 ¼1.6317. Furthermore, we can compute b0 ¼ 0.6872, c0 ¼ 8.8527, d0 ¼  1 and A¼ 5.083640. From Table 1, we know that bifurcation phenomena with the case VIa may appear near (b,t)A(Z,t0). Here, bifurcation critical lines are in Fig. 1.

−0.5

L0 ¼ 1:63170:1789ðbZÞ,

1.5

−1

−1.5

L1 ¼ 1:63176:2119ðbZÞ,

0

200

400

600 t

800

1000

1200

600 t

800

1000

1200

1

L2 ¼ 1:63179:3749ðbZÞ, 0.5

H0 ¼ 1:63176:5329ðbZÞOðbZÞ2 :

x3

L3 ¼ 1:63176:5329ðbZÞ,

0

In the following, let b ¼ Z þ m1 and t ¼ t0 þ m2, (1) If m1 ¼0.012 and m2 ¼ 0.012, then (m1,m2)AD9. We know that system (2.1) has only a zero equilibrium and Fig. 2 shows that the zero solution of system (2.1) is asymptotically stable. (2) If m1 ¼  0.045 and m2 ¼0.014, then (m1,m2)AD1. System (2.1) has two nontrivial equilibria and a trivial equilibrium. Fig. 3 shows that the two nontrivial equilibriums are asymptotically stable, and the bifurcation occurs from the trivial equilibrium. (3) If m1 ¼  0.045 and m2 ¼0.3, then (m1,m2)AD4. System (2.1) has two nontrivial equilibria and a trivial equilibrium. Fig. 4 shows there are two stable nontrivial periodic orbits, the diagram (a) and (b) in Fig. 4 shows the stable periodic orbit at

−0.5

−1

−1.5

0

200

400

Fig. 5. Two attractive quasi-periodic motions coexist, the phase portrait (a) and its corresponding waveform graph (b) and (c) show that trajectories, starting from an arbitrary point in a neighborhood of each small unstable limit cycle (outside of the small limit cycles), are attracted by the big stable limit cycle.

T. Dong et al. / Neurocomputing 97 (2012) 223–232

arbitrary point in a neighborhood of each small unstable limit cycle (outside of the small limit cycles), are attracted by the big stable limit cycle.

6. Conclusions In this paper, we have investigated the Hopf–pitchfork bifurcation of an inertial two-neuron system with time delay. Our contributions include the following: 1. By analyzing the distribution of the eigenvalues of the corresponding transcendental characteristic equation of its linearized equation, we find the critical values for the occurrence of Hopf–pitchfork bifurcation. 2. By using the normal form method [19,20] and the center manifold theorem [21], we have derived the normal form of the reduced system on the center manifold and discussed the Hopf–pitchfork bifurcation with the parameter perturbations in Eq. (2.1), and completely determined the stability and bifurcation of the zero solution near the critical value. Furthermore, we can obtain the coexistence of two asymptotically stable states, two stable periodic orbits and two attractive quasi-periodic motions. Our work is a further study of [7,9], which was helpful in the study of the complex phenomena caused by high co-dimensional bifurcation of delay differential equation.

231

[9] Q. Liu, X. Liao, Dynamics of an inertial two-neuron system with time delay, Nonlinear Dyn. 58 (2009) 573–609. [10] Z. Bo, Q. Song, Stability and Hopf bifurcation analysis of a tri-neuron BAM neural network with distributed delay, Neurocomputing (2011). [11] H. Wang, W. Jiang, Hopf–pitchfork bifurcation in van der Pol’s oscillator with nonlinear delayed feedback, J. Math. Anal. Appl. 368 (2010) 9–18. [12] X. He, C. Li, Y. Shu, Bogdanov–Takens bifurcation in a single inertial neuron model with delay, Neurocomputing 89 (15) (2012) 193–201. [13] T. Dong, X. Liao, H. Li, Stability and Hopf Bifurcation in a Computer Virus Model with Multistate Antivirus, Abstr. Appl. Anal. (2012). [14] X. He, C. Li, Y. Shu, Fold-flip bifurcation analysis on a class of discrete-time neural network, Neural Comput. Appl., http://dx.doi.org/10.1007/s00521-011-0699-y. ˜ ez, J. A´ngel, Hopf–pitchfork singularities in coupled systems, [15] F. Drubi, S. Iba´n Physica D 240 (2011) 825–840. [16] X. Yan, Bifurcation analysis in a simplified tri-neuron BAM network model with multiple delays, Nonlinear Anal.: Real World Appl. 9 (3) (2008) 963–976. [17] K. YA, Elements of Applied Bifurcation Theory, 2nd ed., Springer-Verlag Inc, New York, 1998. [18] J. Guckenheimer, P. Holmes, Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, Springer-Verlag, New York, 1983. [19] L.O. Chua, H. Kokubu, Normal forms for nonlinear vector fields—part I: theory and algorithm, IEEE. Trans. Circuits Syst. 35 (1988) 863–880. [20] L.O. Chua, H. Kokubu, Normal forms for nonlinear vector fields—part II: applications, IEEE Trans. Circuits Syst. 36 (1988) 51–70. [21] J. Carr, Applications of Centre Manifold Theory, Springer-Verlag, New York, 1981. [22] S. Chow, C. Li, D. Wang, Normal Forms and Bifurcation of Planar Vector Fields, Cambridge University Press, New York, 1994. [23] Y. Song, M. Han, J. Wei, Stability and Hopf bifurcation on a simplified BAM neural network with delays, Physica D 200 (2005) 85–204. [24] W. Jiang, J. Wei, Bifurcation analysis in van der Pol’s oscillator with delayed feedback, J. Comput. Appl. Math. 213 (2008) 604–615. [25] X. Wu, L. Wang, Multi-parameter bifurcations of the Kaldor–Kalecki model of business cycles with delay, Nonlinear Anal.: Real World Appl. 11 (2010) 869–887. [26] S. Ruan, J. Wei, On the zeros of transcendental functions with applications to stability of delay differential equations with two delays, Dyn. Contin. Discrete Impuls. Syst. Ser. A: Math. Anal. 10 (2003) 863–874.

Acknowledgment This work was supported in part by the National Natural Science Foundation of China under Grants 60973114, 61170249 , in part by the Research Fund of Preferential Development Domain for the Doctoral Program of Ministry of Education of China under Grant 20110191130005, in part by the Natural Science Foundation project of CQCSTC under Grant 2009BA2024, in part by Changjiang Scholars, and in part by the State Key Laboratory of Power Transmission Equipment & System Security and New Technology, Chongqing University, under Grant 2007DA10512711206.

Tao Dong received the B.S. and M.S. degrees from the University of Electronic Science and Technology of China in 2004 and 2007. He is a lecturer at Chongqing University of Posts and Telecommunications since 2007. Currently, he is working toward the Ph.D. degree in the College of Computer Science at Chongqing University, Chongqing, China. His research interest focuses on neural networks, nonlinear dynamical systems, bifurcation and chaos, and congestion control model.

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Xiaofeng Liao received the B.S. and M.S. degrees in mathematics from Sichuan University, Chengdu, China, in 1986 and 1992, respectively, and the Ph.D. degree in circuits and systems from the University of Electronic Science and Technology of China in 1997. From 1999 to 2001, he was involved in postdoctoral research at Chongqing University, where he is currently a professor. From November 1997 to April 1998, he was a research associate at the Chinese University of Hong Kong. From October 1999 to October 2000, he was a research associate at the City University of Hong Kong. From March 2001 to June 2001 and March 2002 to June 2002, he was a senior research associate at the City University of Hong Kong. From March 2006 to April 2007, he was a research fellow at the City University of Hong Kong. He has published more than 250 international journal and conference papers. His current research interests include neural networks, nonlinear dynamical systems, bifurcation and chaos, and cryptography.

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Tingwen Huang received his B.S. degree in mathematics from Southwest Normal University, Chongqing, China, in 1990, M.S. degree in applied mathematics from Sichuan University, Chengdu, China, in 1993 and Ph.D. degree in mathematics from Texas A&M Univsersity, College Station, Texas, USA, in 2002. He was a lecturer at Jiangsu University, China, from 1994 to 1998, a Visiting Assistant Professor at Texas A&M Univsersity, College Station, USA, from January 2003 to July 2003, from August 2003 to June 2009, he was an Assistant Professor, and from July 2009 to date, an Associate Professor at Texas A&M Univsersity at Qatar, Doha, Qatar. His research areas include neural networks, complex networks, chaos and dynamics of systems and operator semi-groups and their applications.

Huaqing Li received the B.S. degree from Chongqing University of Posts and Telecommunications, Chongqing, China, in 2009. Currently, he is working toward the Ph.D. degree in the College of Computer Science at Chongqing University, Chongqing, China. His research interest focuses on nonlinear dynamical systems, bifurcation and chaos, neural networks and consensus of multi-agent systems.