Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

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Nonlinear Analysis: Real World Applications

journal homepage: www.elsevier.com/locate/nonrwa

Codimension two bifurcation in a delayed neural network withunidirectional couplingXing He a, Chuandong Li a,∗, Tingwen Huang b, Chaojie Li ca College of Computer Science, Chongqing University, Chongqing, 400030, PR Chinab Texas A&M University at Qatar, Doha, P.O. Box 23874, Qatarc School of Science, Information Technology and Engineering, University of Ballarat, Mt Helen, VIC 3350, Australia

a r t i c l e i n f o

Article history:Received 11 July 2012Accepted 18 September 2012

Keywords:Codimension two bifurcationZero–Hopf bifurcationDouble Hopf bifurcationDelayed neural networks

a b s t r a c t

In this paper, a delayed neural network model with unidirectional coupling is considered.Zero–Hopf bifurcation is studied by using the center manifold reduction and the normalform method for retarded functional differential equation. We get the versal unfolding ofthe norm form at the zero–Hopf singularity and show that themodel can exhibit pitchfork,Hopf bifurcation, and double Hopf bifurcation is also found to occur in this model. Somenumerical simulations are given to support the analytic results.

© 2012 Elsevier Ltd. All rights reserved.

1. Introduction

As we know, artificial neural networks (ANN) are an information processing system, which are used to mimic thebehavior of a natural system as closely as possible. In the last three decades, different types of artificial neural networkshave been proposed and developed due to their wide applications, such as associative memory [1–3], pattern recognition[4,5], optimization [6], signal processing [7] and so on. And these applications depend heavily on the network’s dynamics,so there has been increasing interest in investigating the dynamical behaviors of two or more neurons with or withoutdelay. And a great many neural network models [8–23] have been constructed and studied extensively to understand thedynamical behaviors of neurons. In these models, various types of behaviors including stability, Hopf bifurcation, and chaoswere presented. However, most work focused on codimension one bifurcation analysis. In fact, it is well known that delaycan lead to various bifurcations and complex dynamics [24]. Therefore, it is very interesting to investigate codimension twobifurcation of delayed neural networks [25–31].

In this paper, we consider the following delayed neural network with unidirectional coupling [18],u1 (t) = −u1 (t)+ a12f (u2 (t − τ))+ αf (u4 (t − τ)) ,u2 (t) = −u2 (t)+ a21f (u1 (t − τ)) ,u3 (t) = −u3 (t)+ a12f (u4 (t − τ))+ αf (u2 (t − τ)) ,u4 (t) = −u4 (t)+ a21f (u3 (t − τ)) ,

(1)

where αij, i, j = 1, 2, denote the connection weights within the individual network and in the literature, for aij > 0(< 0),the j-th neuron is called excitatory (inhibitory otherwise), α measures the coupling strength between two networks, τrepresents the time delay due to the finite switching speed of amplifiers and f : R → R is the amplification function.Throughout this paper, we assume f (0) = f ′′ (0) = 0, f ′ (0) = β , f ′′′ (0) = 0. In [18], using the theory of functional

∗ Corresponding author.E-mail addresses: hexingdoc@hotmail.com (X. He), licd@cqu.edu.cn (C. Li), tingwen.huang@qatar.tamu.edu (T. Huang).

1468-1218/$ – see front matter© 2012 Elsevier Ltd. All rights reserved.doi:10.1016/j.nonrwa.2012.09.010

1192 X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

differential equations, the authors studied absolute stability and Hopf bifurcation of system (1), also discussed the spatio-temporal patterns of system (1) by using the symmetric bifurcation theory with representation theory of Lie groups. Exceptfor the dynamics of system (1) in [18], it is interesting to further find out what kind of new dynamics this system has.The study carried out in the present paper may contribute to understand codimension two bifurcation in the delayed neuralnetworkwith unidirectional coupling.We use τ and a21 as bifurcation parameters. System (1) exhibits zero–Hopf singularitywhen two parameters vary in a neighborhood of the critical values. By using the normal formmethod for retarded functionaldifferential equation [32,33], we obtain the normal form to study its dynamical behaviors. It is shown that system (1)can display pitchfork bifurcation and Hopf bifurcation. Moreover, we have investigated other dynamical behaviors, suchas double Hopf bifurcation.

This paper is organized as follows. In the next section, we discuss the existence of zero–Hopf bifurcation. In Section 3,we analyze the normal form and unfolding for zero–Hopf bifurcation in the delayed neural network with unidirectionalcoupling, and the normal form are used to predict zero–Hopf bifurcation diagrams. In Section 4, the position in the parameterspace is explicitly found, where the points of a double Hopf bifurcation for the coupled network system are located. Somenumerical simulations are given to support the analytic results in Section 5. Section 6 summarizes the main conclusions.

2. The existence of zero–Hopf bifurcation

In this section,we discuss the distribution of eigenvalues of system (1) at the origin, and obtain the existence of zero–Hopfbifurcation. A point of zero–Hopf bifurcation of a fixed point occurs if the corresponding characteristic equation has a singlezero root and a pair of purely imaginary roots, and no other root of the characteristic equation has a zero real part. Thelinearization equation of system (1) at the origin is

u1 (t) = −u1 (t)+ a12βu2 (t − τ)+ αβu4 (t − τ) ,u2 (t) = −u2 (t)+ a21βu1 (t − τ) ,u3 (t) = −u3 (t)+ a12βu4 (t − τ)+ αβu2 (t − τ) ,u4 (t) = −u4 (t)+ a21βu3 (t − τ) ,

and the corresponding characteristic equation is

F (λ) =λ2 + 2λ+ 1 − a21β2 (a12 + α) e−2λτ λ2 + 2λ+ 1 − a21β2 (a12 − α) e−2λτ

= 0. (2)

Lemma 1 (see [18]). λ = 0 is a single zero of Eq. (2) if and only ifa21αβ2

= 1− a12a21β2 anda12a21β2, a21αβ2

= (1, 0).

Theorem 1. Ifa21αβ2

= 1 − a12a21β2 and2a21β2a12 − 1

> 1, then there exist τ0 < τ1 < · · · < τj < · · ·, such thatEq. (2) has a pair of purely imaginary roots when τ = τj, j = 0, 1, . . .

Proof. From Eq. (2), when a21αβ2= 1 − a12a21β2, we obtain

F (λ) = △+ △− =λ2 + 2λ+ 1 − e−2λτ λ2 + 2λ+ 1 −

2a21β2a12 − 1

e−2λτ

= 0. (3)

Substituting λ = iω(ω > 0) into λ2 + 2λ+ 1 − e−2λτ= 0 and equating both the real and imaginary parts to zero yields

−ω2+ 1 − cos (2ωτ) = 0,

2ω + sin (2ωτ) = 0,

resulting in ω2= 0 or ω2

= −2, which is meaningless. Substituting λ = iω(ω > 0) into λ2 + 2λ + 1 −2a21β2a12 − 1

e−2λτ

= 0, we also obtain−ω2

+ 1 −2a21β2a12 − 1

cos (2ωτ) = 0,

2ω +2a21β2a12 − 1

sin (2ωτ) = 0,

resulting in ω2= |2a21β2a12 − 1| − 1. Thus, the critical value τj is given by

τj =

12ω

arccos

1 + 2a21β2a122a21β2a12 − 1

+ 2jπ, a21β2a12 < 0,

12ω

2 (j + 1) π − arccos

2a21β2a12 − 31 − 2a21β2a12

, a21β2a12 > 1,

for j = 0, 1, 2 · · ·. When a21αβ2= a12a21β2

− 1, we obtain the same Eq. (3). This completes the proof of Theorem 1. �

X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202 1193

Lemma 2 (see [34]). Consider the exponential polynomialP(λ, e−λτ1 , . . . e−λτm) = λn+p(0)1 λ

n−1+· · ·+p(0)n−1λ+p(0)n +[p(1)1 λ

n−1+· · ·+p(1)n−1λ+p(1)n ]e−λτ1 +· · ·+[p(m)1 λn−1

+· · ·+

p(m)n−1λ + p(m)n ]e−λτm . where τi ≥ 0(i = 1, 2, . . . ,m) and p(i)j (i = 0, 1, . . . ,m; j = 1, 2 · · · n) are constants. As (τ1, τ2, . . . τm)vary, the sum of the order of zeros of P(λ, e−λτ1 , . . . e−λτm) on the open right half-plane can change only if a zero appears on oracross the imaginary axis.

Theorem 2. Ifa21αβ2

= 1 − a12a21β2, a21β2a12 < 0, τ = τj, j = 0, 1, . . . , then all the roots of Eq. (2) have negative realparts except a single zero root and a pair of purely imaginary roots ±iω.

Proof. Obviously, Eq. (2) have a single zero root and a pair of purely imaginary roots ±iω if a21αβ2= 1− a12a21β2, τ = τj.

For τ = 0, F (λ) =λ2 + 2λ

λ2 + 2λ+ 2 − 2a21β2a12

= 0 has a zero root and three roots with negative real parts if

a21β2a12 < 0. Using Lemma 2, we obtain the conclusion. As a result, a zero–Hopf bifurcation occurs at τ = τj, j = 0, 1, 2 · · ·.�

3. Normal form and unfolding for zero–Hopf bifurcation

In this section, we investigate zero–Hopf bifurcation by using the method [32,33]. From Theorem 2, we know that,at the origin, the characteristic equation of system (2) has a single zero root and a pair of purely imaginary roots ifa21 = a∗

21 =1

β2(α+a12)or a21 = a∗

21 =1

β2(a12−α), and τ = τ0. So we treat (a21, τ ) as bifurcation parameters near (a21, τ0).

Rescaling the time by t → t/τ to normalize the delay, system (1) can be written asu1 (t) = −τu1 (t)+ a12τ f (u2 (t − 1))+ ατ f (u4 (t − 1)) ,u2 (t) = −τu2 (t)+ a21τ f (u1 (t − 1)) ,u3 (t) = −τu3 (t)+ a12τ f (u4 (t − 1))+ ατ f (u2 (t − 1)) ,u4 (t) = −τu4 (t)+ a21τ f (u3 (t − 1)) .

(4)

Let τ = τ0 + µ1, a21 = α∗

21 + µ2, and expand the function f . Then system (4) can be written as

u1(t) = (τ0 + µ1){−u1(t)+ a12(βu2(t − 1)+f ′′′(0)3!

u32(t − 1))

+α(βu4(t − 1)+f ′′′(0)3!

u34(t − 1))} + h.o.t,

u2(t) = (τ0 + µ1){−u2(t)+ (a∗

21 + µ2)(βu1(t − 1)+f ′′′(0)3!

u31(t − 1))} + h.o.t,

u3(t) = (τ0 + µ1){−u3(t)+ a12(βu4(t − 1)+f ′′′(0)3!

u34(t − 1))

+α(βu2(t − 1)+f ′′′(0)3!

u32(t − 1))} + h.o.t,

u4(t) = (τ0 + µ1){−u4(t)+ (a∗

21 + µ2)(βu3(t − 1)+f ′′′(0)3!

u33(t − 1))} + h.o.t.

(5)

Choose the phase space C = C[−1, 0] ; R4

as the Banach space of the continuous function from [−1, 0] to R4. For any

u ∈ C , define ut (θ) = u (t + θ) and ∥u∥ = sup |u (θ)|−1<θ<0

. Then system (5) becomes

u (t) = A0ut + Fut , (6)

where u =u1, u2,u3, u4

T , A0 and F are operators, given by

A0ϕ = ϕ,D (A0) =

ϕ ∈ C1 [−1, 0] , R4

: ϕ (0) =

0

−1dη (θ) ϕ (θ)

and

dη (θ) =

−τ0δ (θ) a12βτ0δ (θ + 1) 0 αβτ0δ (θ + 1)a∗

21βτ0δ (θ + 1) −τ0δ (θ) 0 00 αβτ0δ (θ + 1) −τ0δ (θ) a12βτ0δ (θ + 1)0 0 a∗

21βτ0δ (θ + 1) −τ0δ (θ)

,with δ (θ) being the Dirac-delta function, and

Fut =

0,−1 ≤ θ < 0,L1 (µ) ut + G (ut , µ) , θ = 0

1194 X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

where

L1 (µ) ut =

a12βµ1u2 (−1)− µ1u1 (0)+ αβµ1u4 (−1)a∗

21βµ1u1 (−1)− µ1u2 (0)+ τ0βµ2u1 (−1)a12βµ1u4 (−1)− µ1u3 (0)+ αβµ1u2 (−1)a∗

21βµ1u3 (−1)− µ1u4 (0)+ τ0βµ2u3 (−1)

,

G (ut , µ) = f ′′′ (0)

τ0a123!

u32 (−1)+

τ0α

3!u34 (−1)

τ0a∗

21

3u31 (−1)

τ0a123!

u34 (−1)+

τ0α

3!u32 (−1)

τ0a∗

21

3u33 (−1)

.

From Section 2, under some conditions, we know that the characteristic Eq. (2) have a single zero root and a pair of purelyimaginary roots ±iω, and all other eigenvalues have negative real parts. Therefore the phase space C can be decomposed byΛ = {±iω, 0}, and C = P ⊕ Q where

Q = {ϕ ∈ C : ⟨ϕ,ψ⟩ = 0, ∀ψ ∈ P∗} , P is the generalized eigenvalue space associated withΛ and P∗ the space adjointwith P . Further, one can define

A∗

0ψ = −ψ, DA∗

0

=

ψ ∈ C1 [0, 1] , R4∗

: ψ (0) = −

0

−1dη (θ)ψ (−θ)

where A∗

0 is dual operator of A0, and the bilinear form

⟨ψ, ϕ⟩ = ψ (0) ϕ (0)−

0

−1

θ

0ψ (ξ − θ) dη (θ) ϕ (ξ) dξ, ∀ψ ∈ Q ,∀ϕ ∈ C .

Thus, it is not difficult to verify that the bases for P and P∗ are

Φ (θ) = (ϕ1 (θ) , ϕ2 (θ) , ϕ3 (θ)) =

eiωτ0θ e−iωτ0θ (a12 + α) β

a∗

21β

iω + 1eiωτ0(θ−1) a∗

21β (1 + iω)ω2 + 1

e−iωτ0(θ−1) 1

−eiωτ0θ −e−iωτ0θ (a12 + α) β

−a∗

21β

iω + 1eiωτ0(θ−1)

−a∗

21β (1 + iω)ω2 + 1

e−iωτ0(θ−1) 1

,and

Ψ (s) =

ψ1 (s)ψ2 (s)ψ3 (s)

=

d1e−iωτ0s

d1 (1 − iω)a∗

21βe−iωτ0(1+s)

−d1e−iωτ0s −d1 (1 − iω)

a∗

21βe−iωτ0(1+s)

d1eiωτ0sd1 (1 + iω)

a∗

21βeiωτ0(1+s)

−d1eiωτ0s −d1 (1 + iω)

a∗

21βeiωτ0(1+s)

d2 d2 (a12 + α) β d2 d2 (a12 + α) β

,where

⟨Ψ (s) ,Φ (θ)⟩ = I, d1 =1

4 + 4τ0 (1 + iω), d2 = 4β (a12 + α) (1 + τ0) , −1 ≤ θ ≤ 0, 0 ≤ s ≤ 1.

Thus the dual bases satisfy

Φ = Φ

iω 0 00 −iω 00 0 0

and − Ψ =

iω 0 00 −iω 00 0 0

Ψ .

Let u = Φz + y, namely

u1 (θ) = eiωτ0θ z1 + e−iωτ0θ z2 + (a12 + α) βz3 + y1 (θ) ,

u2 (θ) =a∗

21β

iω + 1eiωτ0(θ−1)z1 +

a∗

21β (1 + iω)ω2 + 1

e−iωτ0(θ−1)z2 + z3 + y2 (θ) ,

u3 (θ) = −eiωτ0θ z1 − e−iωτ0θ z2 + (a12 + α) βz3 + y3 (θ) ,

u4 (θ) = −a∗

21β

iω + 1eiωτ0(θ−1)z1 −

a∗

21β (1 + iω)ω2 + 1

e−iωτ0(θ−1)z2 + z3 + y4 (θ) .

X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202 1195

Thenu1 (0) = z1 + z2 + (a12 + α) βz3 + y1 (0) , u1 (−1) = e−iωτ0z1 + eiωτ0z2 + (a12 + α) βz3 + y1 (−1) ,

u2 (0) =a∗

21β

iω + 1e−iωτ0z1 +

a∗

21β (1 + iω)ω2 + 1

eiωτ0z2 + z3 + y2 (0) ,

u2 (−1) =a∗

21β

iω + 1e−2iωτ0z1 +

a∗

21β (1 + iω)ω2 + 1

e2iωτ0z2 + z3 + y2 (−1) ,

u3 (0) = −z1 − z2 + (a12 + α) βz3 + y3 (0) , u3 (−1) = −e−iωτ0z1 − eiωτ0z2 + (a12 + α) βz3 + y3 (−1) .

u4 (0) = −a∗

21β

iω + 1e−iωτ0z1 −

a∗

21β (1 + iω)ω2 + 1

eiωτ0z2 + z3 + y4 (0) ,

u4 (−1) = −a∗

21β

iω + 1e−2iωτ0z1 −

a∗

21β (1 + iω)ω2 + 1

e2iωτ0z2 + z3 + y4 (−1) .

From Refs. [32,33], the dynamical behavior of system (5) is determined byz = Jz + Ψ (0) F (Φz + y, µ)+ h.o.t. (7)

On the center manifold, system (7) can be written asz = Jz + Ψ (0) F (Φz, µ)+ h.o.t. (8)

Substituting Ψ (0),Φ into system (8), one obtainsz1 = iωz1 + d1F 12 + d1E1e−iωτ0F 2

2 − d1F 32 − d1E1e−iωτ0F 4

2 + h.o.t,z2 = −iωz2 + d1F 1

2 + d1E2eiωτ0F 22 − d1F 3

2 − d1E2eiωτ0F 42 + h.o.t,

z3 = d2F 12 + d2 (a12 + α) βF 2

2 + d2F 32 + d2 (a12 + α) βF 4

2 + h.o.t,(9)

where

E1 =1 − iωa∗

21β, E2 =

1 + iωa∗

21β, H1 =

a∗

21β

iω + 1, H2 =

a∗

21β (1 + iω)ω2 + 1

,

F 12 = a12βµ1(H1e−2iωτ0z1 + H2e2iωτ0z2 + z3)− µ1(z1 + z2 + (a12 + α)βz3)+ αβµ1(−H1e−2iωτ0z1

−H2e2iωτ0z2 + z3)+τ0f ′′′(0)

3[a12(H1e−2iωτ0z1 + H2e2iωτ0z2 + z3)3 + α(−H1e−2iωτ0z1 − H2e2iωτ0z2 + z3)3],

F 22 = a∗

21βµ1(e−iωτ0z1 + eiωτ0z2 + (a12 + α)βz3)− µ1(H1e−iωτ0z1 + H2eiωτ0z2 + z3)

+ τ0βµ2(e−iωτ0z1 + eiωτ0z2 + (a12 + α)βz3)+τ0f ′′′(0)a∗

21

3(e−iωτ0z1 + eiωτ0z2 + (a12 + α)βz3)3,

F 32 = a12βµ1(−H1e−2iωτ0z1 − H2e2iωτ0z2 + z3)− µ1(−z1 − z2 + (a12 + α)βz3)

+αβµ1(H1e−2iωτ0z1 + H2e2iωτ0z2 + z3)+τ0f ′′′(0)

3[a12(−H1e−2iωτ0z1 − H2e2iωτ0z2 + z3)3

+α(H1e−2iωτ0z1 + H2e2iωτ0z2 + z3)3],

F 42 = a∗

21βµ1(−e−iωτ0z1 − eiωτ0z2 + (a12 + α)βz3)− µ1(−H1e−iωτ0z1 − H2eiωτ0z2 + z3)+ τ0βµ2(−e−iωτ0z1

− eiωτ0z2 + (a12 + α)βz3)+τ0f ′′′(0)a∗

21

3(−e−iωτ0z1 − eiωτ0z2 + (a12 + α)βz3)3.

Following the computation of the normal form for functional differential equations introduced by Refs. [32,33], we getthe normal form with versal unfolding on the center manifoldz1 = iωz1 + k1µ1z1 + k2µ2z1 + k3z21z2 + k4z1z23 + h.o.t,

z2 = −iωz2 + k1µ1z2 + k2µ2z2 + k3z1z22 + k4z2z23 + h.o.t,z3 = k5µ1z3 + k6µ2z3 + k7z33 + k8z1z2z3 + h.o.t,

(10)

wherek1 = 2d1

βH1 (a12 − α) e−2iωτ0 + E1

a∗

21β − H1e−2iωτ0 − 1

,

k2 = 2d1E1τ0βe−2iωτ0 , k3 = d1f ′′′ (0) τ0E1a∗

21 + (a12 − α)H21H2

e−2iωτ0

k4 = d1f ′′′ (0) τ0E1a∗

21 (a12 + α)2 β2+ (a12 − α)H1

e−2iωτ0 ,

k5 = 2d2 (a12 + α) βa∗

21β2 (a12 + α)− 1

, k6 = 2d2τ0 (a12 + α)2 β3,

k7 =τ0d2f ′′′ (0)

3(a12 + α)

1 + a∗

21 (a12 + α)3 β4 ,k8 = 2τ0d2f ′′′ (0) (a12 + α)

H1H2 + a∗

21 (a12 + α) β2 .Thus we can get the following result.

1196 X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

Theorem 3. On the center manifold, system (4) is locally topologically equivalent to the normal form Eq. (10).

Let z1 = γ cosθ − iγ sinθ , z2 = γ cosθ + iγ sinθ and z3 = z3. Then Eq. (10) becomesγ = h1γ + h2γ3+ h3z23γ + h.o.t,

z3 = h4z3 + h5z33 + h6γ2z3 + h.o.t,

θ = −ω + [Im (k1) µ1 + Im (k2) µ2] + h.o.t,(11)

where

h1 = Re (k1) µ1 + Re (k2) µ2, h2 = Re (k3) , h3 = Re (k4) ,h4 = k5µ1 + k6µ2, h5 = k7, h6 = k8.

Truncating higher order terms and removing the azimuthal term, making the transformation [35]: γ = γ√

|h2|, z3 =

z3√

|h5|, we obtain (for simplicity, still using γ , z3)γ = γ

δ1 + γ 2

+ bz23,

z3 = z3δ2 + cγ 2

+ dz23,

(12)

whereδ1 = h1sgn (h2) = [Re (k1) µ1 + Re (k2) µ2] sgn {Re (k3)} , b =

h3|h5|

=Re(k4)|k7|

,

δ2 = h4sgn (h5) = (k5µ1 + k6µ2) sgn (k7) , c =h6|h2|

=k8

|Re(k3)|, d = sgn (h5) = sgn (k7) .

The complete bifurcation diagrams of system (12) can be found in [35]. Here we just briefly list some results.(i) System (12) undergoes a pitchfork bifurcation at the origin on the curvesP1 = {(δ1, δ2) : δ1 = 0, δ2 = 0} and P2 = {(δ1, δ2) : δ2 = 0, δ1 = 0}.(ii) System (12) undergoes a pitchfork bifurcation at the non-trivial equilibrium on the curvesP3 = {(δ1, δ2) : δ2 = cδ1} and P4 =

(δ1, δ2) : δ2 =

dbδ1.

(iii) System (12) undergoes Hopf bifurcation at the non-trivial equilibrium on the curve

H =

(δ1, δ2) : δ2 =

d(1−c)b−d δ1

.

Applying the above results and using the expressions of δ1, δ2, we obtain the following result.

Theorem 4. For sufficiently small µ1, µ2,(i) system (4) undergoes a pitchfork bifurcation at the origin on the curves

P1 =

(µ1, µ2) : µ2 =

Re(k1)Re(k2)

µ1, k5µ1 + k6µ2 = 0and

P2 =

(µ1, µ2) : µ2 =

k5k6µ1, Re (k1) µ1 + Re (k2) µ2 = 0

,

(ii) system (4) undergoes a pitchfork bifurcation at the non-trivial equilibrium on the curvesP3 = {(µ1, µ2) : (k5µ1 + k6µ2) Re (k3) sgn (k7) = k8 [Re (k1) µ1 + Re (k2) µ2]} andP4 = {(µ1, µ2) : (k5µ1 + k6µ2) Re (k4) sgn (k7Re (k3)) = k7 [Re (k1) µ1 + Re (k2) µ2]},(iii) system (4) undergoes Hopf bifurcation at the non-trivial equilibrium on the curve

H =

(µ1, µ2) : (k5µ1 + k6µ2) =

sgn{Re(k3)}−k8

Re(k3)Re(k4)|k7|

−sgn(k7)[Re (k1) µ1 + Re (k2) µ2]

.

From [35], there are twelve distinct types of unfoldings by the different signs of b, c , d, d − bc. In Section 5, from thenumerical simulations, we focus on the type, d = −1, b > 0, c < 0, d − bc > 0, at which type system has complexdynamic behavior.

4. Double Hopf bifurcation

It is known that in some cases, it is possible to show the presence of points at which the characteristic equation hastwo pairs of pure imaginary roots, ±iω1, ±iω2. As such points commonly occur where two curves of Hopf bifurcationcross, we refer to them as points of double Hopf bifurcation. In this section, we discuss double Hopf bifurcation ofsystem (1).

Lemma 3 (see [18]). Ifa12a21β2, a21αβ2

∈ D1 ∪ D3, Eq. (3) has one or two purely imaginary pairs of zero at the

critical τ , where D1 =

a12a21β2, a21αβ2

|a21αβ2

> 1 − a12a21β2, D3 =

a12a21β2, a21αβ2

| 1 + a12a21β2 <a21αβ2

< 1 − a12a21β2.

X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202 1197

Fig. 1. The curves of Hopf bifurcation of system (1) for a21 = −0.5, a12 ∈ [1, 3].

Fig. 2. Waveform plot of the variable u of system (1) for µ1 = 0.133427, µ2 = 0.01.

Fig. 3. Phase portraits of system (1) for µ1 = 0.133427, µ2 = 0.01.

1198 X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

Fig. 4. Phase portraits of system (1) for µ1 = −0.5, µ2 = −0.01.

Fig. 5. Waveform plot of the variable u of system (1) for α = 3.1, a12 = 0.99, τ = 0.81.

Fig. 6. Phase portraits of system (1) for α = 3.1, a12 = 0.99, τ = 0.81.

X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202 1199

Fig. 7. Phase portraits of system (1) for α = 3.1, a12 = 1.01, τ = 0.79.

Fig. 8. Waveform plot of the variable u of system (1) for α = 3.1, a12 = 0.95, τ = 0.84.

From Section 2, when △+

= 0, Eq. (3) has a pair of imaginary toots. The critical τ and ω are as follow

τ1 =1

2ω1

arccos

−a21β2 (a12 + α)− 2−a21β2 (a12 + α)

+ 2kπ, a21β2 (a12 + α) < −1,

or

τ1 =1

2ω1

2 (k + 1) π − arccos

a21β2 (a12 + α)− 2−a21β2 (a12 + α)

, a21β2 (a12 + α) > 1,

ω1 =

a21β2 (a12 + α)− 1, k = 0, 1, . . .

(13)

and when △−

= 0, the critical τ and ω are as follows

τ2 =1

2ω2

arccos

−a21β2 (a12 − α)− 2−a21β2 (a12 − α)

+ 2lπ, a21β2 (a12 − α) < −1,

or

τ2 =1

2ω2

2 (l + 1) π − arccos

a21β2 (a12 − α)− 2−a21β2 (a12 − α)

, a21β2 (a12 − α) > 1,

ω2 =

a21β2 (a12 − α)− 1, l = 0, 1, . . . .

(14)

1200 X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202

Fig. 9. Two-dimensional phase portraits of system (1) α = 3.1, a12 = 0.95, τ = 0.84.

According to the theory of functional differential equation, double Hopf points occur at these curve when τ1 = τ2, and wehave the following result.

Theorem 5. Under the conditiona12a21β2, a21αβ2

∈ D1 ∪ D3 and

ω1ω2

=

√|a21β2(a12+α)|−1,

√|a21β2(a12−α)|−1

=mn , If

mn is rational (irrational)

number, system (1) undergoes resonant (non-resonant) double Hopf bifurcation. The corresponding value of τ , ω1, ω2 is givenby (13) and (14).

For system (1), we fix a21 = −0.5, a12 ∈ [1, 3] and its step size is 0.5. The curves of Hopf bifurcation for differentparameter values are given in Fig. 1, and the intersection between a red line and a black line is a double Hopf bifurcationpoint for the same parameters.

5. Numerical simulations

In this section, we give some examples to verify the theoretical result, and discuss complex phenomenon nearcodimension two bifurcation point.

Example 1. For system (1), we fix a12 = 0.5, a21 = −0.5, α = −2.5. f (x) = tanh(x). Then we can calculate ω = 0.7071,τ0 = 1.7862, b = 1.2925, c = −1.5511, d = −1, k1 = −0.0663− 0.0736i, k2 = 0.6513− 0.2953i, k3 = 0.2714− 0.1230i,k4 = 0.9770 − 0.4429i, k5 = 0, k6 = −318.5187, k7 = −265.4323, k8 = −371.6052. From condition (iii) of Theorem 4,system (1) undergoesHopf bifurcation on the curveH = {(µ1, µ2) : µ1 = 13.3427µ2}. Chooseµ2 = 0.01,µ1 = 0.133427.A stable limit cycle exists through the Hopf bifurcation in Figs. 2 and 3. In fact, it is well known that the time delay can lead tocomplex dynamics near the zero–Hopf point. In the following, we set µ2 = −0.01, Fig. 4 shows the quasi-periodic motionof system (1) if µ1 = −0.5.

Example 2. For system (1), we fix a12 = 1, a21 = −0.5, α = 3.1, f (x) = tanh(x). Then we obtain ω1 = 1.0247,ω2 = 0.2236, τ1 = τ2 = 0.7923 and ω1

ω2=

√21, and we choose a12, τ as bifurcation parameters. If a12 = 0.99, τ = 0.81,

a stable limit cycle exists through the Hopf bifurcation in Figs. 5 and 6. If a12 = 1.01, τ = 0.79, there exist quasi-periodicmotions in system (1) in Fig. 7. If a12 = 0.95, τ = 0.84, there may exist global bifurcation structure in Figs. 8–10. It is shownthat a strange attractor exists between one equilibrium and another equilibrium.

6. Conclusions

In this paper,we have investigated zero–Hopf bifurcation of a delayed neural networkwith unidirectional coupling. Usingthe normal form theory and the center manifold reduction, we are able to predict their corresponding bifurcation diagramssuch as Hopf, pitchfork bifurcation. Moreover, we also have investigated other dynamical behaviors, such as double Hopfbifurcation. There are still abundant and complex dynamical behaviors to be completely and thoroughly investigated andexploited. We will further study global bifurcation near the codimension two bifurcation point in the near future.

X. He et al. / Nonlinear Analysis: Real World Applications 14 (2013) 1191–1202 1201

Fig. 10. Three-dimensional phase portraits of system (1) α = 3.1, a12 = 0.95, τ = 0.84.

Acknowledgments

The work is supported by the National Natural Science Foundation of China Grant No. 60974020 and the FundamentalResearch Funds for the Central Universities of China (Project No. CDJZR10 18 55 01); the work is also supported by NPRPGrant 4-1162-1-181 from the Qatar National Research Fund (a member of the Qatar Foundation).

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