Hopf bifurcation and spatio-temporal patterns in a hierarchical network with delays and Z2×Zn symmetry

Neurocomputing 168 (2015) 475–487

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Neurocomputing journal homepage: www.elsevier.com/locate/neucom

Hopf bifurcation and spatio-temporal patterns in a hierarchical network with delays and Z2
Zn symmetry Haijun Hu a, Yanxiang Tan a,b, Chuangxia Huang a,c,n a

School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410114, PR China College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, PR China c Academy of Mathematics and Systems Science, Chinese Academy of Science, Beijing 10090, China b

art ic l e i nf o

a b s t r a c t

Article history: Received 13 December 2014 Received in revised form 26 March 2015 Accepted 20 May 2015 Communicated by Yang Tang Available online 30 May 2015

A hierarchical network composed of two interacting rings each of which consists of n identical cells with an unidirectional coupling is the topic of this paper. We present a detailed discussion about the linear stability of the equilibrium by analyzing the associated characteristic equation. The local Hopf bifurcation and spatio-temporal patterns of bifurcating periodic oscillations are also given by employing the symmetric Hopf bifurcation theory for delay differential equations. In particular, by using the normal form theory and the center manifold theorem, we derive the formula determining the direction of the Hopf bifurcation and the stability of the bifurcated periodic orbits. An example with numerical simulations is presented to illustrate our theoretical results. & 2015 Elsevier B.V. All rights reserved.

Keywords: Hierarchical network Hopf bifurcation Normal form Symmetry Time delay

1. Introduction There has been a flurry of research activity on neural networks since Hopfield [1] constructed a simplified neural network model. In this model, each neuron is represented by a linear circuit consisting of a resistor and a capacitor and is connected to the other neurons via sigmoidal activation functions. This model can be regarded as an information processing device that is inspired by the way biological nervous systems, such as the brain, process information simultaneously. However, the information transmission between neurons is not instantaneous and so it is more reasonable and realistic for the model to incorporate the factor of time delays (see [2]). A variety of research carried on neural network models with delays has shown that delays can lead to interesting dynamics in various ways. For more details, we refer to [3–12], and references therein. In addition, due to the complexity of dynamics of neural networks, some recent works (e.g., [13–18]) have focused on networks with the same time delay, the small scale or simple architectures. Ring networks have been found in a variety of neural architectures such as cerebellum [19], and even in the fields of chemistry

n Corresponding author at: School of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410114, PR China. E-mail addresses: [email protected] (H. Hu), [email protected] (C. Huang).

http://dx.doi.org/10.1016/j.neucom.2015.05.080 0925-2312/& 2015 Elsevier B.V. All rights reserved.

and electrical engineering. In fact, ring networks are of a limited biological relevance, and may be regarded as building blocks for networks with more realistic connection topologies. Among many models of neural network, ring networks can lead to many interesting patterns of oscillation. Thus, they can be studied to gain some insight into the mechanisms underlying the behavior of recurrent network [2,20]. In recent years, a ring structure with nearestneighbor (unidirectional or bidirectional) coupling between the elements has received a great deal of attention; a significant body of research has been carried out (see [15–17,21–25] and references therein). Some of these studies have concerned lower dimensional systems. For example, Guo [25] studied a tri-neuron ring model with unidirectional coupling u_ i ðtÞ ¼  μui ðtÞ þ f ðui þ 1 ðt  τÞÞ;

iðmod 3Þ;

ð1:1Þ

where u_ ¼ du=dt, ui(t) represents the activation of the ith neuron at time t, μ 4 0 represents the decay rate of the activation, f represents the activation function, τ Z 0 is the signal transmission delay. The vast majority of previous works have just considered the individual network but not investigated the interactions between multiple networks. In fact, numerous natural and artificial systems possess a hierarchic structure or functioning and can be naturally described by coupled sub-network. Coupled networks of nonlinear dynamical systems can exhibit rich dynamics, such as synchronization, symmetric bifurcation, chaos (see [26–31]). The rich dynamics arising from the interaction of simple networks can help scientists analyze the collective behavior of complex systems. For example, the brain may be conceived as a dynamic network of

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coupled neurons. To describe the complicated interaction between billions of neurons in a large neural network, the neurons are generally lumped into highly connected sub-networks [32]. These sub-network interactions (pathological synchronization) may bring about serious problems such as Parkinsons disease, essential tremor, and epilepsy [33,34]. Inspired by the above works, we consider in this paper a twolevel hierarchical system, which consists of two coupled modules of interacting nonlinear neuron oscillators with time delays and Zn symmetry. This network is depicted schematically in Fig. 1, and given by the following system of delay differential equations (DDEs): (

x_ 0j ðtÞ ¼  x0j ðtÞ þ f ðx0j ðt  τÞÞ þ gðx0;j þ 1 ðt  τÞÞ þ hðx1j ðt  τÞÞ; x_ 1j ðtÞ ¼  x1j ðtÞ þ f ðx1j ðt  τÞÞ þ gðx1;j þ 1 ðt  τÞÞ þ hðx0j ðt  τÞÞ;

ð1:2Þ

where j ¼ 0; 1; …; n  1ðmod nÞ, the activation functions f ; g; h A C 1 ðR; RÞ satisfying f ð0Þ ¼ gð0Þ ¼ hð0Þ ¼ 0. In this model, the individual elements are represented by a scalar equation, composed of a linear decay term and a nonlinear, time delayed self-feedback. In model (1.2), each sub-network can be considered as an identical ring module in which n elements are coupled in such a way that the invariance under cyclic permutations is attained. Noticing that the decay rate μ in system (1.1) can be normalized through the time transformation, system (1.2) is a natural extension of system (1.1). System (1.2) is also a particularly simple example of a symmetric system exhibiting a hierarchical structure with two levels: a “macro” level concerning the interactions between the groups and a “micro” level concerning the interactions within the groups. The overall symmetry of system (1.2) can then be represented as a product of permutation groups, Z2
Zn , which allows us to study the dynamics analytically. Furthermore, the symmetry implies generally a certain spatial invariant of the dynamical systems. For example, the spatio-temporal patterns of bifurcating periodic solutions can be characterized precisely according to symmetric Hopf bifurcation theory, which is due to the pioneering work of Wu [35] (based on the topological methods and theorem by Golubitsky [36]). The main difference from the models considered in [30,31] is that model (1.2) is a large-scale network, each sub-network of which is composed of arbitrary n neurons with an unidirectional ring structure. This adds some complications to the analysis and computation but, as we shall show, also allows for more interesting dynamics of the system. In this paper, we are interested in

studying how the time delay can affect the stability of two-level hierarchical system (1.2). We are concerned about the occurrence of bifurcating periodic solutions when the delay τ passes through a critical value, and spatio-temporal patterns of the bifurcating periodic oscillations depending on the Z2
Zn symmetry. In addition, the stability of these periodic solutions is clearly important in applications, but also poses significant computational challenges. We also manage to obtain some formula about the direction of the Hopf bifurcation and the stability of the bifurcated periodic solution by using normal form method and center manifold introduced by Faria and Magalháes [37,38]. The outline of this paper is as follows. In Section 2, we discuss the linear stability of the equilibrium by analyzing the distribution of roots of the associated characteristic equation. The local Hopf bifurcation and spatio-temporal patterns of it are addressed in Section 3. Section 4 is devoted to the direction of Hopf bifurcation and the stability of the bifurcating periodic solutions. An example and numerical simulations are presented to illustrate the results in Section 5. Finally, a brief discussion is drawn in Section 6.

2. Linear stability analysis It is obvious that system (1.2) admits the trivial solution x^ ¼ 0. The linearization of (1.2) at the origin is given by ( x_ 0j ðtÞ ¼  x0j ðtÞ þ ax0j ðt  τÞ þ bx0;j þ 1 ðt  τÞ þ cx1j ðt  τÞ; ð2:1Þ x_ 1j ðtÞ ¼  x1j ðtÞ þ ax1j ðt  τÞ þ bx1;j þ 1 ðt  τÞ þ cx0j ðt  τÞ; 0

0

where j mod n, and a ¼ f ð0Þ, b ¼ g 0 ð0Þ, c ¼ h ð0Þ. Letting xðtÞ ¼ ðx00 ðtÞ; x01 ðtÞ; …; x0;n  1 ðtÞ; x10 ðtÞ; x11 ðtÞ; …; x1;n  1 ðtÞÞT A R2n and xt ðθÞ ¼ xðt þ θÞ A Cð½  τ; 0; R2n Þ, system (2.1) can be rewritten as _ ¼ Lτ xt ; xðtÞ

ð2:2Þ

where the linear operator Lτ : Cð½  τ; 0; R Þ-R 2n

Lτ φ ¼  φð0Þ þ M φð  τÞ;

2n

is given by ð2:3Þ

M ¼ circðM 1 ; M 2 Þ is a circle block matrix, M 1 ¼ circða; b; 0; …; 0Þ is a circulant matrix of order n, and M 2 ¼ c Idn , Idn denotes the identity matrix of order n. It is well-known that for each fixed delay τ, the linear system (2.1) generates a strongly continuous semigroup of linear operators with the infinitesimal generator AðτÞ given by _; AðτÞφ ¼ φ

φ A DomðAðτÞÞ; _ A Cð½  τ; 0; R2n Þ; φ _ ð0Þ ¼ Lτ φg: DomðAðτÞÞ ¼ fφ A Cð½  τ; 0; R2n Þ; φ ð2:4Þ

We recall that AðτÞ has only the point spectrum, and the spectrum σ ðAðτÞÞ consists of eigenvalues which are solutions of the characteristic equation det Δðτ; λÞ ¼ 0;

ð2:5Þ

where the characteristic matrix is given by

Δðτ; λÞ ¼ λ Id2n  Lτ ðeλ Id2n Þ ¼ ðλ þ 1ÞId2n  Me  λτ ; i2π =n

λ A C:

, vq ¼ ð1; χ ; …; χ ðn  1Þq ÞT , and vpq ¼ ðvq ; ð  1Þp vq ÞT . NotLet χ ¼ e ing that Mvpq ¼ ½a þ ð  1Þp c þbχ q vpq , we have n o Δðτ; λÞvpq ¼ λ þ 1  ½a þ ð  1Þp c þ bχ q e  λτ vpq : ð2:6Þ q

Hence, we obtain n1

1

detΔðτ; λÞ ¼ ∏ ∏ Δpq ; Fig. 1. Architecture of model (1.2).

q¼0p¼0

ð2:7Þ

H. Hu et al. / Neurocomputing 168 (2015) 475–487

to τ, we have

where

Δpq ¼ λ þ1  ½a þð  1Þp c þ bχ q e  λτ :

ð2:8Þ

In order to analyze the distribution of zeros of det Δðτ; λÞ, we introduce an important lemma improving the relevant results in [35,39]. Lemma 2.1. Let z ¼ Reiθ , 0 r θ o 2π . For any non-negative integer s, define mappings: n o τbs ; τ~ s : zj z ¼ Reiθ A C; R 41 -R

τbs ðzÞ ¼

477

1 R pffiffiffiffiffiffiffiffiffiffiffiffiffi R2  1

θ arccos þ 2sπ

;

τ~ s ðzÞ ¼

1 2π  θ  arccos þ2sπ pffiffiffiffiffiffiffiffiffiffiffiffiffiR : R2  1

We have the following conclusions about Q ðλÞ ¼ λ þ 1 ze  λτ :

ð2:9Þ

(i) If Rr 1, then Q ðλÞ has no purely imaginary zeros for all τ Z 0. pffiffiffiffiffiffiffiffiffiffiffiffiffi (ii) If R 4 1, then Q ðλÞ has a purely imaginary zero i R2  1 when τ ¼ τbs , where s ¼ 1; 2; …, if θ A ½0; arccos R1Þ, else s ¼ 0; 1; 2; …; Q ðλÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi has a purely imaginary zero  i R2  1 when τ ¼ τ~ s , where s ¼ 1; 2; …, if θ A ð2π  arccos 1R; 2π Þ, else s ¼ 0; 1; 2; … . Moreover,

Q ðλÞ has no purely imaginary zeros when τ is any other values. (iii) If R 4 1, then for each fixed s satisfied τbs Z 0, there exist δ 4 0 and smooth mapping λ : ðτbs  δ; τbs þ δÞ-C such that Q ðλðτÞÞ ¼ 0 for pffiffiffiffiffiffiffiffiffiffiffiffiffi 4 0; for all τ A ðτbs  δ; τbs þ δÞ, λðτbs Þ ¼ i R2  1 and ddτRe λðτÞj τ ¼ b τ s

each fixed s satisfied τ~ s Z 0, the similar conclusion also holds.

Proof. On one hand, for arbitrary β 4 0 we have Q ðiβÞ ¼ iβ þ 1  Reiðθ  τβÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 ¼ 1 þ β ei arccos 1= 1 þ β  Reiðθ  τβÞ : Thus, Q ðiβ Þ ¼ 0 if and only if 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi 2 > > 1 þ β ¼ R; > < 1 arccosqffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ θ  τβ þ 2sπ ; > > > 2 : 1þβ

s A Z:

λ0 ðτbs Þ ¼

τ s Þb τs  zλðτbs Þe  λðb  λðτbs Þ½λðτbs Þ þ 1 ¼ : τ s Þb τs 1 þ τbs ½λðτbs Þ þ1 1 þ τbs ze  λðb

Therefore, R2  1

0

Re λ ðτbs Þ ¼

ð1 þ τbs Þ2 þ τbs ðR2  1Þ 2

4 0:

For each fixed s satisfied τ~ s Z 0, the proof is similar and thus omitted. Therefore, the conclusion ðiiiÞ holds. So far, the proof of the lemma is completed.□ Remark 2.1. We include the proof of Lemma 2.1 for the completeness of the presentation. For example, it is easy to see that if z is an imaginary number, then fτbs ðzÞg \ fτ~ s ðzÞg ¼ ∅, implying that Q ðλÞ pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffi has only one purely imaginary zero i R2 1 (  i R2  1) when τ ¼ τbs (τ~ s ). Thus, the presentation of Lemma 2.1 could be regarded as a revised version of the result in [35]. For the sake of convenience, we introduce two sets:

Ω ¼ f0; 1g
f0; 1; …; n  1g; 







Ω0 ¼ ðp; qÞ A Ω; a þð  1Þp c þ bχ q  4 1 : Applying Lemma 2.1 to each factor of (2.7), we can obtain the following conclusion. Lemma 2.2. (i) If a þ ð 1Þp c þ b cos 2qnπ o1, 8 ðp; qÞ A Ω and Ω0 ¼ ∅, then all roots of the characteristic equation (2.5) have negative real parts for all τ Z 0. (ii) If there exists ðp; qÞ A Ω0 . Define   τpqs ¼ τbs a þð  1Þp c þ bχ q ; qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi   βpq ¼ a þ ð  1Þp c þ bχ q 2  1: Then at τ ¼ τpqs Z0, the characteristic equation (2.5) has a pair of purely imaginary roots 7 iβpq ; for each fixed s satisfied τpqs Z 0, there exist δ 4 0 and a smooth mapping λ : ðτpqs  δ; τpqs þ δÞ-C such that λðτpqs Þ ¼ iβpq and det Δðτ; λðτÞÞ ¼ 0 for all τ A ðτpqs  δ; τpqs þ δÞ. Moreover, ddτRe λðτÞj τ ¼ τpqs 4 0. Proof. (i) We note that when τ ¼ 0, the each factor have only one zero point: a þ ð  1Þp c þ bχ q  1, and

Δpq of (2.7)

2qπ 1 o 0: n

On the other hand, for arbitrary β o 0 we have

Reða þ ð  1Þp c þ bχ q  1Þ ¼ a þ ð  1Þp c þ b cos

Q ðiβÞ ¼ iβ þ 1  Reiðθ  τβÞ qffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi 2 2 ¼ 1 þ β eið2π  arccos 1= 1 þ β Þ  Reiðθ  τβÞ :

In addition, the condition a þð  1Þp c þ b cos 2qnπ o 1, 8 ðp; qÞ A Ω implies that 0 is not a root of (2.5) for all τ Z 0. By Ω0 ¼ ∅, i.e.,  a þ ð  1Þp c þ bχ q  r1, 8 ðp; qÞ A Ω and Lemma 2.1(i), we also show that (2.5) has no purely imaginary roots for all τ Z 0. Therefore, the conclusion (i) follows.

Thus, Q ðiβ Þ ¼ 0 if and only if 8 qffiffiffiffiffiffiffiffiffiffiffiffiffi > < 1 þ β 2 ¼ R; > : 2π  arccos

1 pffiffiffiffiffiffiffiffiffi ffi 1þβ

2

¼ θ  τβ  2sπ ;

s A Z:

Therefore, (i) and (ii) hold. If R 4 1, since for each fixed s satisfied τbs Z 0,   d pffiffiffiffiffiffiffiffiffiffi Q ðλÞ ¼ 1 þ zτe  λτ j pffiffiffiffiffiffiffiffiffiffi λ ¼ i R2  1;τ ¼ b τs dλ λ ¼ i R2  1;τ ¼ b τs pffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ 1 þ τbs ði R2  1 þ 1Þ ¼ a 0; there exist δ 4 0 and smooth mapping λ : ðτbs  δ; τbs þ δÞ-C such pffiffiffiffiffiffiffiffiffiffiffiffiffi that Q ðλðτÞÞ ¼ 0, λðτbs Þ ¼ i R2  1. Differentiating (2.9) with respect

(ii) Noticing that a þ ð  1Þp c þ bχ q ¼ a þ ð  1Þp c þ bχ n  q , we have if ðp; qÞ A Ω0 , then ðp; n  qÞ A Ω0 . Moreover, if q¼ 0 and    a þ ð  1Þp c þbχ q 4 0, then τbs a þ ð  1Þp c þ bχ q ¼ τ~ s  1 a þ ð  1Þp c þ     bχ n  q Þ, else τbs a þ ð  1Þp c þ bχ q ¼ τ~ s a þ ð  1Þp c þ bχ n  q . Thus, it follows from Lemma 2.1(ii)–(iii) that the conclusion (ii) holds. This completes the proof.□ It is well-known that if all roots of the characteristic equation (2.5) have negative real parts, then the trivial equilibrium of system (1.2) is asymptotically stable; if the characteristic equation (2.5) has a root with positive real part, then the trivial equilibrium of system (1.2) is unstable. Therefore, we can obtain the following conclusions about the linear stability of system (1.2) from Lemmas 2.1–2.2.

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H. Hu et al. / Neurocomputing 168 (2015) 475–487

Theorem 2.1. If a þ ð  1Þp c þb cos 2qnπ o 1, 8 ðp; qÞ A Ω and Ω0 ¼ ∅, then for all τ Z 0, the equilibrium x^ ¼ 0 of system (1.2) is asymptotically stable. Theorem 2.2. If there exists ðp; qÞ A Ω such that a þ ð 1Þp c þb cos 2qnπ 4 1, then for all τ Z 0, the equilibrium x^ ¼ 0 of system (1.2) is unstable. Theorem 2.3. If a þð  1Þp c þ b cos 2qnπ o 1, 8 ðp; qÞ A Ω and Ω0 a ∅, then the equilibrium x^ ¼ 0 of system (1.2) is asymptotically stable when τ A ½0; τ0 Þ, the equilibrium x^ ¼ 0 of system (1.2) is unstable when τ 4 τ0 , where τ0 ¼ minðp;qÞ A Ω0 fτpq0 g. 3. The local existence and spatio-temporal patterns of periodic solutions In this section, we assume that Ω0 a ∅. For each fixed ðp; qÞ A Ω0 and each fixed s, let τn ¼ τpqs . By Lemma 2.2(ii), Aðτn Þ has a pair of purely imaginary eigenvalues 7iβpq . For the sake of simplicity, we also require ðHÞ the eigenvalues 7 iβ pq of Aðτn Þ are simple, and all other eigenvalues Aðτn Þ are not integer multiple of 7iβpq . Under assumption ðHÞ, by Lemma 2.2(ii) and the well-known Hopf bifurcation theorem (see [40]), we can draw that when τ passes the critical value τn , system (1.2) undergoes Hopf bifurcation, that is, there exists a branch of periodic solution with the period ωn ðτÞ bifurcated from the equilibrium x^ ¼ 0. In what follows, we aim to analyze the spatio-temporal patterns of the bifurcated periodic solutions. Define a mapping F : Cð½  τ; 0; R2n Þ-R2n by ðF ðϕÞÞij ¼  ðϕÞi;j ð0Þ þ f ððϕÞij ð  τÞÞ þ gððϕÞi;j þ 1 ð  τÞÞ þ hððϕÞi þ 1;j ð  τÞÞ; iðmod 2Þ; jðmod nÞ; where

ϕ ¼ ðϕ00 ; ϕ01 ; …; ϕ0;n  1 ; ϕ10 ; ϕ11 ; …; ϕ1;n  1 ÞT A Cð½ τ; 0; R2n Þ;  F ðϕÞ ¼ ðF ðϕÞÞ00 ; ðF ðϕÞÞ01 ; …; ðF ðϕÞÞ0;n  1 ; ðF ðϕÞÞ10 ; ðF ðϕÞÞ11 ; …; ðF ðϕÞÞ1;n  1

T

A R2n :

Lemma 3.1. Let ρ2 and ρn denote the generator of cyclic group Z2 and Zn respectively, and define the action of Z2
Zn on R2n by ððρ2 ; 1Þ  xÞi;j ¼ xi þ 1;j ;

ðð1; ρn Þ  xÞi;j ¼ xi;j þ 1 ;

where the subscripts i; j are written modulo 2 and n respectively. Then F is Z2
Zn -equivariant. Proof. For ϕ A C and iðmod 2Þ; jðmod nÞ, we have ðF ððρ2 ; 1ÞϕÞÞij ¼  ððρ2 ; 1ÞϕÞij ð0Þ þ f ðððρ2 ; 1ÞϕÞij ð  τÞÞ þ gðððρ2 ; 1ÞϕÞi;j þ 1 ð  τÞÞ þhðððρ2 ; 1ÞϕÞi þ 1;j ð  τÞÞ ¼  ðϕÞi þ 1;j ð0Þ þf ððϕÞi þ 1;j ð  τÞÞ þ gððϕÞi þ 1;j þ 1 ð  τÞÞ þ hððϕÞij ð  τÞÞ ¼ ððρ2 ; 1ÞF ðϕÞÞij ; ðF ðð1; ρn ÞϕÞÞij ¼  ðð1; ρn ÞϕÞij ð0Þ þ f ððð1; ρn ÞϕÞij ð  τÞÞ þ gððð1; ρn ÞϕÞi;j þ 1 ð  τÞÞ þhððð1; ρn ÞϕÞi þ 1;j ð  τÞÞ

where for θ A ½  τ; 0,

ϵ1 ðθÞ ¼ Refeiβpq θ vpq g ¼ cos ðβpq θÞRefvpq g  sin ðβpq θÞImfvpq g; ϵ2 ðθÞ ¼ Imfeiβpq θ vpq g ¼ sin ðβpq θÞRefvpq g þ cos ðβpq θÞImfvpq g: Proof. It follows from the proof of Lemma 2.2 that the eigenspace of Aðτn Þ associated with 7 iβ pq is spanned by eiβpq θ vpq and e  iβpq θ v pq . Hence, the space has the real basis fϵ1 ; ϵ2 g. In addition, the condition ðHÞ makes sure that the eigenspace Aðτn Þ associated iβ pq is of dimension 1, and the root λ ¼ iβ pq of characteristic

equation Δðτn ; λÞ ¼ 0 is simple. According to the folk theorem in functional differential equations (see [41]), U 7 iβpq ðAðτn ÞÞ must

coincide with the eigenspace of Aðτn Þ associated with 7 iβpq . This completes the proof.□ Let Γ ¼ Z2
Zn and ω ¼ β2π , and let Pω be the Banach space of pq all continuous

ω-periodic functions x : R-R2n . Then for the circle

group S , Γ
S1 acts on Pω by  ω ðγ ; eiθ Þ  xðtÞ ¼ γ  x t þ θ ; ðγ ; eiθ Þ A Γ
S1 ; 2π 1

x A Pω :

Denote by SP ω the subspace of Pω consisting of all solutions of system (2.1) with τ ¼ τn . Then

ω-periodic

SP ω ¼ fx1 ϵ1 ðtÞ þ x2 ϵ2 ðtÞj x1 ; x2 A Rg; where

ϵ1 ðtÞ ¼ Refeiβpq t vpq g ¼ cos ðβpq tÞRefvpq g  sin ðβpq tÞImfvpq g; ϵ2 ðtÞ ¼ Imfeiβpq t vpq g ¼ sin ðβpq tÞRefvpq g þ cos ðβpq tÞImfvpq g: Therefore, for ϵ1 ðtÞ and ϵ2 ðtÞ we have the following properties. Lemma k ¼ 1; 2.

3.3. ðρ2 ; 1Þ  ϵk ¼ ϵk ðt þ pω=2Þ,

ð1; ρn Þ  ϵk ¼ ϵk ðt þ qω=nÞ,

Proof. For iðmod 2Þ; jðmod nÞ and t A R, note that  ϵ1i;j ðtÞ ¼ cos βpq t þ ipπ þ 2jqπ =n ;  ϵ2i;j ðtÞ ¼ sin βpq t þ ipπ þ 2jqπ =n ; we have    ðρ2 ; 1Þ  ϵ1 ðtÞ i;j ¼ ϵ1i þ 1;j ðtÞ ¼ cos β pq t þ ði þ1Þpπ þ 2jqπ =n ;  ¼ cos β pq t þipπ þ 2jqπ =n þ pπ ! pπ 1 ; ¼ ϵi;j t þ

βpq

   ðρ2 ; 1Þ  ϵ2 ðtÞ i;j ¼ ϵ2i þ 1;j ðtÞ ¼ sin β pq t þ ði þ 1Þpπ þ2jqπ =n ;  ¼ sin βpq t þ ipπ þ 2jqπ =n þ pπ ! pπ 2 : ¼ ϵi;j t þ

βpq

Therefore

¼  ðϕÞi;j þ 1 ð0Þ þf ððϕÞi;j þ 1 ð  τÞÞ

ðρ2 ; 1Þ  ϵk ¼ ϵk ðt þ pω=2Þ;

þ gððϕÞi þ 1;j þ 2 ð  τÞÞ þ hððϕÞi þ 1;j þ 1 ð  τÞÞ

We can also prove similarly that

¼ ðð1; ρn ÞF ðϕÞÞij :

ð1; ρn Þ  ϵk ¼ ϵk ðt þ qω=nÞ;

k ¼ 1; 2:

k ¼ 1; 2:□

Therefore, F is Z2
Zn -equivariant. This completes the proof.□ Lemma 3.2. Assume that ðHÞ holds, then the generalized eigenspace U 7 iβpq ðAðτn ÞÞ consists of eigenvector of Aðτn Þ associated with 7 iβ pq . Moreover,

In order to characterize the spatial–temporal symmetry of a bifurcation of periodic solutions of system (1.2), we consider a subgroup of Γ
S1

U 7 iβpq ðAðτn ÞÞ ¼ fa1 ϵ1 þ a2 ϵ2 ; a1 ; a2 A Rg;

Σ ¼ 〈ðρ2 ; 1; e  ipπ Þ; ð1; ρn ; e  i2qπ=n Þ〉:

H. Hu et al. / Neurocomputing 168 (2015) 475–487

For the subgroup Σ, the fixed-point set   FixðΣ ; SP ω Þ ¼ x A SP ω j ðγ ; θÞx ¼ x; 8 ðγ ; θÞ A Σ is a subspace of SP ω . Lemma 3.3 implies that FixðΣ ; SP ω Þ ¼ SP ω . Therefore, we can apply the equivariant Hopf bifurcation theorem for functional differential equations (see [35]) to draw the following result on the spatio-temporal patterns of the bifurcated periodic solutions of system (1.2). Theorem 3.1. Assume that there exists ðp; qÞ A Ω0 and ðHÞ is satisfied. Then near each τn , there exists a branch of periodic solutions bifurcated from the equilibrium x^ ¼ 0. More precisely, there exist εn 4 0, δn 4 0 and a C1-smooth mapping ðτn ; ωn ; xn Þ : ½0; εn Þ
½0; 2π -R
R þ
CðR; R2n Þ, such that for each θ A ½0; 2π ; α A ð0; εn Þ, xn ¼ xn ðt; α; θÞis an ωn ¼ ωn ðα; θÞ-periodic solution of system (1.2) with τ ¼ τn þ τn ðα; θÞ, and



 pωn q ωn ¼ xni;j þ 1 t  ; iðmod 2Þ; jðmod nÞ; xni;j ðtÞ ¼ xni þ 1;j t  2 n

ωn ð0; θÞ ¼

2π ; βpq

τn ð0; θÞ ¼ 0;

 2jqπ  θ þ oðj α j Þ xni;j ðt; α; θÞ ¼ α cos β pq t þ ipπ þ n

as α-0: ð3:1Þ

2π

Furthermore, for j τ  τn j o δ , j ω  β j o δ , every ω  -periodic pq n pωn solution  x(t)n of system (1.2) with j xt j o δ , xi;j ðtÞ ¼ xi þ 1;j t  2 ¼ xi;j þ 1 t  qω must be of the above type. n n

n

n

n

4. Direction and stability of the bifurcation In this section, we will discuss the Hopf bifurcation described in Theorem 3.1 in more details, that is, we will work out an algorithm for determining the direction of bifurcation and the stability of the bifurcating periodic solutions. Due to it, we compute the normal form of the system on the center manifold associated with the pair of purely imaginary solutions. Throughout this section, it is necessary to assume that f ; g; h A C 3 ðR; RÞ: For the sake of computation, we rewrite (1.2) as the following system of functional differential equations with the transformation t↦t=τ ( u_ 0j ðtÞ ¼  τu0j ðtÞ þ τf ðu0j ðt  1ÞÞ þ τgðu0;j þ 1 ðt  1ÞÞ þ τhðu1j ðt  1ÞÞ; u_ 1j ðtÞ ¼  τu1j ðtÞ þ τf ðu1j ðt  1ÞÞ þ τgðu1;j þ 1 ðt  1ÞÞ þ τhðu0j ðt  1ÞÞ; ð4:1Þ where j ¼ 0; 1; …; n  1ðmod nÞ. Let τ ¼ τn þ μ. Then μ ¼ 0 is the bifurcation value of the system  (4.1). We choose the phase space of system (4.1) as C ½  1; 0; C2n , where for convenience in computation we use C

2n

2n

instead of R .

479

Note that the Riesz representation allows us to represent the linear operator Lμ as Z 0 Lμ ϕ ¼ dηðθ; μÞϕðθÞ; ϕ A C; ð4:5Þ 1

where ηð; μÞ : ½  1; 0-R2n
R2n is a matrix-valued function of bounded variation defined by (  ðτn þ μÞId2n ; θ ¼ 0; ηðθ; μÞ ¼  ðτn þ μÞM δðθ þ 1Þ; θ A ½  1; 0Þ ðδðθÞ is the Dirac functionÞ: ð4:6Þ For each fixed (p,q), letting βn ¼ τn βpq ; Λ0 ¼ fiβn ;  iβn g, recall that the characteristic matrix of linearized system of (4.2) is

 λ Δn ðτ; λÞ ¼ τΔ τ; ; λ A C; ð4:7Þ

τ

then det Δ ðτn ; 7iβ n Þ ¼ 0. Hence, by Lemma 2.2 and the assumption ðHÞ, we note that the non-resonance conditions for Λ0 hold. Next, for ϕ A C, we define a operator 8 > < dϕðθÞ ; θ A ½  1; 0Þ; ð4:8Þ Aμ ϕðθÞ ¼ R dθ > 0 :  1 dηðs; μÞϕðsÞ; θ ¼ 0: n

Denote C n ≔Cð½0; 1; C2nn Þ, where C2nn is the 2n-dimensional space of complex row vectors. For ψ A C n , the adjoint operator An0 of A0 is defined by 8 > <  dψ ðsÞ ; s A ð½0; 1; An0 ψ ðsÞ ¼ R ds ð4:9Þ > 0 T :  1 dη ðξ; 0Þψ ð  ξÞ; s ¼ 0; where ηT represents the transpose of η. Define the so-called adjoint bilinear form in C n
C as Z 0 Z θ 〈ψ ðsÞ; ϕðθÞ〉 ¼ ψ ð0Þϕð0Þ  ψ ðξ  θÞ dηðθ; 0ÞϕðξÞ dξ; ψ A C n ; ϕ A C: 1

0

It is easy to see that 7iβn are the eigenvalues of A0. Let φ1 ðθÞ and φ2 ðθÞ denote the eigenvectors of A0 associated with the eigenvalues iβ n and  iβn respectively, that is, A0 φ1 ðθÞ ¼ iβ n φ1 ðθÞ;

A0 φ2 ðθÞ ¼ iβ n φ2 ðθÞ:

By (2.6) and (4.7), we have

Δn ðτn ; iβn Þvpq ¼ 0; thus we can choose

φ1 ðθÞ ¼ vpq eiβ θ ; n

φ2 ðθÞ ¼ v pq e  iβ θ ;

θ A ½  1; 0:

n

Similarly, we can also choose

ψ 1 ðξÞ ¼ DvTpq eiβ ξ ; n

ψ 2 ðξÞ ¼ Dv Tpq e  iβ ξ ; n

ξ A ½0; 1;

Letting uðtÞ ¼ ðu00 ðtÞ; u01 ðtÞ; …; u0;n  1 ðtÞ; u10 ðtÞ; u11 ðtÞ; …; u1;n  1 ðtÞÞT and ut ðθÞ ¼ uðt þ θÞ A C system (4.1) is equivalent to the following equation of operator form:

as the eigenvectors of A0 associated with the eigenvalues iβ n and iβ n respectively, where D is a constant. Using the formal adjoint theory for DDEs (see [40]), the phase C is decomposed by Λ0 as

_ ¼ Lμ ut þ Fðut ; μÞ; uðtÞ

C ¼ P  Q;

ð4:2Þ

where Lμ φ ¼  ðτn þ μÞφð0Þ þ ðτn þ μÞM φð  1Þ;

ð4:3Þ

and Fðφ; μÞ ¼

τn þ μ

τn þ μ

2

6

M″ ð0Þðφð  1ÞÞ2 þ

 4 M‴ ð0Þðφð  1ÞÞ3 þ Oðφ Þ ð4:4Þ

where MðuÞ denotes a 2n
2n matrix-valued function defined by MðuÞ≔circðM1 ðuÞ; M2 ðuÞÞ, where M1 ðuÞ ¼ circðf ðuÞ; gðuÞ; 0; …; 0Þ is a circulant matrix of order n, M2 ðuÞ ¼ hðuÞ  Idn .

n

where P is the generalized eigenspace for A0 associated with the eigenvalues in Λ0, Q ¼ fφ A C∣〈ψ ðsÞ; φðθÞ〉 ¼ 0 for all ψ A P n g, and the dual space Pn is the generalized eigenspace for An0 associated with the eigenvalues Λ0. Furthermore, the dual bases ΦðθÞ and Ψ ðξÞ of P and Pn respectively are given by the following form:

ΦðθÞ ¼ ðφ1 ðθÞ; φ2 ðθÞÞ;

Ψ ðξÞ ¼ ðψ 1 ðξÞ; ψ 2 ðξÞÞT :

According to the condition 〈Φ; Ψ 〉 ¼ Id2 , we can choose D¼

1 : 2nð1 þ τn þiβn Þ

480

H. Hu et al. / Neurocomputing 168 (2015) 475–487

As the statement in [37,38], to discuss the normal form of system (4.2), it is necessary to enlarge the phase space C in such a way that (4.2) is written as an abstract ordinary differential system. An adequate phase space to accomplish this is the space BC defined by BC≔fψ : ½  1; 0-C2n ∣ψ is continuous on ½  1; 0Þ; ( lim ψ ðθÞ A C2n g; θ-0

with the sup norm. The elements of BC are of the form ψ ¼ φ þ X 0 α, where φ A C; α A C2n , and ( θ A ½  1; 0Þ; 0; ð4:10Þ X 0 ðθ Þ ¼ Id2n ; θ ¼ 0; so that BC is identified with C
C2n with j φ þX 0 α j ¼ j φ j C þ j α j C2n . The canonical projection π : BC-P is defined by

π ðφ þ X 0 αÞ ¼ Φ½ðΨ ; φÞ þ Ψ ð0Þ α;

φ A C; α A C2n :

Hence BC can be decomposed by

Λ0 as

the

norm

At step k, by a change of variables 1 ^ þ U k ðz^ Þ; ðz; yÞ ¼ ðz^ ; yÞ k!

ð4:16Þ

where z; z^ A C2 ; y; y^ A Q 1 ; U k ¼ ðU 1k ; U 2k Þ A V 3k ðC2 Þ
V 3k ðQ 1 Þ, becomes 8 k X > 1 1 1 1 > > z_ ðtÞ ¼ Bz þ g ðz; y; μÞ þ f~ ðz; y; μÞ þ h:o:t ., > > ℓ! ℓ ðk þ 1Þ! k þ 1 < ℓ¼2 ℓ¼2

ð4:17Þ 1 2 where g k ¼ f~k  Mk U k ; g k ¼ ðg 1k ; g 2k Þ; f~ k ¼ ðf~ k ; f~ k Þ, and the operator 1 2 Mk ¼ ðMk ; Mk Þ is defined as follows:

Obviously, Q ⊊Ker π . We then write (4.2) as _ ¼ L0 ut þ F~ ðut ; μÞ; uðtÞ

ð4:11Þ

M1k : V 3k ðC2 Þ-V 3k ðC2 Þ;

M2k : V 3k ðQ 1 Þ-V 3k ðKer π Þ;

ðM1k h1 ÞðzÞ ¼ Dz h1 ðzÞBz  Bh1 ðzÞ;

where F~ ðut ; μÞ ¼ Lμ ut  L0 ut þ Fðut ; μÞ: According to the above decomposition of BC, ut can be written as ut ¼ ΦzðtÞ þ yt , where zðtÞ A C2 , yt A C 1 \ Ker π ¼ C 1 \ Q ≔Q 1 , C1 is the subset of C consisting of continuously differentiable functions. We note that the operator π can commute with the differential operator and the linear operator Lμ. Let A1Q be the restriction of A on Q1, i.e.,

ðM2k h2 ÞðzÞ ¼ Dz h2 ðzÞBz  AQ 1 ðh2 ðzÞÞ:

Repeating the above step, we can obtain the following normal form of system (4.14) 8 P 1 1 > > g ℓ ðz; y; μÞ þ ⋯; z_ ðtÞ ¼ Bz þ > < ℓ Z 2ℓ! ð4:18Þ P 1 2 > > _ ¼ AQ 1 y þ g ℓ ðz; y; μÞ þ ⋯: > : yðtÞ ℓ Z 2ℓ! Let e1 ¼ ð1; 0ÞT ; e2 ¼ ð0; 1ÞT . By the definition of M1ℓ , for any z ¼ ðz1 ; z2 Þ A C2 ; μ A C and r ¼ ðr 1 ; r 2 ; r 3 Þ A N30 satisfied j rj ¼ ℓ, we have

AQ 1 : Q 1  Ker π -Ker π such that _ þX 0 ½L0 φ  φ _ ð0Þ: AQ 1 φ ¼ φ

ð4:12Þ

ð4:13Þ

where B ¼ diagðiβn ;  iβn Þ. For a normed space Y, V m ℓ ðYÞ denotes the linear space of homogeneous polynomials of degree ℓ in m variables z1 ; z2 ; …; zm with coefficients in Y , i.e., ( X   cr xr  zr ¼ zr1 zr2 ⋯zrm ; r ¼ ðr 1 ; r 2 ; …; r m Þ A Nm ; cr A Y ; V m ðYÞ ¼ 1

2

m

M1ℓ ðzr11 zr22 μr3 ek Þ ¼ iβn ðr 1  r 2 þ ð 1Þk Þzr11 zr22 μr3 ek ;

k ¼ 1; 2; ℓ Z 2:

Hence,

Hence, system (4.11) is equivalent to 8 < z_ ðtÞ ¼ Bz þ Ψ ð0Þ F~ ðΦz þy; μÞ; _ ¼ AQ 1 y þ ðI  π ÞX 0 F~ ðΦz þ y; μÞ; z A C2 ; y A Q 1 ; : yðtÞ

j rj ¼ ℓ

(4.15)

k > X 2 1 2 1 > > _ ¼ AQ 1 y þ > g ðz; y; μÞ þ f~ ðz; y; μÞ þ h:o:t ., yðtÞ > : ℓ! ℓ ðk þ 1Þ! k þ 1

BC ¼ P  Ker π :

ℓ

of orders 2; …; k  1 have already been performed, leading to 8 kX 1 > 1 1 1 1 > > z_ ðtÞ ¼ Bz þ g ℓ ðz; y; μÞ þ f~ k ðz; y; μÞ þ h:o:t ., > > ℓ! k! < ℓ¼2 ð4:15Þ kX 1 > 1 2 1 ~2 > > _ > g f yðtÞ ¼ A y þ ðz; y; μ Þ þ ðz; y; μ Þ þ h:o:t .. 1 > Q : ℓ! ℓ k! k ℓ¼2

0

with the norm   X  X  r jc j : cr z  ¼  ∣r∣ ¼ ℓ  jr j ¼ ℓ r Y According to the Taylor formulas, (4.13) can be written as 8 P 1 1 > > f ℓ ðz; y; μÞ; > z_ ðtÞ ¼ Bz þ < ℓ Z 2ℓ! P 1 2 > > _ ¼ A1Q y þ f ℓðz;y;μÞ ; z A C2 ; yA Q 1 : > : yðtÞ ℓ Z 2ℓ!

KerðM12 Þ ¼ spanfμz1 e1 ; μz2 e2 g: KerðM13 Þ ¼ spanfz21 z2 e1 ; μ2 z1 e1 ; z11 z22 e2 ; μ2 z2 e2 g: Especially, we have V 3ℓ ðC2 Þ ¼ ImðM1ℓ Þ  KerðM1ℓ Þ: Therefore, 1 g 1ℓ ðz; 0; μÞ ¼ ðI  P 1ℓ Þf~ ℓ ðz; 0; μÞ;

P 1ℓ

U 1ℓ ðz; μÞ ¼ ðM1ℓ Þ  1 P 1ℓ f~ ℓ ðz; 0; μÞ;

Following the approach in [37,38], the normal form can be obtained by a recursive process of changes of variables. At a step k, the terms of order k Z 2 are computed from the terms of the same order in the original equation and from the terms of lower orders already computed in previous steps. We assume that steps

ℓ Z 2;

V 3ℓ ðC2 Þ-ImðM1ℓ Þ

: is the canonical projection. According where, to the relevant theorem in [37], we can choose Uj2 such that g 2ℓ ðz; 0; μÞ ¼ 0, ℓ Z 2. Therefore, the flow on a local center manifold for system 4.11 at zero is given by 1 1 z_ ðtÞ ¼ BzðtÞ þ g 12 ðz; 0; μÞ þ g 13 ðz; 0; μÞ þ h:o:t .. 2 3!

ð4:14Þ

1

ð4:19Þ

which is in normal form. Based on the discussion above, we next compute the concrete expressions of the second and third order terms g12 and g13 in (4.19), respectively. In order to simplify the notation, we introduce the operator Θ : V 3ℓ ðCÞ-V 3ℓ ðC2 Þ, ! cxr11 xr22 μr3 Θðcxr11 xr22 μr3 Þ ¼ ; c A C; r ¼ ðr 1 ; r 2 ; r 3 Þ A N30 ; jr j ¼ ℓ: cxr11 xr22 μr3 First, note that ðΦð  1ÞzÞ2 ¼ z21 e  2iβn v0;2q þ z22 e2iβn v 0;2q þ 2z1 z2 v00 ;

H. Hu et al. / Neurocomputing 168 (2015) 475–487

ðΦð  1ÞzÞ3 ¼ z31 e  3iβn vp;3q þ z32 e3iβn v p;3q þ 3z21 z2 e  iβn vpq þ 3z1 z22 eiβn v pq ;  ðℓÞ ðℓÞ MðℓÞ ð0Þvpq ¼ f ð0Þ þ ð  1Þp h ð0Þ þ g ðℓÞ ð0Þχ q vpq ; ℓ Z 1;  ðℓÞ ðℓÞ MðℓÞ ð0Þv pq ¼ f ð0Þ þ ð  1Þp h ð0Þ þ g ðℓÞ ð0Þχ  q v pq ; ℓ Z 1; 8 n T > < 2nðD; 0Þ ; q a 0; ; 2 n  n  n o Ψ ð0Þvpq ¼ > : 2nðD; DÞT ; ðp; qÞ A ð1; 0Þ; 1; ; 0; ; 2 2 8 n T > < 2nð0; DÞ ; q a 0; ; 2 n  n  n o Ψ ð0Þv pq ¼ > : 2nðD; DÞT ; ðp; qÞ A ð1; 0Þ; 1; ; 0; ; 2 2



8  n 2n > T > ; 0; ; ; ðp; qÞ 2 = ð0; 0Þ; 0; ð0; 0Þ > < 3 3 

 Ψ ð0Þv0;2q ¼ n 2n > > 2nð0; DÞT ; ðp; qÞ A > ; 0; ; 0; : 3 3 8  n 2n  > T > ; 0; ; ðp; qÞ2 = ð0; 0Þ; 0; > < ð0; 0Þ ; 3 3 

 Ψ ð0Þv 0;2q ¼ n 2n > T > > ; 0; ; 0; : 2nðD; 0Þ ; ðp; qÞ A 3 3 ( p a0 or q a 0; ð0; 0ÞT ; Ψ ð0Þv00 ¼ 2nðD; DÞT ; p ¼ 0; q ¼ 0;

1 1 τn f ðz; 0; μÞ ¼ Ψ ð0Þ½Lμ ðΦzÞ  L0 ðΦzÞ þ M00 ð0ÞðΦð 1ÞzÞ2  2 2 2

τn

Að101Þ ¼ Að011Þ ¼

τn

″

;

Að200Þ ¼ Að020Þ ¼ τ n ðf ð0Þ þ h ð0Þ þ g ð0Þχ Þe 2q

 2iβn

;

″

Að110Þ ¼ 2τn ðf ð0Þ þ h ð0Þ þ g ″ ð0ÞÞ: Therefore, the second order term in ðz1 ; z2 ; μÞ of the normal form is given by 1 1 2 g 2 ðz; 0;

μÞ ¼

1 1 2 ProjKerðM12 Þ f 2 ðz; 0;

μÞ ¼ Θð2nDAð101Þ μz1 Þ:

where  1 1 U 12 ðz; 0Þ ¼ U 12 ðz; μÞμ ¼ 0 ¼ ðM12 Þ  1 ProjImðM1 Þ f 2 ðz; 0; 0Þ ¼ ðM12 Þ  1 f 2 ðz; 0; 0Þ; 2

and U 22 ðz; 0Þ ¼ U 22 ðz; μÞj μ ¼ 0 is determined by the equation 2

Since h i 1 f 3 ðz; 0; 0Þ ¼ Ψ ð0Þ τn M‴ ð0Þðz1 e  iβn vpq þ z2 eiβn v pq Þ3 ;

ℏðzÞ ¼

and write

ℏ20 z21 þ ℏ11 z1 z2 þ ℏ02 z22 :

Since  1 1 f 2 ðz; y; μÞ ¼ f 2 ðz; 0; μÞ þ Ψ ð0Þ 2Lμ ðΦzÞ  2L0 ðΦzÞ  þ τn M″ ð0Þð2Φð  1Þzyð  1Þ þ y2 ð  1ÞÞ ; we have 1  ðDy f Þy ¼ 0;μ ¼ 0 ðℏÞ ¼ 2τ n Ψ ð0ÞM00 ð0ÞΦð  1Þzℏð  1Þ;

ð4:22Þ

2

h i 1 ProjS ðDy f 2 Þℏ ðz; 0; 0Þ  h ″ ″ ¼ Θ 2τn D f ð0Þ þ h ð0Þð  1Þp þg ″ ð0Þχ q e  iβn vT00 h11 ð  1Þ  i ″ ″ þ f ð0Þ þ h ð0Þð  1Þp þ g ″ ð0Þχ  q eiβn v T0;2q h20 ð 1Þ z21 z2 : ð4:23Þ Moreover, by the definition of the operator (4.20) is equivalent to

M22 ; AQ 1

and

π, Eq.

Dz ℏðzÞBz  A1Q ðℏðzÞÞ ¼ ðI  π ÞX 0 ½τn M00 ð0ÞðΦð  1ÞzÞ2 ;

ð4:24Þ

ℏ_ ð11Þ ðθÞ ¼ Að110Þ ΦΨ ð0Þv00 ; ℏ_ ð11Þ ð0Þ  L0 ℏð11Þ ¼ Að110Þ v00 :

ð4:26Þ

1 1 f ðz; 0; μÞ ¼ Θð2nDAð101Þ μz1 Þ; 2 2     (b) if ðp; qÞ A ð1; 0Þ; ð1; n2Þ; ð0; n2Þ ; ðp; qÞ2 = ð0; n3Þ; ð0; 2n 3 Þ , we have 1 1 f ðz; 0; μÞ ¼ Θð2nDðAð101Þ μz1 þ Að011Þ μz2 ÞÞ: 2 2 Thus, U 12 ðz; 0Þ ¼ ð0; 0ÞT ;

In what follows, we compute the cubic term for μ ¼ 0, that is, work out the special expression of the following: h 1 1 g 13 ðz; 0; 0Þ ¼ ProjS f 3 ðz; 0; 0Þ þ 32 ProjS ðDz f 2 Þðz; 0; 0ÞU 12 ðz; 0Þ i 1 þ ðDy f 2 Þðz; 0; 0ÞU 22 ðz; 0Þ ;

ðM22 U 22 Þðz; 0Þ ¼ f 2 ðz; 0; 0Þ:

Define ℏ ¼ ℏðzÞðθÞ by ℏðzÞ ¼

U 22 ðz; 0Þ,

In what follows, we distinguish three cases.  Case (i): If p ¼1, or p ¼0 and q2 = 0; n3; 2n 3 , then (a) if   n n n 2n ðp; qÞ2 = ð1; 0Þ; ð1; 2Þ; ð0; 0Þ; ð0; 2Þ; ð0; 3Þ; ð0; 3 Þ , we have

where ″

‴ ‴ f ð0Þ þ ð  1Þp h ð0Þ þ g ‴ ð0Þχ q De  iβn z21 z2 : ð4:21Þ

(

¼ Ψ ð0Þvpq Að101Þ μz1 þ Ψ ð0Þv pq Að011Þ μz2 1 1 1 þ Ψ ð0Þv0;2q Að200Þ z21 þ Ψ ð0Þv 0;2q Að020Þ z22 þ Ψ ð0Þv00 Að110Þ z1 z2 ; 2 2 2

″



where ℏ_ ¼ ddθℏ. From this system, we obtain ℏ02 ¼ ℏ20 ; ℏ11 ¼ ℏ11 , and two initial value problems (IVPs), ( ℏ_ ð20Þ ðθÞ  2iβ n ℏð20Þ ðθÞ ¼ Að200Þ ΦΨ ð0Þv0;2q ; ð4:25Þ ℏ_ ð20Þ ð0Þ  L0 ℏð20Þ ¼ Að200Þ v0;2q ;

¼ Ψ ð0Þ μ½  Φð0Þz þ M Φð  1Þz þ Ψ ð0ÞM00 ð0ÞðΦð  1ÞzÞ2 2 h  ¼ Ψ ð0Þ μ  ðz1 vpq þ z2 v pq Þ þ a þ ð  1Þpc þ b χ q z1 e  iβn vpq  i þ a þ ð  1Þpc þ b χ  q z2 eiβn v pq h  τn ″ ″ ″ ″ þ Ψ ð0Þ f ð0Þ þ h ð0Þ þ g ″ ð0Þχ 2q z21 e  2iβn v0;2q þ f ð0Þ þ h ð0Þ 2 i   ″ ″ þ g ″ ð0Þχ  2q z22 e2iβn v 0;2q þ 2 f ð0Þ þ h ð0Þ þg ″ ð0Þ z1 z2 v00

″

1

namely, to ( _  Dz ℏðzÞBz ¼ ΦΨ ð0Þ½τn M00 ð0ÞðΦð 1ÞzÞ2 ; ℏðzÞ _ ℏðzÞð0Þ  L0 ℏðzÞ ¼ τn M00 ð0ÞðΦð  1ÞzÞ2 ;

thus

iβ n

ProjS f 3 ðz; 0; 0Þ ¼ 6nτn Θ

481

consequently h i 1 ProjS ðDz f 2 Þðz; 0; 0ÞU 12 ðz; 0Þ ¼ ð0; 0ÞT :

ð4:27Þ

In this case, the IVPs (4.25) and (4.26) become (

ℏ_ ð20Þ ðθÞ  2iβ n ℏð20Þ ðθÞ ¼ 0; ℏ_ ð20Þ ð0Þ  L0 ℏð20Þ ¼ Að200Þ v0;2q ; ( ℏ_ ð11Þ ðθÞ ¼ 0; ℏ_ ð11Þ ð0Þ  L0 ℏð11Þ ¼ Að110Þ v00 :

ð4:20Þ Solving the two IVPs above, we get ℏð20Þ ðθÞ ¼

Að200Þ e2iβn θ v0;2q ; 2iβ n þ τn ða þcÞτn e  2iβn  bτn e  2iβn χ 2q

ð4:28Þ

482

ℏð11Þ ðθÞ ¼

H. Hu et al. / Neurocomputing 168 (2015) 475–487

Að110Þ v00 : ð1  a  b  cÞτn

ð4:29Þ

Therefore, together with (4.21), (4.23), (4.27), (4.28) and (4.29), we obtain 1 1 g ðz; 0; 0Þ ¼ ΘðB2 z21 z2 Þ; 3! 3 where

 ‴ ‴ B2 ¼ nτn f ð0Þ þð  1Þp h ð0Þ þ g ‴ ð0Þχ q De  iβn  ″ ″ nD f ð0Þ þ ð  1Þp h ð0Þ þ g ″ ð0Þχ q Að110Þ e  iβn þ 1  a  b c  ″ ″ nDτn f ð0Þ þ ð  1Þp h ð0Þ þ g ″ ð0Þχ  q Að200Þ e  iβn þ 2iβ n þ τn  τn ða þ b þ cÞe  2iβn  ‴ ‴ ¼ nτn f ð0Þ þ ð  1Þp h ð0Þ þ g ‴ ð0Þχ q De  iβn   ″ ″ ″ ″ 2nDτn f ð0Þ þ ð  1Þp h ð0Þ þg ″ ð0Þχ q f ð0Þ þ h ð0Þ þ g ″ ð0Þ e  iβn þ 1 a  b c   ″ ″ ″ p ″ 2 nDτn f ð0Þ þ ð  1Þ h ð0Þ þ g ″ ð0Þχ  q f ð0Þ þ h ð0Þ þ g ″ ð0Þχ 2q e  3iβn þ : 2iβn þ τn  τ n ða þ b þ cÞe  2iβn   Case (ii): If p ¼0 and q A n3; 2n 3 , then 8 n 1 > qa ; < Θð2nDðAð101Þ μz1 þ 2Að020Þ z22 ÞÞ; 1 1 2 f ðz; 0; μÞ ¼ n > 2 2 : Θð2nDðAð101Þ μz1 þ Að011Þ μz2 þ 12Að020Þ z22 ÞÞ; q ¼ : 2 Therefore,

 2nD U 12 ðz; 0Þ ¼ Θ  Að020Þ z22 : 3iβn

ð4:30Þ

In this case, the IVPs (4.25) and (4.26) become ( ℏ_ ð20Þ ðθÞ  2iβn ℏð20Þ ðθÞ ¼ 2nDv p;q A200 e  iβn θ ; ℏ_ ð20Þ ð0Þ  L0 ℏð20Þ ¼ Að200Þ v0;2q ; ( ℏ_ ð11Þ ðθÞ ¼ 0; ℏ_ ð11Þ ð0Þ  L0 ℏð11Þ ¼ Að110Þ v00 :

ℏð11Þ ðθÞ ¼

Að200Þ e2iβn θ v0;2q ; 2iβn þ τn  ða þ cÞτn e  2iβn  bτn e  2iβn χ 2q

Að110Þ v00 : ð1  a  b  cÞτn

Therefore, h i 1 ProjS ðDz f 2 Þðz; 0; 0ÞU 12 ðz; 0Þ 

2

 4n 2 ¼Θ  D2 Að200Þ Að110Þ þ DDðA2ð110Þ þ Að200Þ Að020Þ Þ z21 z2 : 3 iβ n ð4:33Þ In this case, the IVPs (4.25) and (4.26) become ( ℏ_ ð20Þ ðθÞ  2iβ n ℏð20Þ ðθÞ ¼ 4nReðDeiβn θ ÞAð200Þ v00 ; ℏ_ ð20Þ ð0Þ  L0 ℏð20Þ ¼ Að200Þ v00 ; ℏ_ ð11Þ ðθÞ ¼ 4nReðDeiβn θ ÞAð110Þ v00 ; ℏ_ ð11Þ ð0Þ L0 ℏð11Þ ¼ Að110Þ v00 :

Solving the two IVPs above, we obtain " # e2iβn θ 2nD iβn θ 2nD  iβn θ Að200Þ v00 ; ℏð20Þ ðθÞ ¼  e  e 3iβn 2iβn þ τn  τn ðaþ b þ cÞe  2iβn iβn

ℏð11Þ ðθÞ ¼



 1 2n þ ðDeiβn θ De  iβn θ Þ Að110Þ v00 : ð1  a  b  cÞτn iβn

ð4:35Þ

Therefore, together with (4.21), (4.23), (4.33), (4.34) and (4.35), we get 1 1 g ðz; 0; 0Þ ¼ ΘðB2 z21 z2 Þ; 3! 3

2n DAð200Þ e  iβn θ v pq 3iβn

þ

Case (iii): If p ¼0 and q ¼0, then



1 1 1 1 f 2 ðz; 0; μÞ ¼ Θ 2nD Að101Þ μz1 þ Að011Þ μz2 þ Að020Þ z22 þ Að200Þ z21 2 2 2  1 þ Að110Þ z1 z2 Þ; 2



 2nD 1 1 U 2 ðz; 0Þ ¼ Θ : Að200Þ z21  Að110Þ z1 z2  Að020Þ z22 3 iβ

ð4:34Þ

Solving the two IVPs above, we obtain ℏð20Þ ðθÞ ¼ 

 2n2 DD τ2n g ″ ð0Þðχ  2q  χ  q Þ ‴ ‴ ¼ nτn f ð0Þ þ h ð0Þ þ g ‴ ð0Þχ q De  iβn þ 3iβ n   ″ ″ ″ ″ 2nDτn f ð0Þ þ h ð0Þ þ g ″ ð0Þχ q f ð0Þ þh ð0Þ þ g ″ ð0Þ e  iβn þ 1abc   ″ ″ ″ ″ 2 ″ nDτn f ð0Þ þ h ð0Þ þg ð0Þχ  q f ð0Þ þ h ð0Þ þ g ″ ð0Þχ 2q e  3iβn : þ 2iβn þ τn  τn ða þb þ cÞe  2iβn

(

and

! h i 8n2 DD 1 Að200Þ Að020Þ z21 z2 : ProjS ðDz f 2 Þðz; 0; 0ÞU 12 ðz; 0Þ ¼ Θ 3iβn

 ″ ″ nDτn f ð0Þ þh ð0Þ þ g ″ ð0Þχ  q Að200Þ e  iβn  þ g ″ ð0Þχ  q Að200Þ e2iβn þ 2iβ n þ τn  τn ða þ b þ cÞe  2iβn

where ð4:31Þ

ð4:32Þ

Therefore, together with (4.21), (4.23), (4.30), (4.31) and (4.32), we have 1 1 g ðz; 0; 0Þ ¼ ΘðB2 z21 z2 Þ; 3! 3 where  2n2 DD ‴ ‴ Að200Þ Að020Þ B2 ¼ nτn f ð0Þ þh ð0Þ þ g ‴ ð0Þχ q De  iβn þ 3iβn  ″ ″ nD f ð0Þ þ h ð0Þ þ g ″ ð0Þχ q Að110Þ e  iβn 2n2 DD τ2  ″ ″ n  þ f ð0Þ þ h ð0Þ 1  a  b c 3iβ n

 n2 D ‴ ‴  DAð200Þ Að110Þ þ DA2ð110Þ B2 ¼ nτn f ð0Þ þ h ð0Þ þg ‴ ð0Þ De  iβn þ iβ n   2 1 1 2n þ DAð200Þ Að020Þ þ nDA2ð110Þ e  iβn þ ðDe  iβn 3 2 ð1 a  b cÞτn iβn " i 1 e  2iβn iβ n iβ n  De Þ þ nDAð110Þ Að200Þ e 2 2iβ n þ τn  τn ða þ b þ cÞe  2iβn #  2nD  iβn 2nD iβn ‴ ‴ e  e  ¼ nD f ð0Þ þ h ð0Þ þ g ‴ ð0Þ τn e  iβn iβ n 3iβ n "  2 2e  iβn ″ ″ þ nD f ð0Þ þ h ð0Þ þ g ″ ð0Þ τ2n
ð1  a  b cÞτn #  3iβn e þ : 2iβ n þ τn  τn ða þ b þ cÞe  2iβn

H. Hu et al. / Neurocomputing 168 (2015) 475–487

It follows from the discussion above that the normal form of system (4.1) on the center manifold becomes z_ ¼ Bz þ

2niβ n

τn

Dz1 μ

!

 Dz2 μ

þ

!

B2 z21 z2 B2 z1 z22

þOðjzjμ2 þ jzj4 Þ:

ð4:36Þ

Changing to real coordinates w, where z1 ¼ w1  iw2 ; z2 ¼ w1 þ iw2 , and then to polar coordinates ðρ; ξÞ, w1 ¼ ρ cos ξ; w2 ¼ ρ sin ξ, Eq. (4.36) can become the following form: 8   < ρ_ ¼ K 1 μρ þ K 2 ρ3 þOðμ2 ρ þ ðρ; μÞ4 Þ;   : ξ_ ¼  β n þ Oððρ; μÞÞ;

τn

ImðDÞ ¼

ðβ n Þ2

τn ½ð1 þ τn Þ2 þ ðβn Þ2 

To supplement our theoretical results, we let n ¼3 and consider a particular Hopfield model with Z2
Z3 symmetry 8 x_ 00 ðtÞ ¼ > > > > > x_ 01 ðtÞ ¼ > > > > < x_ 02 ðtÞ ¼ > > x_ 10 ðtÞ ¼ > > > > x_ 11 ðtÞ ¼ > > > : x_ ðtÞ ¼

 x00 ðtÞ þatanhðx00 ðt  τ ÞÞþ 0:5tanhðx01 ðt  τÞÞ þ ctanhðx10 ðt  τ ÞÞ;  x01 ðtÞ þatanhðx01 ðt  τ ÞÞþ 0:5tanhðx02 ðt  τÞÞ þ ctanhðx11 ðt  τ ÞÞ;  x02 ðtÞ þatanhðx02 ðt  τ ÞÞþ 0:5tanhðx00 ðt  τÞÞ þ ctanhðx12 ðt  τ ÞÞ;  x10 ðtÞ þatanhðx10 ðt  τ ÞÞþ 0:5tanhðx11 ðt  τÞÞ þ ctanhðx00 ðt  τ ÞÞ;  x11 ðtÞ þatanhðx11 ðt  τ ÞÞþ 0:5tanhðx12 ðt  τÞÞ þ ctanhðx01 ðt  τ ÞÞ;  x12 ðtÞ þatanhðx12 ðt  τ ÞÞþ 0:5tanhðx10 ðt  τÞÞ þ ctanhðx02 ðt  τ ÞÞ:

ð5:1Þ

ð4:37Þ

2nβ

2nβn

5. An example and numerical simulations

12

where K 1 ¼  τn n ImðDÞ; K 2 ¼ ReðB2 Þ. It is well-known that the sign of K 1 K 2 determines the direction of the Hopf bifurcation, the sign of K2 determines the stability of nontrivial periodic orbits inside the center manifold (see [42,43]). Note that K1 ¼ 

483

4 0;

Clearly, the origin is an equilibrium of system (5.1). The characteristic equation of linearization of (5.1) at the origin is given by " # pffiffiffi! 1 3  λτ   λτ  λ þ 1  a 7c  þ i λþ1 ½λ þ1  ða 7c þ 0:5Þe e 4 4 # pffiffiffi! 3  λτ 1  a7c i ¼ 0: e 4 4 Hence, Theorems 2.1– Theorems 2.3 deduce the following result about the stability of the equilibrium of system (5.1).

thus we obtain the main result of this section.

Corollary 5.1. For system (5.1), we have

Theorem 4.1. Assume that ðp; qÞ A Ω0 and ðHÞ holds. Then near each τn 4 0, system (1.2) undergoes a Hopf bifurcation. Moreover, if K 2 4 0ðK 2 o0Þ, then the Hopf bifurcation is supercritical(subcritical), i.e., the bifurcating periodic solutions exist for τ 4 τn ðτ o τn Þ, and the bifurcating periodic orbits on the center manifold are stable (unstable).

Remark 4.1. Note that when the characteristic equation has at least one root with positive real part, the bifurcating periodic orbits must be unstable in the phase space, even though they are stable on the center manifold. Therefore, according to Theorem 2.3, the stability of bifurcating periodic orbits on the center manifold implies the same stability of them in the phase space only if a þ ð 1Þp c þ b cos 2qnπ o 1 for 8 ðp; qÞ A Ω, and τn ¼ τ0 .

pffiffiffiffi (i) If 1 4 13 o a 7 c o 12, then the origin is absolutely stable, that is, stable for all τ Z 0. (ii) If a þ c 4 12 or a c 4 12, then the origin pffiffiffiffi is unstable. pffiffiffiffi (iii) Assume that a 7 c o 12. If a þ c o 1 4 13 or a  c o 1 4 13, then the origin is conditionally stable, that is, the origin is stable when τ A ½0; τ0 Þ and unstable when τ 4 τ0 . Moreover, when τ crosses τ0, system (5.1) undergoes Hopf bifurcation near the origin,

0.3 0.2 0.1 0 −0.1 −0.2

2.5

−0.3

2

−0.4

1.5

0

10

20

30

40

60

70

80

90

100

60

70

80

90

100

0.6

0.5

0.4

0 −0.5

0.2

−1

0

−1.5

−0.2

−2 −2.5 −2

50 time t

1

−0.4 −1.5

−1

−0.5

0

0.5

1

1.5

2

Fig. 2. Region of local stability of the origin in a–c plane. Red region (D1) corresponds to the absolutely stable region, blue region (D2) corresponds to the conditionally stable region and margin region (D3) corresponds to the unstable region. (For interpretation of the references to color in this figure caption, the reader is referred to the web version of this paper.)

−0.6 −0.8

0

10

20

30

40

50 time t

Fig. 3. When ða; cÞ ¼ ð0:1;  0:3Þ, the origin of (5.1) is asymptotically stable. Here, τ ¼ 6.

484

H. Hu et al. / Neurocomputing 168 (2015) 475–487

0.6

where

a  j cj  0:25 1 arccosqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi  arccosqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ða j cj  0:25Þ2 þ 0:1875 ða j cj  0:25Þ2 þ 0:1875 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi τ0 ¼ : ða j cj  0:25Þ2  0:8125

0.5 0.4 0.3

ð5:2Þ

0.2 0.1 0 −0.1 −0.2 −0.3 −0.4

0

50

100

150

200

250

300

350

400

time t 0.6

This corollary implies that the a–c plane can be divided into three regions according to the stability of the origin (see Fig. 2). Next, we perform numerical simulations of the solution of system (5.1) with several parameter values (a,c) in different regions respectively, and the same initial value





  1 π 1 2π 1 2 4π ; sin ðθ þ π Þ; sin θ þ ; ϕ0 ðθÞ ¼ sin θ þ ; sin θ þ 6 2 3 3 3 3 3

 T 5 5π sin θ þ ; sin θ : 6 3

0.4 0.2

0.3

0

0.25 0.2

−0.2

0.15

−0.4

0.1 0.05

−0.6

0

−0.8

0

50

100

150

200

250

300

350

400

−0.05 −0.1

time t

Fig. 4. When ða; cÞ ¼ ð0:2;  0:4Þ, the origin of system (5.1) becomes unstable. Here, τ ¼ 6.

−0.15 −0.2 1950

1955

1960

1965

1970

1975

1980

1985

1990

1995

2000

1980

1985

1990

1995

2000

1980

1985

1990

1995

2000

time t 0.3

0.3

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

−0.05

−0.05

−0.1

−0.1

−0.15

−0.15

−0.2 0

20

40

60

80

100

120

140

160

180

200

time t

−0.2 1950

1955

1960

1965

1970

1975 time t

0.6

0.3 0.25

0.4

0.2

0.2

0.15

0

0.1 0.05

−0.2

0

−0.4

−0.05 −0.1

−0.6

−0.15 −0.8 0

20

40

60

80

100

120

140

160

180

200

time t

Fig. 5. When ða; cÞ ¼ ð  1; 0:2Þ and τ ¼ 1:6 o τ0 , the origin of system (5.1) is asymptotically stable.

−0.2 1950

1955

1960

1965

1970

1975 time t

Fig. 6. When ða; cÞ ¼ ð  1; 0:2Þ and τ ¼ 1:8τ0 , the origin of system (5.1) becomes unstable and there exists a periodic solution with spatio-temporal symmetry.

H. Hu et al. / Neurocomputing 168 (2015) 475–487

Let ða; cÞ ¼ ð0:1;  0:3Þ A D1 , and τ ¼ 6. The numerical simulation is given in Fig. 3. Let ða; cÞ ¼ ð0:2;  0:4Þ A D3 , and the same delay τ ¼ 6. The numerical simulation is given in Fig. 4. Let ða; cÞ ¼ ð  1; 0:2Þ A D2 . We can compute τ0 ¼ 1:7631 by (5.2). Choose τ ¼ 1:6 o τ0 , the numerical simulation is given in Fig. 5. Choose τ ¼ 1:8 4 τ0 , the numerical simulation is given in Fig. 6. These also confirm Theorem 3.1, because in this case we have p ¼ 1; q ¼ 1,



 ωn 2 ωn ¼ x02 t  ; x00 ðtÞ ¼ x01 t  3 3



 n n ω 2ω ¼ x12 t  ; x10 ðtÞ ¼ x11 t  3 3





 ωn ωn ωn x00 ¼ x10 t  ; x01 ¼ x11 t  ; x02 ¼ x12 t  ; 2 2 2

485

Hence, it follows from Theorem 4.1 that the Hopf bifurcation is super-critical and the nontrivial periodic orbit is stable, which are exactly consistent with what Fig. 7 shows.

6. Discussion In the present paper, we have studied the rich dynamics about a hierarchical network of 2n identical cells with delays and Z2
Zn -symmetry. The derivative values of the activation functions f ; g and h at the origin affect significantly the sizes of the absolutely stable region, unstable region and conditionally stable region (see Theorems 2.1 and 2.3). If the derivative value ða; b; cÞ in the conditionally stable region, we also have shown that the time delay τ is able to alter the dynamics of system (1.2) and periodic orbits via Hopf bifurcation occur when τ passes through a critical value. The phase-locked oscillations of bifurcating periodic solutions at the “macro” level and the “micro” level are depicted accurately with the help of the symmetric bifurcation theory of DDEs. Furthermore, we have computed the normal forms directly by using the method due to Faria and Magalháes, to derive the formula determining the direction of bifurcation and the stability of the bifurcating periodic solutions for this large-scale neural network with 2n neurons. Although the structure of system (1.2) is a little simple, it would be of great significance for applications to understand possible mechanisms behind the observed behavior in natural and artificial hierarchically structured networks.

where ωn is the period of the bifurcated periodic solution. ″ ‴ In addition, noting that tanh ð0Þ ¼ 0; tanh ð0Þ ¼  2, from the algorithm in Section 4, we can easily obtain  ‴ ‴ B2 ¼ 3τ0 f ð0Þ þ ð  1Þp h ð0Þ þ g ‴ ð0Þχ q De  iβn pffiffiffi! 1 e  iτ0 β11 3 ¼  τ0  1  0:2  þ i 4 4 1 þ τ0 þ iτ0 β 11

τ0 ðiβ11 þ 1Þ 1 þ τ0 þ iτ0 β 11 ¼  0:75099 0:16932i; K 2 ¼ ReðB2 Þ ¼  0:75099 o 0: ¼

0.4

0.4 0.3

0.3

0.2

0.2

0.1 0.1

0

0

−0.1 −0.2

−0.1

−0.3

−0.2

−0.4 −0.3 −0.4 −0.4

−0.5 −0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

−0.6 −0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

Fig. 7. When ða; cÞ ¼ ð  1; 0:2Þ and τ ¼ 1:8, the projection of the orbit of (5.1) with the initial value ϕ0 ðθÞ on different coordinate planes.

0.4

486

H. Hu et al. / Neurocomputing 168 (2015) 475–487

Besides, we would like to point out that the assumption ðHÞ does not hold for some particular systems. Firstly, we notice that the purely imaginary eigenvalues of the infinitesimal generator AðτÞ of model (1.2) may be multiple in some special cases. For instance, if b¼c and n is even, by 2.8 we have

Δ10 ¼ Δ0;n=2 ¼ λ þ 1  ae  λτ : In this case, if j aj 4 1, then the characteristic equation has purely pffiffiffiffiffiffiffiffiffiffiffiffi ffi imaginary eigenvalues 7 a2  1 of multiplicity 2 when τ ¼ τbs ðaÞ Z0. Secondly, consider the following system with Z2
Z12 -equivariance 8 pffiffi 3 > _ > 0j ðt  τ ÞÞ > x 0j ðtÞ ¼ x0j ðtÞ þð1  2 pÞtanhðx > ffiffi > > < þ 2tanhðx0;j þ 1 ðt  τÞÞ þ 3tanhðx1j ðt  τÞÞ; 2 pffiffi ð6:1Þ > _ 1j ðtÞ ¼ x1j ðtÞ þð1  23Þtanhðx1j ðt  τÞÞ x > > > p ffiffi > > : þ 2tanhðx1;j þ 1 ðt  τÞÞ þ 3tanhðx0j ðt  τÞÞ; 2 where j ¼ 0; 1; …; 11ðmod 12Þ. By (2.7), we can show that the characteristic equation of linearization of (6.1) at the origin is given by 11

1

∏ ∏ Δpq ¼ 0;

q¼0p¼0

where Δpq ¼ λ þ 1  ½1  pute

pffiffi pffiffi p 3 3 iq6π  λτ . 2 þ ð 1Þ 2 þ 2e e

It is easy to com-

Δ11 ¼ λ þ 1  ð1 þ iÞe  λτ ; Δ03 ¼ λ þ 1  ð1 þ 2iÞe  λτ ; Δ1;11 ¼ λ þ 1  ð1  iÞe  λτ ; Δ09 ¼ λ þ 1  ð1  2iÞe  λτ : In this case, Lemma 2.2 implies that the infinitesimal generator AðτÞ has two pairs of purely imaginary eigenvalues 7 i and 7 2i when τ ¼ 2sπ ðs ¼ 0; 1; …Þ. Therefore, system (6.1) undergoes a double Hopf bifurcation with 1:2 resonance when τ passes through these critical values. There are probably complex dynamics in these particular systems, and we leave them for our future work.

Acknowledgments The authors are grateful to the editor and three anonymous reviewers for their constructive comments, which led to a significant improvement of our original manuscript. The work is partially supported by National Natural Science Foundation of China (Nos. 11326116, 11401051, 11101053, 71471020), China Postdoctoral Science Foundation, the Hunan Provincial Natural Science Foundation (No. 2015JJ3013), Scientific Research Fund of Hunan Provincial Education Department (No. 15A003). References [1] J.J. Hopfield, Neurons with graded response have collective computational properties like two-state neurons, Proc. Natl. Acad. Sci. U.S.A. 81 (1984) 3088– 3092. [2] X. Yang, Can neural networks with arbitrary delays be finite-timely synchronized? Neurocomputing 143 (2014) 275–281. [3] X. Yang, Q. Zhu, C. Huang, Lag stochastic synchronization of chaotic mixed time-delayed neural networks with uncertain parameters or perturbations, Neurocomputing 74 (2011) 1617–1625. [4] P.V.D. Driessche, X. Zou, Global attractivity in delayed Hopfield neural network models, SIAM J. Appl. Math. 58 (1998) 1878–1890. [5] J. Cao, M. Dong, Exponential stability of delayed bidirectional associative memory networks, Appl. Math. Comput. 135 (2003) 105–112. [6] S. Arik, V. Tavsanoglu, Global asymptotic stability analysis of bidirectional associative memory neural networks with constant time delays, Neurocomputing 68 (2005) 161–176. [7] Q. Song, Z. Wang, J. Liang, Analysis on passivity and passification of T–S fuzzy systems with time-varying delays, J. Intell. Fuzzy Syst. 24 (1) (2013) 21–30.

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H. Hu et al. / Neurocomputing 168 (2015) 475–487

Haijun Hu received the B.S. and Ph.D. degrees from Hunan University, Changsha, PR China, both in Mathematics/Applied Mathematics, in 2003 and 2010, respectively. Currently, he is working for Changsha University of Science and Technology, PR China, and he is a visiting scholar at the Department of Applied Mathematics, Western University, London, Canada. His research interests are focused on dynamics of artificial neural networks, the bifurcation and chaos of delay differential equations.

Yanxiang Tan was born in Hunan Province, China, in 1980. He received the M.S. degree from School of Mathematics and Computer Science, Changsha University of Science and Technology, Changsha, China, in 2006. He is currently working in School of Mathematics and Computer Science, Changsha University of Science and Technology, he is now also pursuing the Ph.D. degree in College of Mathematics and Econometrics, Hunan University, Changsha, China. His current research interests include dynamics of neural networks, stability theory of functional differential equations, and monotone dynamical systems.

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Chuangxia Huang received the B.S. degree in Mathematics in 1999 from National University of Defense Technology, Changsha, China. From September 2002, he began to pursue his M.S. degree in Applied Mathematics at Hunan University, Changsha, China, and from April 2004, he pursued his Ph.D. degree in Applied Mathematics in advance at Hunan University. He received the Ph.D. degree in June 2006. He is currently a Professor of Changsha University of Science and Technology, Changsha, China. He is the author of more than 50 journal papers. His research interests include dynamics of neural networks, and stability theory of functional differential equations.