# Multiple Nonlinear Oscillations in a𝔻3×𝔻3-Symmetrical Coupled System of Identical Cells with Delays

Authors: Haijun Hu; Li Liu; Jie Mao
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

A coupled system of nine identical cells with delays and𝔻3×𝔻3-symmetry is considered. The individual cells are modelled by a scalar delay differential equation which includes linear decay and nonlinear delayed feedback. By analyzing the corresponding characteristic equations, the linear stability of the equilibrium is given. We also investigate the simultaneous occurrence of multiple periodic solutions and spatiotemporal patterns of the
bifurcating periodic oscillations by using the equivariant bifurcation theory of delay differential equations combined with representation theory of Lie groups. Numerical simulations are carried out to illustrate our theoretical results.


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## Body

## 1. Introduction

Over the past decades, symmetry has become a topic of considerable attention in the research of nonlinear dynamical systems [1–11]. In general, the symmetry reflects a certain spatial invariant of the dynamical systems. The work of Golubitsky et al. [1] shows that systems with symmetry can lead to multiple patterns of oscillation, which are predictable based on the theory of equivariant bifurcations. It is well known that the introduction of time delays into some systems is more reasonable and realistic [12]. Wu and coworkers [3, 13, 14] extended the theory of equivariant Hopf bifurcation to delay differential equations.An artificial neural network is an information processing device that is inspired by the way biological nervous systems, such as the brain, process information simultaneously. It has many applications in different areas including pattern recognition, associative memory, signal processing, and combinatorial optimization. There has been an increasing interest in the investigation of neural networks (see, e.g., [4–6, 9–11, 15]) since Hopfield [16] constructed a simplified neural network model. Ring networks have been found in a variety of neural structures such as cerebellum [17] and even in chemistry and electrical engineering. In the field of neural networks, rings are studied to gain insight into the mechanisms underlying the behavior of recurrent networks [15, 18]. The dynamical behavior of ring networks has been investigated in more detail. For example, in order to understand which patterns occur in a particular system, Huang and Wu [4] studied the following ring neural network of three identical neurons with delayed feedback:
(1)x˙i(t)=-xi(t)+f(xi(t-τ))-[g(xi-1(t-τ))+g(xi+1(t-τ))],i(mod3),
where x˙=dx/dt, xi(t) represents the state of the ith neuron at time t, f represents the nonlinear self-feedback function, g is the connection function between neurons, and τ≥0 is the time delay. Afterward, some researchers have been studying many ring networks with 𝔻n-symmetry (see [5–8]). However, previous work just has considered the individual network but not investigated the interactions between multiple networks.In fact, a wide variety of natural and artificial systems possess a hierarchic structure or functioning and can naturally be modeled by coupled subnetwork. For example, the brain may be conceived as a dynamic network of coupled neurons. In order to describe the complicated interaction between billions of neurons in large neural network, the neurons are often lumped into highly connected subnetworks [19]. Coupled networks of nonlinear dynamical systems can exhibit rich dynamics, such as synchronization, symmetric bifurcation, and chaos. The spatio-temporal dynamics of systems of several coupled nonlinear oscillators is presently receiving great attention and a significant body of research has been carried out [9–11, 20, 21]. It must be pointed out that the hierarchical network of neuronal oscillators with 𝔻3×𝔻3-symmetry investigated in [20, 21] is described by a system of ordinary differential equations (ODEs), and the effect of time delays is not considered.Motivated by the above ideas, in this paper, we consider the two-level hierarchical system which is composed of three coupled modules of interacting nonlinear neuron oscillators with time delays, modeled by the following system of delay differential equations (DDEs):(2)x˙i,j(t)=-xi,j(t)+f(xi,j(t-τ))+g(xi,j-1(t-τ))+g(xi,j+1(t-τ))+h(xi-1,j(t-τ))+h(xi+1,j(t-τ)),i,j=0,1,2(mod3),
where f,g,h∈C1(ℝ,ℝ), f(0)=g(0)=h(0)=0, and h represents the connection function between different modules. The individual elements are represented by a scalar equation, consisting of a linear decay term and a nonlinear time-delayed self-feedback. The architecture of the model is given in Figure 1.Figure 1
Architecture of model (2).It is easy to see that all cells are identical, all couplings within each group are identical, and all groups are identically coupled to each other in this model. Therefore, these lead to a𝔻3×𝔻3-symmetry of the associated system. the model is a natural extension of system (1) and is 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. On the other hand, system (2) can be regarded as a special example of the general Hopfield neural networks with delays [16].Although model (2) is a little simple, it would be of great significance for applications to have a detailed analysis and then to understand possible mechanisms behind the observed behaviour. In this paper, our main purpose is to reveal how the time delay can affect the stability of system (2), the simultaneous occurrence of multiple periodic solutions, and spatio-temporal patterns of the bifurcating periodic oscillations depending on the 𝔻3×𝔻3-symmetry.The rest of the paper is organized as follows. In Section2, the associated characteristic equation is analyzed and the linear stability of the equilibrium is given. In Section 3, we discuss the existence of multiple branches of periodic oscillations and their spatio-temporal patterns with the help of symmetric bifurcation theory of delay differential equations coupled with representation theory of Lie groups. An example and numerical simulations are presented to illustrate the results in Section 4. In Section 5, a brief discussion is drawn to conclude this paper.


## 2. Distribution of Characteristic Roots and Linear Stability

It is clear that (2) admits the trivial solution, x^=0. The linearization of (2) at this equilibrium point is given by
(3)x˙i,j(t)=-xi,j(t)+axi,j(t-τ)+bxi,j-1(t-τ)+bxi,j+1(t-τ)+cxi-1,j(t-τ)+cxi+1,j(t-τ),i,j=0,1,2(mod3),
where a=f′(0),b=g′(0), and c=h′(0). Regarding τ as the parameter, let A(τ) denote the infinitesimal generator of the semigroup generated by linear system (3). We first determine when A(τ) has a pair of purely imaginary eigenvalues.The characteristic matrix of (3) is
(4)Δ(τ,λ)=(λ+1)Id9-Me-λτ,λ∈ℂ,
where Idn denotes the identity matrix of order n, M=circ(M1,cId3,cId3) is a circle block matrix, and M1=circ(a,b,b) is a circulant matrix of order 3. Then we have the following lemma.Lemma 1.
The associated characteristic equation of (3) is
(5)detΔ(τ,λ)=∏q=02‍∏p=02‍Δpq=0,
where Δpq=λ+1-(a+2bcos(2qπ/3)+2ccos(2pπ/3))e-λτ.Proof.
Letχ=ei(2π/3),vq=(1,χq,χ2q)T, and vpq=(vq,χpvq,χ2pvq)T. Then
(6)Mvpq=(a+2bcos2qπ3+2ccos2pπ3)vpq.
Hence
(7)Δ(τ,λ)vpq=[λ+1-(a+cχp+cχ-p+bχq+bχ-q)e-λτ]vpq=[λ+1-(a+2bcos2qπ3+2ccos2pπ3)e-λτ]vpq.
The conclusion is obtained.Note that(8)Δ00=λ+1-(a+2b+2c)e-λτ,Δ01=Δ02=λ+1-(a+2c-b)e-λτ,Δ10=Δ20=λ+1-(a+2b-c)e-λτ,Δ11=Δ12=Δ21=Δ22=λ+1-(a-b-c)e-λτ,
we have
(9)detΔ(τ,λ)=Δ00(Δ01)2(Δ10)2(Δ11)4.
In this paper, for the sake of simplicity, we consider only the case where characteristic equation (5) may have a pair of purely imaginary roots of multiplicity 4, that is, focus on the distribution of zeros of the factor Δ11. The other cases are simpler and can be handled in a similar way, and we omit them. Therefore, throughout this paper we suppose(H)|a+2b+2c|<1
,|a+2c-b|<1, |a+2b-c|<1, a-b-c<-1.The following result about the distribution of the characteristic roots is obtained.Lemma 2.
Assume that(H) holds. Define
(10)τs=arc cos(1/(a-b-c))+2sπ(a-b-c)2-1,s=0,1,2,…,(11)β=(a-b-c)2-1.
Then(i)
for allτ⩾0, all zeros of the factors Δ00,Δ10, and Δ01 have negative real parts,(ii)
whenτ∈[0,τ0), all roots of the characteristic equation (5) have negative real parts; when τ∈(τs,τs+1), the characteristic equation (5) has exactly 2s+2 roots with positive real parts; the other roots have negative real parts; at (and only at) τ=τs, A(τ) has a pair of purely imaginary eigenvalues ±iβ of multiplicity 4, and all other eigenvalues of A(τ) are not integer multiples of ±iβ,(iii)
for each fixeds∈ℕ, there exist δ>0 and C1-smooth mapping λ:(τs-δ,τs+δ)→ℂ such that λ(τs)=iβ and λ(τ)+1-(a-b-c)e-τλ(τ)=0 for all τ∈(τs-δ,τs+δ). Moreover, (d/dτ)Reλ(τ)|τ=τs>0,(iv)
the generalized eigenspaceU±iβ(A(τs)) consists of eigenvector of A(τs) associated with ±iβ. Moreover,
(12)U±iβ(A(τs))={∑q=12‍∑p=12‍(ypqϵpq+zpqςpq),ypq,zpq∈ℝ,p,q=1,2},where
(13)ϵpq(θ)=Re{eiβθvpq}=cos(βθ)Re{vpq}-sin(βθ)Im{vpq},ςpq(θ)=Im{eiβθvpq}=sin(βθ)Re{vpq}+cos(βθ)Im{vpq},forθ∈[-τ,0].Proof.
Forγ∈ℝ, let qγ(λ)=λ+1-γe-λτ. If λ=α+iβ is a zero of qγ(λ), then 1+α+iβ=γe-(α+iβ)τ, from which it follows that
(14)(1+α)2+β2=γ2e-2ατ.
Thus, we claim that all the zeros of qγ(λ) have negative real parts provided that |γ|<1. Noticing that (H) holds and applying the above discussions to the factors Δ00,Δ10, and Δ01, we obtain conclusion (i).
In what follows, we consider the distribution of zeros of the factorΔ11. For τ=0, Δ11=0 becomes λ=a-b-c-1<-2. Let iβ(β>0) be a root of Δ11=0, then
(15)iβ+1-(b+c-a)ei(π-τβ)=1+β2eiarccos(1/1+β2)-(b+c-a)ei(π-τβ)=0.
Thus,
(16)1+β2=b+c-a,arccos11+β2=π-τβ+2sπ,s∈ℤ.
Therefore, Δ11=0 has a pair of purely imaginary roots ±iβ if and only if
(17)β=(a-b-c)2-1,τ=arccos(1/(a-b-c))+2sπ(a-b-c)2-1∶=τs,forsomes∈ℕ.
Clearly, for each fixed s, τs⩾0. Since
(18)ddλΔ11(λ)|λ=iβ,τ=τs=1+(a-b-c)τe-λτ|λ=iβ,τ=τs=1+τs(iβ+1)≠0,
there exist δ>0 and C1-smooth mapping λ:(τs-δ,τs+δ)→ℂ such that λ(τs)=iβ and λ(τ)+1-(a-b-c)e-τλ(τ)=0 for all τ∈(τs-δ,τs+δ). Differentiating Δ11=0 with respect to τ, we have
(19)λ′(τs)=-(a-b-c)λ(τs)e-λ(τs)τs1+(a-b-c)τse-λ(τs)τs=-λ(τs)[λ(τs)+1]1+τs[λ(τs)+1].
Therefore,
(20)Reλ′(τs)=β2(1+τs)2+τs2β2>0.
So far, the proof of (ii) and (iii) is complete (for more details see XI.2 in [22]). It remains to verify (iv).
Note thatv11=v22¯,v12=v21¯; it follows from the proof of Lemma 1 and the above discussions that the eigenspace of A(τs) associated with ±iβ is spanned by eiβθv11,eiβθv12,eiβθv21,eiβθv22,e-iβθv11, e-iβθv12, e-iβθv21, and e-iβθv22. Hence, the space has the real basis {ϵ11,ς11,ϵ12,ς12,ϵ21,ς21,ϵ22,ς22}. On the other hand, the eigenspace A(τs) associated with iβpq is of dimension 4, and the multiplicity of the characteristic root λ=iβpq is also 4. According to the folk theorem in functional differential equations (see [23]), U±iβ(A(τs)) must coincide with the eigenspace of A(τs) associated with ±iβ. Therefore, conclusion (iv) is correct. This completes the proof.From Lemma2 (ii), we can draw the following conclusion about the linear stability of system (2).Theorem 3.
If the assumption(H) is satisfied, then the equilibrium x^=0 of system (2) is asymptotically stable for τ∈[0,τ0); the equilibrium x^=0 is unstable for τ>τ0.


## 3. Multiple Patterns of Oscillation

It follows from Lemma2 that A(τ) has a pair of purely imaginary eigenvalues ±iβ of multiplicity 4 and all other eigenvalues of A(τ) are not integer multiples of ±iβ at τ=τs(s=0,1,…). Thus, the Hopf bifurcation may provide some asynchronous periodic solutions at each τs.The symmetry of a system is important in determining the patterns of oscillation. In order to explore the symmetry of system and analyze the spatio-temporal patterns of the bifurcated periodic solutions, we need the following definition.Let𝔻3 denote the dihedral group of order 6, which is generated by cyclic group ℤ3 together with the flip κ of order 2. Define the action of Γ:=𝔻3×𝔻3 on ℝ9 by
(21)((ρ,1)·x)i,j=xi+1,j,((1,ρ)·x)i,j=xi,j+1,((κ,1)·x)i,j=x3-i,j,((1,κ)·x)i,j=xi,3-j,i,jmod3,
where ρ is the generator of ℤ3 and κ denotes the flip.Lemma 4.
System (2) is Γ-equivalent.Proof.
Let mappingℱ:C([-τ,0],ℝ9)→ℝ9 be
(22)(ℱ(ϕ))i,j=-(ϕ)i,j(0)+f((ϕ)i,j(-τ))+g((ϕ)i,j-1(-τ))+g((ϕ)i,j+1(-τ))+h((ϕ)i-1,j(-τ))+h((ϕ)i+1,j(-τ)),i,j(mod3),
where
(23)ϕ=(ϕ00,ϕ01,ϕ02,ϕ10,ϕ11,ϕ12,ϕ20,ϕ21,ϕ22)T∈C([-τ,0],ℝ9),ℱ(ϕ)=((ℱ(ϕ))00,(ℱ(ϕ))01,(ℱ(ϕ))02,(ℱ(ϕ))10,(ℱ(ϕ))11,(ℱ(ϕ))12,(ℱ(ϕ))20,(ℱ(ϕ))21,(ℱ(ϕ))22)T∈ℝ9.
Then
(24)(ℱ((ρ,1)ϕ))i,j=-((ρ,1)ϕ)i,j(0)+f(((ρ,1)ϕ)i,j(-τ))+g(((ρ,1)ϕ)i,j-1(-τ))+g(((ρ,1)ϕ)i,j+1(-τ))+h(((ρ,1)ϕ)i-1,j(-τ))+h(((ρ,1)ϕ)i+1,j(-τ))=-(ϕ)i+1,j(0)+f((ϕ)i+1,j(-τ))+g((ϕ)i+1,j-1(-τ))+g((ϕ)i+1,j+1(-τ))+h((ϕ)i,j(-τ))+h((ϕ)i+2,j(-τ))=((ρ,1)ℱ(ϕ))i,j,(ℱ((κ,1)ϕ))i,j=-((κ,1)ϕ)i,j(0)+f(((κ,1)ϕ)i,j(-τ))+g(((κ,1)ϕ)i,j-1(-τ))+g(((κ,1)ϕ)i,j+1(-τ))+h(((κ,1)ϕ)i-1,j(-τ))+h(((κ,1)ϕ)i+1,j(-τ))=-(ϕ)3-i,j(0)+f((ϕ)3-i,j(-τ))+g((ϕ)3-i,j-1(-τ))+g((ϕ)3-i,j+1(-τ))+h((ϕ)4-i,j(-τ))+h((ϕ)2-i,j(-τ))=((κ,1)ℱ(ϕ))i,j.
Similarly, we can prove that
(25)(ℱ((1,ρ)ϕ))i,j=((1,ρ)ℱ(ϕ))i,j,(ℱ((1,κ)ϕ))i,j=((1,κ)ℱ(ϕ))i,j.
Therefore, ℱ is 𝔻3×𝔻3-equivalent. This completes the proof.Lemma 5.
LetΓ act on ℝ4 by
(26)(ρ,1)·(x1x2x3x4)=(-120-3200-120-32320-1200320-12)(x1x2x3x4),(1,ρ)·(x1x2x3x4)=(-12-320032-120000-12-320032-12)(x1x2x3x4),(κ,1)·(x1x2x3x4)=(x1x2-x3-x4),(1,κ)·(x1x2x3x4)=(x1-x2x3-x4).
Then ℝ4 is an absolutely irreducible representation of Γ, and the restricted action of Γ on KerΔ(τ,iβ) is isomorphic to the action of  Γ on ℝ4⊕ℝ4.Proof.
It is straightforward to verify the absolute irreducibility of the representation ofΓ on ℝ4 by the definition (see [1]). Note that
(27)KerΔ(τ,iβ)={∑p=12‍∑q=12‍(apq+ibpq)vpq,apq,bpq∈ℝ,p,q=1,2}.
Define J:KerΔ(τ,iβ)→ℝ4⊕ℝ4 by
(28)∑p=12‍∑q=12‍(apq+ibpq)vpq↦B(a11,b11,a12,b12,a21,b21,a22,b22)T,
where the matrix
(29)B=(10101010010-1010-101010-10-1-101010-1001010101-1010-1010-10-1010100-101010-1).
The nonsingularity of the matrix B implies that J:KerΔ(τ,iβ)≅ℝ4 is a linear isomorphism. It is easy to see that
(30)(ρ,1)·(∑p=12‍∑q=12‍(apq+ibpq)vpq)=∑p=12‍∑q=12‍(apq+ibpq)ei((-1)p+12π/3)vpq,(1,ρ)·(∑p=12‍∑q=12‍(apq+ibpq)vpq)=∑p=12‍∑q=12‍(apq+ibpq)ei((-1)q+12π/3)vpq,(κ,1)·(∑p=12‍∑q=12‍(apq+ibpq)vpq)=(a21+ib21)v11+(a22+ib22)v12+(a11+ib11)v21+(a12+ib12)v22,(1,κ)·(∑p=12‍∑q=12‍(apq+ibpq)vpq)=(a12+ib12)v11+(a11+ib11)v12+(a22+ib22)v21+(a21+ib21)v22.
Therefore, a straightforward calculation shows that
(31)J((ρ,1)·(∑p=12‍∑q=12‍(apq+ibpq)vpq))=(ρ,1)·J(∑p=12‍∑q=12‍(apq+ibpq)vpq),J((1,ρ)·(∑p=12‍∑q=12‍(apq+ibpq)vpq))=(1,ρ)·J(∑p=12‍∑q=12‍(apq+ibpq)vpq),J((κ,1)·(∑p=12‍∑q=12‍(apq+ibpq)vpq))=(κ,1)·J(∑p=12‍∑q=12‍(apq+ibpq)vpq),J((1,κ)·(∑p=12‍∑q=12‍(apq+ibpq)vpq))=(1,κ)·J(∑p=12‍∑q=12‍(apq+ibpq)vpq).
This concludes the proof.Letω=2π/β and let Pω be the Banach space of all continuous ω-periodic functions x:ℝ→ℝ9. Then for the circle group S1,  Γ×S1 acts on Pω by
(32)(γ,eiθ)·x(t)=γ·x(t+ω2πθ),(γ,eiθ)∈Γ×S1,x∈Pω.
Denote by SPω the subspace of Pω consisting of all ω-periodic solutions of system (3) with τ=τs. Then
(33)SPω={∑q=12‍∑p=12‍(ypqϵpq+zpqςpq),ypq,zpq∈ℝ,p,q=1,2},
where
(34)ϵpq(t)=Re{eiβtvpq}=cos(βt)Re{vpq}-sin(βt)Im{vpq},ςpq(t)=Im{eiβtvpq}=sin(βt)Re{vpq}+cos(βt)Im{vpq}.
Therefore, for ϵpq(t),ςpq(t)(p,q=1,2), we have the following properties.Lemma 6.
(35)
(
ρ
,
1
)
·
ϵ
p
q
=
ϵ
p
q
cos
2
p
π
3
-
ς
p
q
sin
2
p
π
3
,
(
ρ
,
1
)
·
ς
p
q
=
ς
p
q
cos
2
p
π
3
+
ϵ
p
q
sin
2
p
π
3
,
(
1
,
ρ
)
·
ϵ
p
q
=
ϵ
p
q
cos
2
q
π
3
-
ς
p
q
sin
2
q
π
3
,
(
1
,
ρ
)
·
ς
p
q
=
ς
p
q
cos
2
q
π
3
+
ϵ
p
q
sin
2
q
π
3
,
(
κ
,
1
)
·
ϵ
p
q
=
ϵ
3
-
p
,
q
,
(
κ
,
1
)
·
ς
p
q
=
ς
3
-
p
,
q
,
(
1
,
κ
)
·
ϵ
p
q
=
ϵ
p
,
3
-
q
,
(
1
,
κ
)
·
ς
p
q
=
ς
p
,
3
-
q
.Proof.
Fori,j(mod3) and t∈ℝ, note that
(36)ϵi,jpq(t)=cos(βt+2ipπ3+2jqπ3),ςi,jpq(t)=sin(βt+2ipπ3+2jqπ3),
we have
(37)((ρ,1)·ϵpq(t))i,j=ϵi+1,jpq(t)=cos(βt+2(i+1)pπ3+2jqπ3)=cos(βt+2ipπ3+2jqπ3+2pπ3)=ϵi,jpqcos2pπ3-ςi,jpqsin2pπ3,((ρ,1)·ςpq(t))i,j=ςi+1,jpq(t)=sin(βt+2(i+1)pπ3+2jqπ3)=sin(βt+2ipπ3+2jqπ3+2pπ3)=ςi,jpqcos2pπ3+ϵi,jpqsin2pπ3,((κ,1)·ϵpq(t))i,j=ϵ3-i,jpq(t)=cos(βt+2(3-i)pπ3+2jqπ3)=cos(βt+2i(3-p)π3+2jqπ3)=ϵi,j3-p,q,((κ,1)·ςpq(t))i,j=ϵ3-i,jpq(t)=sin(βt+2(3-i)pπ3+2jqπ3)=sin(βt+2i(3-p)π3+2jqπ3)=ςi,j3-p,q.
Therefore
(38)(ρ,1)·ϵpq=ϵpqcos2pπ3-ςpqsin2pπ3,(ρ,1)·ςpq=ςpqcos2pπ3+ϵpqsin2pπ3,(κ,1)·ϵpq=ϵ3-p,q,(κ,1)·ςpq=ς3-p,q.
Similarly, we can prove that
(39)(1,ρ)·ϵpq=ϵpqcos2qπ3-ςpqsin2qπ3,(1,ρ)·ςpq=ςpqcos2qπ3+ϵpqsin2qπ3,(1,κ)·ϵpq=ϵp,3-q,(1,κ)·ςpq=ςp,3-q.It is clear that ifx is a periodic solution of system (2), then so is (γ,eiθ)x for every (γ,eiθ)∈Γ×S1. The spatial-temporal symmetry of a bifurcation of periodic solutions x(t) can be completely characterized by the isotropy group Σx={(γ,eiθ)∈Γ×S1∣(γ,eiθ)x=x}≤Γ×S1, and it is easy to verify that the isotropy group of (γ,eiθ)x is (γ,eiθ)Σx(γ,eiθ)-1, which is conjugate to Σx. The maximal isotropy subgroups Σm(m=1,2,…,20) of Γ×S1 are listed in Table 1. For each subgroup Σm, the Σm-fixed-point set
(40)Fix(Σm,SPω)={x∈SPω∣(γ,eiθ)x=x,∀(γ,eiθ)∈Σm}
is a subspace of SPω. According to Lemma 6, it is easy to verify that dimFix(Σm,SPω)=2,m=1,2,…,20.Table 1
The maximal isotropy subgroups ofΓ×S1 and associated fixed point subspaces.
No.
Σ
Fix
(
Σ
,
S
P
ω
)
dim
Fix
(
Σ
,
S
P
ω
)
1
〈
(
κ
,
1,1
)
,
(
1
,
κ
,
1
)
〉
(
y
,
z
,
y
,
z
,
y
,
z
,
y
,
z
)
2
2
〈
(
κ
,
1,1
)
,
(
1
,
κ
,
-
1
)
〉
(
y
,
z
,
-
y
,
-
z
,
y
,
z
,
-
y
,
-
z
)
2
3
〈
(
κ
,
1
,
-
1
)
,
(
1
,
κ
,
1
)
〉
(
y
,
z
,
y
,
z
,
-
y
,
-
z
,
-
y
,
-
z
)
2
4
〈
(
κ
,
1
,
-
1
)
,
(
1
,
κ
,
-
1
)
〉
(
y
,
z
,
-
y
,
-
z
,
-
y
,
-
z
,
y
,
z
)
2
5
〈
(
ρ
,
1
,
e
-
2
π
i
/
3
)
,
(
1
,
ρ
,
e
-
2
π
i
/
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(
y
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6
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ρ
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π
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1
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7
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ρ
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8
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(
0,0
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2
9
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ρ
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1
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y
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10
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11
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12
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17
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18
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19
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20
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2
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1
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(
κ
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κ
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1
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y
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-
z
)
2Together with Lemma2, Lemmas 4–6 allow us to apply the equivariant Hopf bifurcation theorem for delay differential equations due to Wu [3] to obtain the following result on the spatio-temporal patterns of the bifurcated periodic solutions.Theorem 7.
Assume that(H) is satisfied. Then for system (2), near each τs(s=0,1,…) there exist 100 distinct branches of periodic solutions bifurcated from the equilibrium x^=0. More precisely, for each isotropy group Σm(m=1,2,…,20) and a chosen basis {δ1,δ2} of Fix(Σm,SPω), there exist α0>0,τ0*>0,σ0>0, and a C1-smooth mapping (τ*,ω*,x*):ℝα02→ℝ×ℝ+×C(ℝ,ℝ9), where ℝα02={α∈ℝ2∣|α|<α0}, such that for each α∈ℝα02, x*=x*(t;α) is an ω*(α)-periodic solution of system (2) with τ=τs+τ*(α), and
(41)γ·x*(t)=x*(t-ω*(α)2πθ),(γ,eiθ)∈Σm,ω*(0)=2πβ,τ*(0)=0,x*(t;α)=(δ1,δ2)α+o(|α|),as|α|⟶0.
Furthermore, for |τ-τs|<τ0*, |ω~-2π/β|<σ0, every ω~-periodic solution x(t) of system (2) with ∥xt∥<σ0, γ·x(t)=x(t-(ω~/2π)θ) for (γ,eiθ)∈Σm and t∈ℝ must be of the above type.


## 4. An Example and Numerical Simulations

As a simple example, we consider the following specific Hopfield model with𝔻3×𝔻3-symmetry:
(42)x˙i,j(t)=-xi,j(t)-0.6tanh(xi,j(t-τ))+0.3tanh(xi,j-1(t-τ))+0.3tanh(xi,j+1(t-τ))+0.4tanh(xi-1,j(t-τ))+0.4tanh(xi+1,j(t-τ)),
where i,j=0,1,2(mod3).Clearly, the origin is an equilibrium of system (42). It is easy to compute a=-0.6,b=0.3,andc=0.4 and verify that the hypothesis (H) holds. According to (10), we obtain
(43)τs=2.9476+7.5641s,s=0,1,….
From Theorem 3, the origin is asymptotically stable if τ<τ0=2.9476. It follows from Theorem 7 that the equivariant Hopf bifurcation occurs at τ0=2.9476 and there exist 100 distinct branches of periodic solutions bifurcated from the origin. To illustrate the analytical results found, we give some numerical simulations. Figure 2 shows that the origin of system (42) is stable when τ=2.6. Figure 3 shows that four periodic orbits occur simultaneously when τ=3. Unfortunately, we cannot verify the existence of all other bifurcated periodic orbits since they may be unstable.Trajectoriesxi,j(i,j=0,1,2) of system (42) when τ=2.6.
(a)
(b)
(c)Figure 3
System (42) has multiple periodic solutions when τ=3.


## 5. Discussion

In this paper, we have studied a coupled system of nine identical cells with delays and𝔻3×𝔻3-symmetry. By choosing the time delay τ as a bifurcation parameter and analyzing the corresponding characteristic equation, we have shown that under some assumption, the equilibrium of the model loses its stability and periodic solutions via Hopf bifurcation occur when τ passes through a critical value. This implies that the time delay can be regarded as a source of instability and oscillatory response of the networks and is able to alter the dynamics of system (2) significantly. Moreover, the spatio-temporal patterns of bifurcating periodic solutions are explored clearly by employing the symmetric bifurcation theory of delay differential equations combined with representation theory of Lie groups. From Theorem 7, we have obtained the conclusion that the small-scale network with a special structure may have a large number of periodic oscillations. Therefore, it is natural that the large-scale network possesses complicated dynamics generally.Further investigations such as the stability, direction and global existence of the periodic solutions bifurcating from the local Hopf bifurcations are essential in order to fully understand the periodic phenomenon of the system. We can compute the normal forms directly by using the method due to Faria and Magalháes [24, 25]. However, this is a complex and prolix task. In addition, we would like to point out that codimension two mode interactions may take place if the assumption (H) is not satisfied. For example, if a+2b+2c=1, |a+2c-b|<1,|a+2b-c|<1, and a-b-c<-1, then system (2) undergoes a fold-Hopf bifurcation when τ passes through a critical value. Therefore, it is possible to study secondary bifurcations and more complex behaviours in this coupled network. We leave them for our future work.



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*Source: 417678-2013-06-03.xml*