3 Spatio-temporal patterns of periodic solutions

Synchronous dynamics of a delayed two-coupled oscillator

Yagbanga Niella Prudence Marthange, Shangzhi Li and Shangjiang Guo

College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, P. R. China Received 8 February 2017, appeared 25 June 2017

Communicated by Ferenc Hartung

Abstract. This paper presents a detailed analysis on the dynamics of a delayed two- coupled oscillator. Linear stability of the model is investigated by analyzing the asso- ciated characteristic transcendental equation. By means of the equivariant Hopf bifur- cation theorem, we not only investigate the effect of time delay on the spatio-temporal patterns of periodic solutions emanating from the trivial equilibrium, but also derive the formula to determine the direction and stability of Hopf bifurcation. Moreover, we illustrate our results by numerical simulations.

Keywords: periodic solution, Hopf bifurcation, stability.

2010 Mathematics Subject Classification: 34K15, 92B20.

1 Introduction

Synchronization phenomena are common in nature (see Nijmeijer and Rodriguez-Angeles [26] and references therein). An important avenue of study in synchronization focuses on coupled oscillators. One classical example is the Kuramoto model [22], which assumes full connectivity of the network. By using a combination of the Lyapunov functional method, ma- trix inequality techniques and properties of Kronecker product, Alofi et al. [1] investigated a so-called power-rate synchronization problem for the collective dynamics among genetic os- cillators with unbounded time-varying delay. Wanget al. [27] investigated the synchronization of coupled Duffing-type oscillators. By means of the residue harmonic balance method, Xiao et al. [29] investigated the approximations to the periodic oscillations of the fractional order van der Pol equation.

The study of the dynamical behavior of oscillating systems is a central issue in physics and in mathematics. These systems provide basic and general results that found major applications not only in physics, but also in all the other branches of science. The harmonic oscillator is the simplest, and more fundamental theoretical model of oscillatory phenomena. Damped and forced oscillators provide, also, very fundamental results in physics and engineering.

BCorresponding author. Email: shangjguo@hnu.edu.cn

In this paper, we study the existence and stability of periodic orbits in a delayed two- coupled harmonic oscillator modelled by the following system of delay differential equations

¨

u_i(t) +u_i(t) +εu˙_i(t) =εf(u_i+1(t−τ)), (1.1) where f ∈ C¹(_R;R) with f(0) = 0, τ ≥ 0 and ε > 0 are constants, and as well as in all subsequent expressions, the index iis taken to modulo 2, so that, for instance, x₃ = x₁. We also assume that each oscillator has no self-feedback and signal transmission is delayed due to the finite switching speed of oscillator. It can be seen that in system (1.1) the growth rate of one oscillator depends on the feedback from the other. Such a network has been found in a variety of neural structures and even in chemistry and electrical engineering. Despite the low number of units, two-oscillator networks with delay often display the same dynamical behaviors as large networks and, can thus be used as prototypes for us to understand the dynamics of large networks with delayed feedback (see, for example, [8,14–16,20,21]).

Here, we emphasize the importance of temporal delays in the coupling between cells, since in many chemical and biological oscillators (cells coupled via membrane transport of ions), the time needed for transport of processing of chemical components or signals may be of considerable length. Since we have symmetric coupling of identical oscillators, (1.1) has the reflection symmetry of interchange of two oscillators. Although model (1.1) is a little simple, it allows us to have a depth analysis and then to gain insight into possible mechanisms behind the observed behavior.

It is easy to see that every continuous ψ= (ψ₁,ψ₂)^T : [−τ, 0] → _R² uniquely determines a solutionu^ψ = (u^ψ₁,u^ψ₂)^T _: [−_τ,∞)→ _R² of (1.1) with u^ψ|_[−τ,0] = ψ. Clearly, ifψ₁ =ψ₂ then the uniquely determined solution satisfiesu^ψ₁ =u^ψ₂ in[−τ,∞)and can be characterized by the scalar delay differential equation

¨

u(t) +u(t) +εu˙(t) =εf(u(t−τ)), (1.2) Such solutions are said to be synchronous. Equation (1.2) has been used to model a variety of other biological and physical phenomena, and studied by many researchers (see, for example, [24,25]). More precisely, the local stability analysis has been discussed by a lot of investigators [3–6,9,10,23] and complex dynamics including limit cycles and tori are also obtained by Campbell [7], Hou and Guo [19], Zhang and Guo [30,31]. The existence of nonconstant periodic solutions of (1.2) has been proved in [2].

Our goal in this paper is to study the existence and stability of periodic orbits of (1.1).

The plan for this paper is as follows. In Section 2, we consider the linear stability of the trivial solution (1.1). Section 3 is devoted to the spatio-temporal patterns of Hopf bifurcated periodic solutions when the the trivial solution lose its stability. In Section 4, we discuss the bifurcation direction and stability of periodic solutions emerging from from the trivial solution. In Section 5, we illustrate our results with some numerical simulations. Finally, some conclusions are made in Section 6.

2 Properties of bifurcated periodic solutions

Let C([−_{τ, 0}]_,R²) denote the Banach space of continuous mapping from [−_{τ, 0}] _{into R}² equipped with the supremum norm kφk = sup_{−τ≤θ≤0}|φ(θ)| for φ ∈ C([−τ, 0],R²). In what follows, if σ ∈ _R, A ≥ 0 and x : [σ−1,σ+ A] → _R² is a continuous mapping, then x_t ∈ C([−_{τ, 0}]_,R²)_, t ∈ [_σ,σ+A], is defined by x_t(θ) = x(t+θ)_for −τ ≤ θ ≤ _{0. For}

any two integers aandb, defineN(a) ={a,a+1, . . .},N(a,b) ={a,a+1, . . . ,b}whena ≤b.

N= N(0).

The linearization of (1.1) at the origin leads to

¨

u_i(t) +u_i(t) +εu˙_i(t) =εαu_i+1(t−τ), i(mod 2). (2.1) where α= f⁰(0). It is well-known that the associated characteristic equation of (2.1) takes the form

det∆(τ,λ) =0, where the characteristic matrix ∆(τ,λ)is given by

∆(τ,λ) = (λ²+1+ελ)Id−αεMe^−λτ, λ∈_C with Id denoting the identity matrix and

M= 0 1

1 0

. By an easy computation, we have

det∆(_τ,λ) = (λ²+₁+ελ)²−(αε)²e^−2λτ. Hence, by factoring the right side of the above equality, we can obtain

det∆(τ,λ) = [λ²+1+ελ−εαexp{−λτ}][λ²+1+ελ+εαexp{−λτ}]. (2.2) Thus, λ∈_Cis a zero of det∆(τ,λ)if and only if there exists a j∈ {0, 1}such that

p_j(τ,λ),^λ2+ελ+1−(−1)^jεαexp{−λτ}=0. (2.3) We know that±iω (ω >0) are a pair of purely imaginary zeros ofp_j(τ,·)if and only ifω satisfies

(1−ω² = (−₁)^jεαcos(τω),

−ω = (−1)^jαsin(τω). (2.4)

It follows from (2.4) that

ω⁴+ (ε²−2)ω²+1−ε²α² =0. (2.5) The number of positive solutions to (2.5) may be zero, one, or two, which is determined by the signs of(ε²+4α²−4)and(ε|α| −1). In fact, the curves ε²+4α² = 4 andε|α|= 1 divide the right half (ε,α)-plane into six regions:

D₁ =(ε,α)∈_R⁺×_R: ε²+4α²<4 , D⁺₂ =(ε,α)∈_R⁺×_R: εα>1 , D⁻₂ =(ε,α)∈_R⁺×_R: εα<−1 , D⁺₃ =

(

(ε,α)∈_R⁺×_R: r

1− ^ε2

4 <α< ¹

ε, ε<√ 2

) , D⁻₃ =

(

(ε,α)∈_R⁺×_R: r

1− ^ε2

4 <−α< ¹

ε, ε<√ 2

) , D₄ =ⁿ(ε,α)∈_R⁺×_R: ε²+4α² >4, ε|α|<1, ε>√

2o .

More precisely, equation (2.5) has two positive solutions ω= β±when(ε,α)∈ D⁺₃ ∪D₃⁻, has exactly one positive solution ω = ω+ when(ε,α) ∈ D₂⁺∪D₂⁻, and has no positive solution when(ε,α)∈ D₁∪D₄, where

β±= s

1−^ε2 2 ±ε

r

α²−1+ ^ε2 4. Thus, the Hopf bifurcation values ofτ_j,k^± are given as follows:

τ_j,k⁺ =















1

β+(arcsin^β_|α⁺_| +2kπ) if(−1)^jα≤ −1,

1

β+(π−arcsin^β_|⁺

α| +2kπ) if −1≤(−1)^jα<0,

1

β+(_2π−_arcsin^β_|⁺

α| +2kπ) _{if 0}<(−₁)^jα≤_1,

1

β+(π+arcsin^β_|α⁺_| +2kπ) if(−1)^jα>1, τ_j,k⁻ =

( ₁

β−(π−arcsin^β_|α⁻_| +2kπ) if(−1)^jα<0,

1

β−(2π−arcsin^β_|α⁻_| +2kπ) if(−1)^jα>0,

fork ∈_N0and j∈ {0, 1}. Thus, we have the following results about the zeros of p_j(τ,λ). Lemma 2.1. If(ε,α)∈ D₁∪D₄, then for each j andτ≥0, p_j(τ,·)has only zero pointsλsatisfying Reλ<0and has no purely imaginary zero point.

Proof. It follows from (ε,α) ∈ D₁∪D₄ that ε|α| < 1. We first notice the fact that there exist at most a finite number of zeros of p_j(τ,λ)in right half-plane for each j∈ {0, 1}. Indeed, for any zeroλof p_j(τ,λ),

|λ²+ελ+1|=ε|α|exp{−τReλ}.

This implies that there is a real number η such that all zeros of p_j(τ,λ) satisfy Reλ < η.

Clearly, p_j(_τ,λ) is an entire function. Hence, there can only be a finite number of zeros of p_j(τ,λ)in any compact set. Namely, there exist only a finite number of zeros in any vertical strip in the complex plane. We can regardλas the continuous function ofτ according to the implicit function theorem. Notice that

p_j(0,λ) =λ²+ελ+1−(−1)^jεα=0,

which has exactly two zero points with negative real parts. Recall the fact that all zeros of p_j(τ,λ) are simple and continuously depend on τ, then there exists a critical value τ0 such that p_j(τ,λ)has only zero points with negative real parts ifτ∈[0,τ₀), and that asτincreases and passes through τ₀, the zero points with positive real parts may appear. Thus, p_j(τ₀,λ) has a pair of purely imaginary zero points±iω, whereω>0 is a solution to (2.5). In view of (ε,α)∈D₁∪D₄, we see thatτ₀ =∞. This completes the proof.

Lemma 2.2. Assume thatε|α|>1, i.e.,(ε,α)∈ D⁺₂ ∪D₂⁻.

(i) p_j(τ,·)has a pair of simple imaginary zero±iβ+at and only atτ=τ_j,k⁺ >0, k∈_N.

(ii) For each fixed pair(j,k)∈ {0, 1} ×_N0such thatτ_j,k⁺ >0, there existδ¹_j,k >0and C¹-mapping λ_j,k : (τ_j,k⁺−δ¹_j,k,τ_j,k⁺+δ¹_j,k)→_Csuch thatλ_j,k(τ_j,k⁺) =iβ+andλ_j,k(τ)is a zero of p_j(τ,λ)for allτ∈(τ_j,k⁺−δ¹_j,k,τ_j,k⁺+δ¹_j,k). Moreover, _dτ^d Re{λ_j,k(τ)}|_τ=τ⁺

j,k >0.

(iii) For each fixed (ε,α) ∈ D₂⁺, p₀(τ,λ) has exactly one zero point with positive real parts when τ ∈ [0,τ_0,0⁺), and exactly 2k+3zero points with positive real partsτ ∈ (τ_0,k⁺,τ_0,k⁺₊₁); p₁(τ,λ) has only zero points with negative real parts when τ ∈ [_0,τ_1,0⁺), and exactly2k+₂ zero points with positive real partsτ∈ (τ_1,k⁺,τ_1,k⁺₊₁).

(iv) For each fixed (ε,α) ∈ D₂⁻, p₁(τ,λ) has exactly one zero point with positive real parts when τ ∈ [0,τ_1,0⁺), and exactly 2k+3zero points with positive real partsτ ∈ (τ_1,k⁺,τ_1,k⁺₊₁); p₀(τ,λ) has only zero points with negative real parts when τ ∈ [0,τ_0,0⁺), and exactly2k+2 zero points with positive real partsτ∈ (τ_0,k⁺,τ_0,k⁺₊₁).

Proof. (i) Letλ=ivbe a zero ofp_j(τ,λ). Then, we getv= β+, then 1−β²₊= (−1)^jεαcos(τβ+) andβ+= (−1)^jαsin(τβ+). Namely,τ=τ_j,k⁺ for somejandk.

(ii) The existence of δ¹_j,k and the mappingλ_j,k follow from the implicit function theorem.

We now differentiate the equality p_j(τ,λ) =0 with respect to τto get d

dτRe

λ_j,k(τ) |_τ⁺

j,k =−Re

( λ_j,k(τ_j,k⁺)(λ²_j,k(τ_j,k⁺)−ελ_j,k(τ_j,k⁺) +1) 2λ_j,k(τ_j,k⁺)−ε+τ_j,k⁺(λ²_j,k(τ_j,k⁺)−ελ_j,k(τ_j,k⁺) +1)

)

= ^β

2+(ε²+2β²₊−2) h

τ_j,k⁺(1−β²₊)−ε i2

+β²₊(2−ετ_j,k⁺)²

= ^εβ

2+

√

ε²+4α²−4 h

τ_j,k⁺(1−β²₊)−ε i2

+β²₊(2−ετ_j,k⁺)²

>0.

This completes the proof.

(iii) Using a similar argument as that in the proof of Lemma2.1, we can regardλ as the continuous function ofτaccording to the implicit function theorem. Ifτ=0 and(ε,α)∈ D₂⁺ (respectively, (ε,α) ∈ D₂⁻), then p₀(τ,λ) (respectively, p₁(τ,λ)) has exactly one zero point with positive real parts but p₁(τ,λ)(respectively,p0(τ,λ)) has only zero points with negative real parts. Recall the fact that all zeros of p_j(τ,λ) are simple and continuously depend on τ, then there exists a critical value τ_j,0 such that the number of zero points of p_j(τ,λ) with positive real parts keeps the same if τ∈[0,τ₀). It follows from conclusions (i) and (ii) that as τincreases and passes through τ₀, only one zero point of p_j(τ,λ), denoted by λ^∗(τ), varies from a complex number with a negative real part to a purely imaginary number and then to a complex number with a positive real part. In fact, the proof of conclusion (i) yields that τ₀= τ_j,0⁺ >0.

We can repeat the same analysis to conclude that there exists next critical valueτ_j,1 such that the number of zero points of p_j(τ,λ) with positive real parts keeps the same if τ ∈ (τ_j,0⁺,τ_j,1), and that asτ increases and passes throughτ_j,1, a new zero point of p_j(_τ,λ) _varies from a complex number with a negative real part to a purely imaginary number and then to a complex number with a positive real part. Similarly, it follows from the proof of conclusion (i) thatτ_j,1 =τ_j,1⁺.

By induction, we can draw the conclusion that the number of zeros ofp_j(τ,λ)with positive real parts increases asτincreases. This completes the proof.

Lemma 2.3. Assume that(ε,α)∈ D⁺₃ ∪D₃⁻.

(i) p_j(τ,·)has a pair of simple imaginary zeros±iβ±at and only atτ=τ_j,k^± >0, k∈_N.

(ii) For each fixed pair(j,k)∈ {0, 1} ×_N0such thatτ_j,k⁺ >0, there existδ¹_j,k >0and C¹-mapping λ_j,k : (τ_j,k⁺−δ¹_j,k,τ_j,k⁺+δ¹_j,k)→_Csuch thatλ_j,k(τ_j,k⁺) =iβ+andλ_j,k(τ)is a zero of p_j(τ,λ)for allτ∈(τ_j,k⁺−δ¹_j,k,τ_j,k⁺+δ¹_j,k). Moreover, _dτ^d Re{λ_j,k(τ)}|_τ=τ⁺

j,k >0.

(iii) For each fixed pair(j,k)∈ {0, 1} ×_N0such thatτ_j,k⁻ >0, there existδ¹_j,k >0and C¹-mapping λ_j,k : (τ_j,k⁻−δ¹_j,k,τ_j,k⁻+δ¹_j,k)→_Csuch thatλ_j,k(τ_j,k⁻) =iβ−andλ_j,k(τ)is a zero of p_j(τ,λ)for allτ∈(τ_j,k⁻−δ¹_j,k,τ_j,k⁻+δ¹_j,k). Moreover, _dτ^d Re{λ_j,k(τ)}|_τ=τ⁻

j,k <0.

(iv) For each fixed j ∈ {0, 1}, there exists a nonnegative integer m_j such that p_j(τ,λ) has exactly one pair of zeros with positive real parts whenτ_j,k⁺₋₁ < τ< τ_j,k⁻₋₁ withτ_j,^±₋₁ = 0, and all zeros of p_j(τ,λ)have negative real parts when τ_j,k⁻₋₁ < τ< τ_j,k⁺ with τ_j,⁻₋₁ = 0, k= 0, 1, 2, . . . ,m_j, and p_j(τ,λ)has only zeros with negative real parts whenτ>τ_0,m⁺

j.

Proof. Using a similar argument as that in the proof of Lemma2.2, we can prove conclusions (i)–(iii). We now prove conclusion (iv). First we notice the fact that

τ_0,0⁺ = ^2π−arcsin^β_α⁺ β+

< ^2π−arcsin^β_α⁻ β−

= τ_0,0⁻, when 0<α≤1, and

τ_0,0⁺ = ^π+arcsin^β_α⁺ β+

< ^2π−arcsin^β_α⁻ β−

= τ_0,0⁻, whenα>_1,

τ_0,0⁺ = ^π−arcsin^β_α⁺ β+

< ^π−arcsin^β_α⁻ β−

=τ_0,0⁻, when−1< α≤0, and

τ_0,0⁺ = ^arcsin

β+ α

β+

< ^π−arcsin^β_α⁻ β−

=τ_0,0⁻,

whenα< −1. Then we haveτ_j,0⁺ <τ_j,0⁻,j=0, 1. It follows fromτ_j,k^±₊₁−τ_j,k^± = ^2π

β± andβ+> β−

that

τ_j,k⁺₊₁−τ_j,k⁺ <τ_j,k⁻₊₁−τ_j,k⁻. Thus, there exists an nonnegative integerm_j such that

τ_j,0⁺ <τ_j,0⁻ <τ_j,1⁺ <τ_j,1⁻ <· · · <τ_j,m⁺

j < τ_j,m⁺

j+1 <τ_j,m⁻

j. Lemma 2.4.

(i) For any fixed (ε,α) ∈ D₁∪D₄ and τ ≥ 0, all solutions λ to the characteristic equation det∆(τ,λ) =0satisfyReλ<0. Furthermore, no Hopf bifurcation occurs at the origin.

(ii) For any fixed(ε,α)∈ D₂⁺∪D⁻₂ andτ≥ 0, the characteristic equationdet∆(τ,λ) =0has at least one solutionλsatisfyingReλ> 0. Furthermore, system (1.1) undergoes Hopf bifurcation at the origin nearτ=τ_j,k⁺, j∈ {0, 1}, k∈_N0.

(iii) For any fixed (ε,α) ∈ D⁺₃ ∪D₃⁻ and τ ≥ 0, all solutions λ to the characteristic equation det∆(τ,λ) = 0satisfyReλ <0when τ∈ [∪^m_k=⁰₀(τ_0,k⁻₋₁,τ_0,k⁺)]∩[∪^m_k=¹₀(τ_1,k⁻₋₁,τ_1,k⁺)]. Further- more, system (1.1)undergoes Hopf bifurcation at the origin nearτ = τ_j,k^±, j ∈ {_{0, 1}}, k ∈ _N0, where m₀and m₁are given in Lemma2.3.

It follows from the above lemma that we have the following results on the linear stability of the equilibrium x^∗=0 of system (1.1).

Theorem 2.5.

(i) If(ε,α) ∈ D₁∪D₄ and τ ≥ 0, then the equilibrium x^∗ = 0 of system (1.1) is stable for all τ≥0.

(ii) If(ε,α)∈ D₂⁺∪D⁻₂, then the equilibrium x^∗ =0of system(1.1)is unstable for allτ≥0.

(iii) If (ε,α) ∈ D₃⁺∪D⁻₃, then the equilibrium x^∗ = 0 of system (1.1) is stable for all τ ∈ [∪^m_k=⁰₀(τ_0,k⁻₋₁,τ_0,k⁺)]∩[∪^m_k=¹₀(τ_1,k⁻₋₁,τ_1,k⁺)], where m₀and m₁ are given in Lemma2.3.

3 Spatio-temporal patterns of periodic solutions

Throughout this section, we always assume that (_ε,α) ∈ D₂⁺∪D₂⁻∪D⁺₃ ∪D⁻₃. Lemmas 2.2 and2.3, together with the Hopf theorem (see, pp. 332 in [18]), imply that a Hopf bifurcation for (1.1) occurs at each τ = τ_j,k^± > 0. Namely, in every neighborhood of (x^∗ = 0,τ^∗ = τ_j,k^±) there is a unique branch of periodic solutions x^j,k(t,τ) with x^j,k(t,τ) → 0 as τ → τ_j,k^±. The periodP^j,k(_γ,τ)_of x^j,k(t,τ)satisfies thatP^j,k(_γ,τ)→_2π/β±asτ→τ_j,k^±.

In what follows, we aim to analyze the spatio-temporal patterns of these bifurcated peri- odic solutions. It is well-known that the symmetry of a system is important in determining the patterns of oscillation that it can support. To explore the possible (spatial) symmetry of the system (1.1), we need to introduce two compact Lie groups. One is the cycle group S¹, the other is Z2, the cyclic group of order 2 (the order of a finite group is the number of the elements it contains). Clearly, we have

Lemma 3.1. Denote byρthe generator of the cyclic subgroupZ2. Define the action ofZ2onR²by ρ·(x₁,x₂) = (x₂,x₁) for all(x₁,x₂)∈_R²_.

Then system(1.1)isZ2-equivariant.

Proof. Define a mappingF: C([−_{τ, 0}]_,R²)→_R² _by

(F(φ))_i =−φ_i(0) +εφ˙_i(0) +εf(φ_i+₁(−τ)) forφ∈C([−τ, 0],R²)andi(mod 2). Then

(F(ρ·φ))_i = −(ρ·φ)_i(0) +ε(ρ·φ^˙)_i(0) +εf((ρ·φ)_i+1(−τ))

= −φ_i+1(0) +εφ˙_i+1(0) +εf(φ_i(−τ))

= (ρ·F(φ))_i

for φ ∈ C([−τ, 0],R²) and i (mod 2). Namely, F is Z2-equivariant. This completes the proof.

Let ω₀ = β±, ω = ^2π

ω0 and P_ω the Banach space of continuous ω-periodic mappings x:R→_R². Z2×S¹acts on Pω by

(δ,e^iθ)·x(t) =δ·x(t+θ), e^iθ ∈_S¹, x∈Pω, δ∈ _Z2,

LetSPω denote the subspace ofPωconsisting of allω-periodic solutions of (1.1) with τ=τ_j,k. Then

SP_ω =ⁿx₁e¹+x₂e²: x₁,x₂∈_R^o,

wheree¹ and e² are 2-dimensional vector functions defined onR with the m-th components defined bye¹_m(t) =cos(ω₀t+ (m−1)jπ)ande²_m(t) =sin(ω₀t+ (m−1)jπ)fort ∈_Rrespec- tively. Note that for allt∈ _Randm∈ {1, 2},

(ρ·e¹(t))_m = e¹_m+1(t) =cos(ω0t+mjπ)

=_cos(ω₀t+ (m−₁)jπ+jπ)

= e¹_m

t+ ^jπ ω₀

,

(ρ·e²(t))_m = e²_m+1(t) =sin(ω₀t+mjπ)

=sin(ω₀t+ (m−1)jπ+jπ)

= e²_m

t+ ^jπ ω₀

. Then we have

ρ·e¹=e¹

t+ ^j 2ω

, ρ·e²=e²

t+ ^j 2ω

. (3.1)

It has been verified in [28] that, under usual non-resonance and transversality conditions, for every subgroupΣ≤ _Z2×_S¹ such that theΣ-fixed-point subspace of SP_ω (i.e., Fix(_Σ,SP_ω) = {x ∈ SP_ω : γx = x for all γ ∈ _Σ}) is of dimension 2, symmetric delay differential equations has a bifurcation of periodic solutions whose spatial-temporal symmetry can be completely characterized byΣ.

Here, we consider the following subgroup ofZ3×_S¹to describe the symmetry of periodic solution of system (1.1) (see [13] for more details):

Σ=h(ρ,e^−i2jω)i.

The two equations in (3.1) imply that theΣ-fixed-point set of SPω is itself, i.e., Fix(_Σ,SPω) = SP_ω. Thus, the general symmetric local Hopf bifurcation theorem (Theorem 2.1 in [28]) enables us we obtain the following result on the existence of smooth local Hopf bifurcations of wave solutions.

Theorem 3.2. Assume that (ε,α) ∈ D₂⁺∪D₂⁻∪D₃⁺∪D₃⁻. Then near each τ_j,k^± > 0, there exists a branch of small-amplitude periodic solutions of (1.1) emerging from the trivial solution x = 0. More precisely, there existε^j,k >0andδ^j,k >0such that for eachθ∈ [0, 2π],α∈ (0,ε^j,k_±), system(1.1)with τ=τ_j,k^±+τ^j,k(α,θ)has a periodic solution x^j,k =x^j,k(t;α,θ)with period ω^j,k = ω^j,k(α,θ)such that

x_i^j,k(t) =x^j,k_i+1

t− ^jω

j,k

2

, i=0, 1 (3.2)

x_i^j,k(t;α,θ) =α h

cosθe¹_i(t) +sinθe²_i(t)ⁱ+o(|α|)

=αcos(ω₀t+ (i−1)jπ−θ) +o(|α|)

as α → 0. The mapping (x^j,k,τ^j,k,ω^j,k) : (0,ε^j,k)×[0, 2π] → C(_R,R³)×_R×_R is continuously differentiable and

ω^j,k(0,θ) = ^2π

ω₀, τ^j,k(0,θ) =0.

Furthermore, if|τ−τ_j,k|<δ^j,kand|ω− ^2π

βj,k|<δ^j,k then everyω-periodic solution of (1.1)satisfying x_i(t) =x_i+1(t−jω^j,k),andsup_t∈R|x(t)|< δ^j,k must be given by x^j,k(t;α,θ)for someα ∈ (0,ε^j,k) andθ∈ [0, 2π).

We call the above periodic solutionsdiscrete waves. They are also called synchronous oscil- lations (if j = 0) or phase-locked oscillations (if j 6= 0) as each neuron oscillates just like others except not necessarily in phase with each other.

4 Properties of bifurcated periodic solutions

Theorem3.2means that in every neighborhood of(x^∗ =0,τ^∗ =τ_j,k^±)there is a unique branch of periodic solutions with the spatio-temporal pattern (3.2). In order to be able to analyze the Hopf bifurcation in more detail, we compute the reduced system on the center manifold associated with the pair of conjugate complex, purely imaginary solutions Λ = {iω₀,−iω₀} of the characteristic equation, whereω₀ = β±. By this reduction we can determine the Hopf bifurcation direction, i.e., to answer the question of whether the bifurcating branch of peri- odic solution exists locally for all τ > τ_j,k^± (supercritical bifurcation) or τ < τ_j,k^± (subcritical bifurcation). Throughout this section, we always assume that the function f satisfies

(P1). f ∈C²(_R,R),u f(u)6=0 whenu6=0.

To simplify the presentation, we first note that with the transformation (w₁,w₂,w₃,w₄) = (u₁, ˙u₁,u₂, ˙u₂),

we can rewrite (1.1) as the following system of delay differential equations

˙

w₁(t) =w₂(t),

˙

w₂(t) = −w₁(t)−εw₂(t) +εf(w₃(t−τ)),

˙

w₃(t) =w₄(t),

w˙₄(t) = −w3(t)−εw₄(t) +εf(w₁(t−τ)).

(4.1)

Recall that the characteristic matrix∆^∗(τ,λ)of the linearization of (4.1) is given by

∆^∗(τ,λ) =









λ −1 0 0

1 λ+ε −αεe^−λτ 0

0 0 λ −₁

−αεe^−λτ 0 1 λ+ε









, λ∈_C,

then

det∆^∗(τ_j,k^±,±iω₀) =0

for allj∈ {0, 1}andk∈_N0. In particular,

∆^∗(τ_j,k^±, iω₀)v_j =0, ∆^∗(τ_j,k^±,−iω₀)v_j =0. (4.2) wherev_j = (1, iω₀,(−1)^j,(−1)^jiω₀)^T. According to Theorem3.2, near eachτ =τ_j,k^±, there ex- ists a branch of small-amplitude periodic solutions of (4.1) bifurcated from the trivial solution u=0. The spatio-temporal pattern of the bifurcated periodic solution takes the form

u^j,k_i (t) =u_i^j,k₊₁

t− ^jω 2

, i(_{mod 2})_,

whereωrepresents its period and is sufficiently near to 2π/ω₀. Our purpose is to compute the reduced system of (4.1) on the center manifold associated with the pair of conjugate complex, purely imaginary solutionsΛ= {iω0,−iω0}of the characteristic equation.

Let us give the Taylor expansion of the right hand side of (4.1). Then we can rewrite (4.1) as

x˙(t) =Lτxt+G(xt,τ) (4.3) with

L_τϕ= (ϕ₂(0),−ϕ₁(0)−εϕ₂(0) +εαϕ₃(−τ),ϕ₄(0),−ϕ₃(0)−εϕ₄(0) +εαϕ₁(−τ))^T and

G(ϕ,τ) = ^εf

00(0)

2 (0,ϕ²₃(−τ), 0,ϕ²₁(−τ))^T + ^εf

000(0)

6 (0,ϕ³₃(−τ), 0,ϕ³₁(−τ))^T+o(|(0,ϕ³₃(−τ), 0,ϕ³₁(−τ))^T|)

for all ϕ = (ϕ₁,ϕ₂,ϕ₃,ϕ₄)^T ∈ C([−τ_j,k^±, 0],R⁴). By the Riesz representation theorem, there exists an 4×4 matrix-valued function η(·,τ) : [−τ, 0] → _R⁴ whose components each have bounded variation and are such that

L_τϕ=

Z ₀

−τ

dη(θ,µ)ϕ(θ) for ϕ∈ C([−τ, 0],R⁴). Next, we define for ϕ∈C¹([−τ, 0]_,R⁴)_,

Aτϕ=

(dϕ/dθ, if θ ∈[−τ, 0),

R0

−τdη(ξ,τ)ϕ(ξ) =Lτϕ, if θ =0. (4.4) Let ϕ_j(θ)be the eigenvector for A_τ^±

j,k associated with iω₀; namely, A_τ^±

j,kϕ_j(θ) =iω₀ϕ_j(θ). (4.5) In view of (4.2), we can choose ϕ_j(θ) =v_je^iω0θ forθ∈ [−τ_j,k^±, 0]. So, the center space atτ=τ_j,k^± and in complex coordinates isX =span{ϕ_j,ϕ_j}. Hence,Φ = (ϕ_j,ϕ_j)is a basis for the center spaceX. The adjoint operator A^∗

τ_j,k^± is defined by A^∗

τ_j,k^±ψ=







−dψ/dξ, if ξ ∈(0,τ_j,k^±], R0

−τ_j,k^± ψ(−t)dη(t,τ_j,k^±), if ξ =0.

Note that the domains of A_τ^±

j,k andA^∗

τ_j,k^± areC¹([−τ_j,k^±, 0],R⁴)andC¹([0,τ_j,k^±],R^4∗), respectively, where for convenience in computation we shall allow functions with rangeC⁴ instead of R⁴. It follows from (4.5) that±iω₀are also eigenvalues forA^∗

τ_j,k^±, and there is a nonzero row-vector functionψ^∗_j(ξ),ξ ∈[0,τ_j,k^±]such that

A^∗_τ±

j,kψ_j =−iω0ψ_j.

Then, Ψ = (ψ_j,ψ_j)^T is a basis for the adjoint space X^∗. In order to construct coordinates to describe the center manifold C_τ^±

j,k near to the origin, we need an inner product as follows:

hψ,ϕi=ψ(0)ϕ(0)−

Z ₀

θ=−τ_j,k^±

Z _θ

ξ=0ψ(ξ−θ)dη(θ,τ_j,k^±)ϕ(ξ)dξ (4.6) forψ∈C([0,τ_j,k^±],R^4∗)andϕ∈C([−τ_j,k^±, 0],R⁴). Then, as usual,

hψ,A_τ^±

j,kϕi=hA^∗_τ± j,kψ,ϕi for (ϕ,ψ) ∈ Dom(A_τ^±

j,k)×Dom(A^∗

τ_j,k^±). We normalize ψ_j by the condition hψ_j,ϕ_ji = 1 and hψ_j,ϕ_ji=0. By direct computation, we obtain that

ψ_j(ξ) =u_je^iω0ξ, whereu_j = D_j(−iω₀+ε, 1,(−1)^j+1(iω₀−ε),(−1)^j)and

D_j = ¹ 2

h

2iω0−ε+ (−1)^jεατ_j,k^±e^{−iω0τj,k±}i−1

.

Let Q={ϕ∈ C¹([−τ_j,k^±, 0],R⁴))|(_Ψ,ϕ) =0}, thenC([−τ_j,k^±, 0],R⁴) =X^LQ. So Eq. (4.3) can be written in the following abstract form

dUt

dt = A_τU_t+X₀G(U_t,τ), (4.7) where

X₀(θ) =

(0, θ ∈[−τ, 0), Id₄, θ =0.

Then by using the decomposition

Ut=2 Re{z(t)ϕ_j}+yt, z(t)∈_C, yt ∈Q¹ := Q^\C¹([−τ_j,k^±, 0],R⁴), we decompose (4.3) as

˙

z=iω₀z+u_jG^∗(2 Re{zϕ_j}+y,τ),

˙ y= A_τ^±

j,ky+ [X₀−_ΦΨ(₀)]G^∗(_{2 Re}{zϕ_j}+y,τ)_, ^(4.8) wherez ∈_C,y∈Q¹, andG^∗(xt,τ) =Lτxt−L_τ^±

j,kxt+G(xt,τ).

As the formulas to be developed for the bifurcation direction and stability are all relative to τ = τ_j,k^± only, we set τ = τ_j,k^± in (4.8) and obtain a center manifold y = W(z, ¯z) with the range in Q. The flow of (4.8) on the center manifold can be written as

U_t =_Φ·(z(t)_{, ¯}z(t))^T+W(z(t)_{, ¯}z(t))_,

where

˙

z(t) =iω₀z(t) +g^(j)(z, ¯z), (4.9) with

g^(j)(z, ¯z) =u_jG^∗(2Re{zϕ_j}+W(z, ¯z),τ)

=g⁽₂₀^j)z²

2 +g⁽₁₁^j)zz¯+g⁽₀₂^j)z¯²

2 +g⁽₂₁^j)z²z¯

2 +· · · . Hence we have

g⁽₂₀^j)= [1+ (−1)^j]D_jεf⁰⁰(0)e^{−2iω0τj,k±}, g⁽₁₁^j)= [1+ (−1)^j]D_jεf⁰⁰(0),

g⁽₀₂^j)= [1+ (−1)^j]D_jεf⁰⁰(0)e^{−2iω0τj,k±}, and

g₂₁^(j)= (−1)^jD_jεf⁰⁰⁰(0)e^{−iω0τj,k±} +¹

2D_jεf⁰⁰(0)^hW_20,3(−τ_j,k^±)e^iω0τj,k± +2W_11,3(−τ_j,k^±)e^{−iω0τj,k±}i +(−1)^j

2 D_jεf⁰⁰(0)^hW_20,1(−τ_j,k^±)e^iω0τj,k± +2W_11,1(−τ_j,k^±)e^{−iω0τj,k±}i .

(4.10)

So in order to compute g⁽₂₁^j), we need to computeW₁₁ = (W_11,1,W_11,2,W_11,3,W_11,4) andW20 = (W_20,1,W_20,2,W_20,3,W_20,4).

SinceW(z(t), ¯z(t))satisfies

W˙ = x˙t−ϕ_jz˙(t)−ϕ¯_jz˙¯(t)

= A_τ±

j,kx_t+X₀G(x_t,τ_j,k^±)−ϕ_jz˙(t)−ϕ¯_jz˙¯(t)

= A_τ^±

j,kW+X₀G(xt,τ_j,k^±)−ϕ_jg(z, ¯z)−ϕ¯_jg¯(z, ¯z)

= A_τ^±

j,kW+_H20^z

2

2 +_H11zz¯+_H02^z¯

2

2 +· · ·

(4.11)

then by using the chain rule

W˙ = ^∂W(z, ¯z)

∂z z˙+^∂W(z, ¯z)

∂z¯ z,˙¯

we have

((_2iω0− A_τn,λ)W₂₀= H₂₀

−A_τn,λW₁₁ =H₁₁. (4.12)

Note that

−ϕ_j(θ)g(z, ¯z)−ϕ¯_j(θ)g¯(z, ¯z) =H₂₀(θ)^z

2

2 +H₁₁(θ)zz¯+H₀₂(θ)^z¯

2

2 +· · · for−τ_j,k^± ≤θ <0, then we have

H20(θ) =−ϕ_j(θ)g₂₀^(j)−ϕ¯_j(θ)g¯02, H₁₁(θ) =−ϕ_j(θ)g₁₁^(j)−ϕ¯_j(θ)g¯₁₁

(4.13)