Hopf-zero bifurcation of Oregonator oscillator with delay

Transcription

1 Cai et al. Advances in Difference Equations 08 08:438 R E S E A R C H Open Access Hopf-zero bifurcation of Oregonator oscillator with delay Yuting Cai Liqin Liu * and Chunrui Zhang * Correspondence: fymm007@63.com Department of Mathematics Northeast Forestry University Heilongjiang China Abstract In this paper we study the Hopf-zero bifurcation of Oregonator oscillator with delay. The interaction coefficient and time delay are taken as two bifurcation parameters. Firstly we get the normal form by performing a center manifold reduction and using the normal form theory developed by Faria and Magalhães. Secondly we obtain a critical value to predict the bifurcation diagrams and phase portraits. Under some conditions saddle-node bifurcation and pitchfork bifurcation occur along M and N respectively; Hopf bifurcation and heteroclinic bifurcation occur along H and S respectively. Finally we use numerical simulations to support theoretical analysis. MSC: 34K8; 35B3 Keywords: Oregonator model; Delay; Hopf-zero bifurcation; Normal form Introduction An oscillatory chemical reaction refers to the reaction system of certain antileather concentration showing relatively stable cyclical changes. In 9 Bray achieved a liquid oscillatory reaction in the experiment. In 964 Zhabotinsky reported some other oscillatory reactions of this nature [4 5 ]. Until the 970s Field Koros and Noyes proposed the Oregonator model based on an in-depth study of the BZ reaction. According to Field and Noyes the BZ reaction is simplified. In 979 Tyson assumed that the concentration of the reactants A =[BrO 3 ]andb =[BrCHCOOH ] is independent of time. Therefore we can give the following reactant concentration equation: dp dt = k 3AQ k PQ + k 5 AP k 4 P dq = k dt 3 AQ k PQ + fk 0BW dw =k dt 5 AP k 0 BW P =[HBrO ] Q =[Br]W = [CeIV]. We will make the following changes in this system: x = αp y = βq z = γ W t = δt α k 4 k 3 A 06 mol/l β = k k 3 A 07 mol/l The Authors 08. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License which permits unrestricted use distribution and reproduction in any medium provided you give appropriate credit to the original authors and the source provide a link to the Creative Commons license and indicate if changes were made.

2 Cai et al. Advances in Difference Equations 08 08:438 Page of γ k 4k 5 B k 3 A 3 0 mol/l δ = k 5 B s. So we can obtain another form of the Oregonator oscillator the so-called Tyson-type oscillator: ε dx = qy xy + x x dt δ dy dt = qy xy + h z dz dt = x z ε = k 5B k 3 A 4 0 δ = ε α β 0 6 q = k k 4 k k h =h. Because δ is much smaller than ε the second formula in the system can be approximated as y h z q + x. This yields a simplified two-dimensional Oregonator model [7]withrespecttox and z: ε dx dt = x x h z x q x+q dz dt = x z. x =[HBrO ] z =CeIV. When electric current is applied the catalyst CeIV is perturbed and other species are not affected see [5]. Since the perturbation term is introduced in the equation dz dt = x z we rewrite this equation in the following form: dz = x z + kzt τ. dt We consider the Oregonator model with delay: ε dx dt = x x h z x q x+q dz dt = x z + kzt τ. ε =4 0 δ =4 0 4 q =8 0 4 andh 0 is an adjustable parameter. Nowadays many scholars study the Hopf bifurcation or Hopf-zero bifurcation in delay differential equations and some results have been obtained see [ ]. However to the best of our knowledge there are no studies on the Hopf-zero bifurcation of Oregonator oscillator with time delay. Therefore it is the far-reaching significance to research the Hopf-zero bifurcation of Oregonator model. The remainder of the paper is organized as follows. In Sect. we provide stability and conditions of existence of the Hopf-zero bifurcation by taking the interaction coefficient and delay as two parameters. In Sect. 3 we use the center manifold theory and normal

3 Cai et al. Advances in Difference Equations 08 08:438 Page 3 of form method [6 3] to investigate the Hopf-zero bifurcation with original parameters. In Sect. 4 we give several numerical simulations to support the analytic results. Finally we draw the conclusion in Sect. 5. Stability and existence of Hopf-zero bifurcation Let x z be an equilibrium point of system.. Obviously x x h z x q z = x k. x+q =0 Then we have x x h k xx q = 0.. x + q There are three roots x =0x = x + andx = x of Eq.. x ± = h k q ± h k q +4q + h k.. Therefore we obtain that system. has three steady-state solutions x z 0 0 and x + z +. Obviously there is a unique positive steady state. Theorem. For any ε >0q >0and h >0x + z + is the unique positive steady state of system.. Proof Let Hx=x x h k x x q x+q.from.and.wegetthatonlyx + satisfies Hx= 0 Hx >0for0<x < x + andhx <0forx > x +.FurthermorewehaveHx + =0and H x + <0. We further mainly study the dynamics of the equilibrium point x + z +. If the characteristic equation of system. has a simple pair of purely imaginary eigenvalues ±iω asimple root 0 and all other roots of the characteristic equation have negative real parts then the Hopf-zero bifurcation will occur. Let x = x x + and z = z z +.Thenwecanvary. as the following equivalent system: dx dt = ε x + x + x x + h z + z + x+x + q x+x + +q dz dt = x z + kzt τ..3 The linearization equation of system.3 at 0 0 is dx dt = a x + a z dz dt = x z + kzt τ.4

4 Cai et al. Advances in Difference Equations 08 08:438 Page 4 of a = ε qh z + q+x +.4is + x + anda = ε qh h x + q+x +. The characteristic equation of system Eλ=λ b λ b kλe λτ + ka e λτ = 0.5 b = a andb = a + a. If λ =0istherootofEq..5 then ka b =0.Ifτ =0thensystem.5becomes Eλ=λ b + kλ =0. Therefore we obtain that if τ =0andb + k < 0 then excluding a single zero eigenvalue all the roots of Eq..5havenegativerealparts. Next we consider the case of τ 0.Letiωwith ω >0bearootofλ b λ b kλe λτ + ka e λτ =0.Then ω b iω b kiωe iωτ + ka e iωτ =0. Separating the real and imaginary parts we get ω b = kω sin ωτ ka cos ωτ b ω = kω cos ωτ + ka sin ωτ..6 It follows that ω satisfies ω 4 + b + b k ω + b k a = 0..7 Suppose m = ω and denote u =b + b k r = b k a.theneq..7becomes m + u m + r = 0..8 Following [7] we consider the following cases: B r <0. Then we find that system.5 has a unique positive root m = u + u 4r. B r >0 u >0. Then system.5 has no positive root. B3 r >0 u <0. In this case if system.5 has real positive roots then k is very large and h is infinitely close to one which is a contradiction. Theorem. For the quadratic Eq..8 we have: i if r <0 then Eq..5 has a unique positive root m = u + u 4r. ii if r >0 then Eq..5 has no positive root. Suppose that Eq..8 has positive roots. Without loss of generality we assume that it has a positive root defined by m. ThenEq..7 hasapositiverootω andω must satisfy

5 Cai et al. Advances in Difference Equations 08 08:438 Page 5 of the equation ω + a b ω 3 + ωa kω + a + + a kω + a =. According to system.6 we obtain Denote cosωτ= ω + a b kω + a sinωτ= ω3 + ωa + a kω + a. ω τ j = arccos β +jπ α 0 ω π arccos β +jπ α 0 β = ω +a b α kω +a = ω3 +ωa +a j =0...Thensystem.5hasapairofpurely kω +a imaginary roots ±iω with τ = τ j andτ = τ j j =0...satisfytheequationsinωτ>0. We get k <0whenB holds and then τ j = ω arccos β +jπ j {0...}. We obtain the transversality conditions as follows. Theorem.3 If r <0then d Re{λτ j} dτ 0. Proof Substituting λτ τ = τ j intoeq..5 we get So dλ dτ = λka e λτ kλ e λτ λ b λ + τkλe λτ ke λτ τa λe. λτ dλ = τk a dτ ka λ λa λ + b e λτ ka λ. Consequently we obtain dre λτ dτ { τk a = Re τ=τ j ka λ λa λ + b e λτ } ka λ = τa k a ka + ω + ω ω 4 + a + b ω + b ω k a + 0. ω Theorem.4 If ka = b b + k <0and r <0then for τ = τ j j =0...system. undergoes a Hopf-zero bifurcation at equilibrium x + z +. τ=τ j

6 Cai et al. Advances in Difference Equations 08 08:438 Page 6 of 3 Normal form for Hopf-zero bifurcation In this section we use the center manifold theory and normal form method [6 3] to study Hopf-zero bifurcations. The normal form of a Hopf-zero bifurcation for a general delay-differential equations has been given in the following two papers: one is for a saddlenode-hopf bifurcation [] and theother is fora steady-state Hopf bifurcation [0]. After scaling t t/τ system.3becomes dx dt = τ ε x + x + x x + h z + z + x+x + q x+x + +q dz dt = τx z + kzt. 3. Let τ = τ +μ k =+ a a +μ μ and μ are bifurcation parameters. Then system 3.canbewrittenas dx dt =τ + μ a x + a z + M dz dt =τ 3. + μ x z ++ a a + μ zt M = ε [ h z + q x + + ]x + qh q+x + ε q+x + xz + qh x + x 3 + ε x + +q 4 ε qh x + +q 3 x z. Choose the phase space C = C[ 0]; R 4 with supremum norm and define X t C by X t θ=xt + θ τ θ 0 and X t = sup X t θ.thensystem3.becomes Ẋt=LμX t + FX t μ 3.3 a x + a z LμX t =τ + μ x z ++ a a + μ zt and τ + μ M FX t μ= 0 Lμϕ = 0 dηθ μϕξ dξ for ϕ [ 0] R4 0 θ =0 ηθ μ= τ + μ A θ 0 τ + μ A + B θ = with A = a a 0 0 and B = 0 + a. a Consider the linear system Ẋt=L0X t.

7 Cai et al. Advances in Difference Equations 08 08:438 Page 7 of Between C and C = C[0 τ] C n the bilinear form is defined by ψs ϕθ = ψ0ϕ0 0 = ψ0ϕ0 = ψ0ϕ0 θ 0 0 θ 0 0 ψξ θ dηθ0ϕξ dξ ψξ θ d Aϕθ+Bϕθ + ϕξ dξ ψξ +Bϕξ dξ ψ C ϕ C ϕθ=ϕ θ ϕ θ ϕ 3 θ C ψs= ψ s ψ s C. ψ 3 s We know that L0 has a pair of purely imaginary eigenvalues ±iω ω >0asimple0 and all other eigenvalues have negative real parts. Let Λ = {iω iω0}letp be the generalized eigenspace associated with ΛandletP is the space adjoint with P.ThenC can be decomposed as C = P QQ = {ϕ C :ψ ϕ = 0 for all ψ P }.Wecanchoose the bases Φ and Ψ for P and P such that Ψ s Φθ = I Φ = ΦJ and Ψ = JΨ J = diagiω iω0. We calculate ΦθandΨ s asfollows: Φθ= a iω a e iwτ θ a iω a e iwτ θ a e iwτ θ e iwτ θ a and Ψ s= D iω a e iwτ s D iω a e iwτ s D e iwτ s D e iwτ s D a D D = a iω a ++τ D = a + a + a τ b. + a a Let usenlarge the space C to the following space: { } BC = ϕ is a continuous function on [ 0 and lim ϕθexists. θ 0 Itselementscanbewrittenasψ = ϕ + X 0 α with ϕ C α C n and 0 θ [ 0 X 0 θ= I θ =0. In BC system3.3 varies an abstract ODE: d dt X t = Au + X 0 Fu μ 3.4

8 Cai et al. Advances in Difference Equations 08 08:438 Page 8 of u CandA is defined by A : C BC Au = u + X 0 [ L0u u0 ] and Fu μ= [ Lμ L 0 ] u + Fu μ. Then the enlarged phase space BC can be decomposed as BC = P Ker π.letx t = Φxt+ ỹθ xt=x x x 3 T namely xθ= a iω a e iwτθ x + a iω a e iwτθ x a x 3 + y θ zθ=e iwτθ x + e iwτθ x + a x 3 + y θ. Let ψ ψ Ψ 0 = ψ ψ = ψ 3 ψ 3 D iω a D D iω a D D a D. Equation 3.4can be expressed as ẋ = Jz + Ψ 0 F Φz + ỹθ μ ỹ = A Q ỹ +I πy 0 F Φz + ỹ0 μ 3.5 ỹθ Q := Q C Ker πanda Q is the restriction of A as an operator from Q to the Banach space Ker π. System 3.5canberewrittenas ẋ = Jx +! f x y μ+ 3! f 3 x y μ+h.o.t. ẏ = A Q y +! f x y μ+ 3! f 3 x y μ+h.o.t. f f 3 ψ F x y μ+ψ F x y μ x y μ= ψ F x y μ+ψ F x y μ ψ 3 F x y μ+ψ 3F x y μ ψ F3 x y μ+ψ F3 x y μ x y μ= ψ F3 x y μ+ψ F3 x y μ ψ 3 F3 x y μ+ψ 3F3 x y μ

9 Cai et al. Advances in Difference Equations 08 08:438 Page 9 of with [ F = μ a a a x + x a x 3 + y 0 + a x + x + a x 3 + y 0 ] iω a iω a [ [ ] hz+ q x + + a a + τ x ε q + x + + x a x 3 + y 0 iω a iω a + qh a a x x + x a x 3 + y 0 + x + a x 3 + y 0 ] ε q + x + iω a iω a F = μ a a x + x a x 3 + y 0 μ x + x + a x 3 + y 0 iω a iω a + + a μ e iwτ x + e iwτ x + a x 3 + y a + τ μ e iwτ x + e iwτ x + a x 3 + y 3! F 3 = τ qhx + ε q + x + 4 F 3 =0. + τ ε qh q + x + 3 a x + iω a a x + iω a x + x + a x 3 + y 0 According to [5] ImM c is spanned by a iω a x a x 3 + y 0 3 a iω a x a x 3 + y 0 { z e z z 3 e z μ i e μ μ e z z e z μ i e z z 3 e 3 z 3 μ i e 3 } i = with e = 0 0 T e = 0 0 T e 3 = 0 0 T. ImM 3 c is spanned by { z 3 e z z z 3 e z z e z z 3e z z 3e 3 z z3 e } 3. Then we get g x0μ=proj ImM cf x0μ=proj S f x0μ+o μ g3 x0μ=proj ImM3 c f 3 x0μ=proj S f 3 x00+o μ x + μ x S and S are spanned respectively by and z μ i e z μ i e z 3 μ i e 3 i = z 3 e z z z 3 e z z e z z 3e z z 3e 3 z z 3 e 3. System 3.5 canbetransformedonthecentermanifoldinthefollowingnormalform: ẋ = Jx + g x0μ+ 6 g 3 x0μ+h.o.t. 3.6

10 Cai et al. Advances in Difference Equations 08 08:438 Page 0 of We need to compute g x0μandg 3 x0μin3.6. We can compute g x0μ: g x0μ= Proj S f x0μ+ϑ u a μ + a μ x + a 3 x x 3 = a μ + a μ x + a 3 x x 3 + ϑ u a μ + a μ x 3 + a 3 x x + a 4 x 3 a = D τ e iωτ a a a = D iω a + a + iω a [ D τ hz + q x + + a 3 = iω a ε q + x + a = a τ D a = a b D [ τ hz + q x + + a a 3 = D ε q + x + a ω a 4 = τ D ε + a a a iω a τ ε [ hz+ q x + + q + x + a + qh q + x + a a e iωτ qhτ εq + x + qh q + x + ]. ] a a a iω a a iω a + a iω a ] Next we compute g 3 x0μ: 6 g 3 x0μ= 6 Proj KerM f 3 x0μ = 6 Proj S f 3 x00+ϑ x u + x μ = 6 Proj S f 3 x Proj S [ Dx f + ϑ x u + x μ. x00u x0+ D y f x00u x0 ] We can get Proj S f 3 x00.since 6 f 3 x00= τ ε qhx D + a x + +q 4 iω a x + a iω a x a x qh a x + +q 3 iω a x + a iω a x a x 3 qhx D + a x + +q 4 iω a x + a iω a x a x qh a x + +q 3 iω a x + a iω a x a x 3 x + x + a x 3 qhx a D + a x + +q 4 iω a x + a iω a x a x qh a iω a x + a iω a x a x 3 x + x + a x 3 x + +q 3

11 Cai et al. Advances in Difference Equations 08 08:438 Page of we have 6 Proj S f3 x00= b x x + b x x 3 b x x + b x x 3 + ϑ u b x x x 3 + b x 3 3 b = τ ε D qhx + a a 3 x + + q 4 ω + a + ω + a iω a + τ ε D qh a a + x + + q 3 iω a ω + a b = τ ε D qhx + a 3 τ x + + q 4 iω a ε D qh a a x + + q 3 iω a b = τ ε a qhx + 6a 3 D x + + q 4 ω + a + τ ε a qh a a D x + + q 3 ω + a b = τ ε a qhx + D x + + q 4 a3 + τ ε a qh D x + + q a a a iω a Next we compte Proj S [D x f x00u x0]. From [5] weknowthatbecausej is a diagonal matrix the operators Mj j are defined in Vj 5C3 so we have a diagonal representation relative to the canonical basis {μ p x q e k : k = 3p N 0 q N3 5 0 p + q = j} of Vj C3 e = 0 0 T e = 00 T e 3 = 0 0 T. Clearly we get Mj μ p x q e k = iω q q + k μ p x q e k k = μ p x q e 3 = iωq q μ p x q e 3 p + q = j. M j So Ker Mj { = span μ p x q e k :q λ=λ k k =3p N 0 q N3 0 p + q = j} with λ =λ λ λ 3 =iω iω 0. The elements of the canonical basis of V 5 C3 are μ μ e μ i e μ i x e μ i x e μ i x 3 e x x e x x 3 e x x 3 e x e x e x 3 e μ μ e μ i e μ i x e μ i x e μ i x 3 e x x e x x 3 e x x 3 e x e x e x 3 e μ μ e 3 μ i e 3 μ i x e 3 μ i x e 3 μ i x 3 e 3 x x e 3 x x 3 e 3 x x 3 e 3 x e 3 x e 3 x 3 e 3 i = the images of which under iω M are μ μ e μ i e 0 μ i x e μ i x 3 e x x e 0 x x 3 e x e 3x e x 3 e μ μ e μ i e μ i x e 0μ i x 3 e x x e x x 3 e 03x e x e x 3 e 0 0 μ i x e 3 μ i x e 3 00x x 3 e 3 x x 3 e 3 x e 3 x e 30 i =.

12 Cai et al. Advances in Difference Equations 08 08:438 Page of Hence U x0=u x μ μ=0 = M ProjImM f x00 D h[a hx + a hx a x 3 m + n a hx + a hx a x 3 x + x + a x 3 ] D h[a hx + a hx a x 3 m + n a hx + a hx a x 3 x + x + a x 3 ] D [a hx + a hx a x 3 m + n a hx + a hx a x 3 x + x + a x 3 ] D h[ma h x a hhx x + a hx x 3 3 a h a x 3 + na hx a h + hx x 3 a hx a a h +x x 3 + a a x 3 ] D h[m 3 a h x a h x + a x 3 +a hhx x a hx x 3 + n 3 a hx + a a h x x 3 a hx + a a hx x 3 a a x 3 + a h + hx x ] D [m a h x a h x a hx x 3 +a hx x 3 + n a hx + a a hx x 3 a hx a a hx x 3 a x x 3 + a x x 3 = M ProjImM τ ε = τ iωε m = hz +q x + + n = qh q+x + q+x + andh = iω a. Therefore we obtain 4 Proj [ S Dx f x00 U x0] = c x x + c x x 3 c x x + c x x 3 c x x x 3 + c x 3 3 c = τ [ D iωε h m a 4 h3 h + D h mn a 3 h3 + D h mn 3a 3 h h 6 + D h n a h h h+d D hhm 3 a4 h h 4 + D D hhmn 3 a3 h h a3 hh + a h + a hh + D D hhn a hh + D D hm a 4 h h + D D hmn a 3 hh D D hmn a a 3 h h a3 h D D hn a a hh a h ]

13 Cai et al. Advances in Difference Equations 08 08:438 Page 3 of c = τ [ D iωε h m a 4 h +D h mna 3 a h + h + D h n a a h + D D hhm 4a 4 hh + D D hhmn 4a a 3 hh +a3 h +a3 h + D D hhn 3 a a h + h+ a a hh + + D D hm 4a 4 h D D hmn 6a a 3 h a3 +D D hn a a h a a ] c = τ iωε [ D D hm 6a 4 h h D D hmn 3a a 3 h h +6a 3 hh +3a3 h D D hn a a hh +a h a a h + a h + D D hm 6a 4 hh D D hmn 3a a 3 hh 3a 3 h 6a 3 hh D D hn a a hh a h + a a h a h] c = τ [ D D iωε hm a 4 h D D hmn a 3 3a a 3 h D D hn a a h a a D D hm a 4 h D D hmn 3a a 3 h a3 + D D hn a a h a a ]. Finally we can compute Proj S [D y f x00u x0].defineh = hxθ=u x0and write h θ hθ= = h h 00 x θ + h 00x + h 00x 3 + h 0x x + h 0 x x 3 + h 0 x x 3 h 00 h 00 h 00 h 0 h 0 h 0 Q.M hx=f x00 decides the coefficients of hwhichisequivalentto D x hjx A Q h=i πx 0 F Φx0. We use the definitions of A Q and π to obtain ḣ D x hjx = ΦθΨ 0F Φx0 h0 Lh = F Φx0 ḣ is the derivative of hθwithrespecttoθ.let F Φx0=A 00 x + A 00x + A 00x 3 + A 0x x + A 0 x x 3 + A 0 x x 3 A ijk C0 i j k i+j+k =. We can compare the coefficients of x x x 3 x x x x 3 x x 3 andwegetthath 00 = h 00 h 0 = h 0 and the following differential equations are satisfied by h 00 h 0 h 0 h 00 respectively: ḣ 00 iωτ h 00 = ΦθΨ 0A 00 ḣ 00 0 Lh 00 =A

14 Cai et al. Advances in Difference Equations 08 08:438 Page 4 of ḣ 0 iωτ h 0 = ΦθΨ 0A 0 ḣ 0 0 Lh 0 =A 0 ḣ 0 = ΦθΨ 0A 0 ḣ 0 0 Lh 0 =A 0 ḣ 00 = ΦθΨ 0A 00 ḣ 00 0 Lh 00 =A τ a hma h + n τ a hma h + n A 00 = A 00 = 0 0 τ a ma na τ a hmh h + nh + hn A 00 = A 0 = 0 0 τ a ma h + nha n τ a ma h + a n h n A 0 = A 0 =. 0 0 Since we have τε mx + nxz F u t 0= 0 f x y0=ψ 0F Φx + y0 which gives = τ ε = τ ε D h D D h D D a D ma hx + a hx a x 3 + y 0 + na hx + a hx a x 3 + y 0 x + x + a x 3 + y 0 0 D h[ma hx + a hx a x 3 + y 0 + na hx + a hx a x 3 + y 0x + x + a x 3 + y 0] D h[ma hx + a hx a x 3 + y 0 + na hx + a hx a x 3 + y 0x + x + a x 3 + y 0] D [ma hx + a hx a x 3 + y 0 + na hx + a hx a x 3 + y 0x + x + a x 3 + y 0] D h[ma hx + a hx a x 3 + na hx + a hx a x 3 + x + x + a x 3 ]h 0 4 D yf y=0μ=0h= τ D h[ma hx + a hx a x 3 ε + na hx + a hx a x 3 + x + x + a x 3 ]h 0. D [ma hx + a hx a x 3 + na hx + a hx a x 3 + x + x + a x 3 ]h 0

15 Cai et al. Advances in Difference Equations 08 08:438 Page 5 of Thus d x 4 Proj S D y f y=0μ=0u = x + d x x 3 d x x + d x x 3 d x x x 3 + d x x 3 3 d = τ ε D hm + na h h h h τ ε D hn h h 00 0 d = τ ε D hm + na h h 00 0 h τ ε D hn h a h 0 0 d = τ ε D hm + na h h h h 0 0 h τ ε D hn a h h h 0 0 d = D hm + na h D hna h So we can obtain b + c + d x 6 g 3 x0μ= x +b + c + d x x 3 b + c + d x x +b + c + d x x 3 + ϑ x μ + x μ. b + c + d x x x 3 +b + c + d x 3 3 Therefore on the center manifold the system ẋ = Jx + g x0μ + 6 g 3 x0μ +h.o.t. becomes ẋ =a μ + a μ x + a 3 x x 3 +b + c + d x x +b + c + d x x 3 + h.o.t. ẋ =a μ + a μ x + a 3 x x 3 +b + c + d x x +b + c + d x x 3 + h.o.t. ẋ 3 =a μ + a μ x 3 + a 3 x x + a 4 x 3 +b + c + d x x x 3 +b + c + d x h.o.t. 3. By changing variables x = ρ iρ x = ρ + iρ x 3 = ρ 3 and introducing the cylindrical coordinates ρ = r cos θ ρ = r sin θ ρ 3 = γ r >0system3.becomes ṙ = α μr + β rγ + β 30 r 3 + β rγ + h.o.t. γ = α μγ + m 0 r + m 0 γ + m r γ + m 03 γ 3 + h.o.t. θ = ω +Im[a ]μ + Im[a ]μ α μ=re[a ]μ + Re[a ]μ β = Re[a 3 ] β 30 = Re[b + c + d ] β = Re[b + c + d ] α μ=a μ m 0 = a 3 m 0 = a 4 m = b + c + d m 03 = b + c + d.

16 Cai et al. Advances in Difference Equations 08 08:438 Page 6 of Therefore we can get the system in the plane r γ : ṙ = α μr + β rγ + β 30 r 3 + β rγ + h.o.t. ζ = α μγ + m 0 r + m 0 γ + m r γ + m 03 γ 3 + h.o.t. 3. From [5]we know that Eq.3.becomes ṙ =α μ+β δ + β δ r +β +β δrγ + β 30 r 3 + β rγ γ =α μδ + m 0 δ + m 03 δ 3 +α μ+m 0 δ +3m 03 δ γ +m 0 + m δr +m 0 +3m 03 δγ + m r γ + m 03 γ Choose δ = δμsuchthat α μ+m 0 δ +3m 03 δ =0. To simplify the above system we only discuss the case of m 0 0m Clearly for small α μ the equation has two real roots. We take 3m δ = 03 [ m 0 + m 0 3m 03α μ] if m 0 >0 3m 03 [ m 0 m 0 3m 03α μ] if m 0 <0. Then δ = δμ is differentiable at μ = 0 andδ0 = 0. Define k = α μ+β δ + β δ k = α μδ + m 0 δ + m 03 δ 3 a = β +β δ b = m 0 + m δ c = m 0 +3m 03 δ and choose x = r y = γ.theneq.3.3becomes ẋ = k x + axy + β 30 x 3 + β xy ẏ = k + bx + cy + γ x y + γ 03 y Let x c x y b y t c b t and η = k c b η = k c b. Then system 3.4becomes ẋ = η x + Bxy + d x 3 + d xy ẏ = η + ηx y y + d 3 x y + d 4 y B = a c 0 η = sgnbc

17 Cai et al. Advances in Difference Equations 08 08:438 Page 7 of and d = β 30 c c b d b β = d 3 = m c c c b d b m03 4 =. c According to [5] we assume that K 3 = η B + d + B d + ηd 3 +3d 4 0. For small η and η the qualitative behavior of 3.5 near 0 0 is the same as that of the following system see [9]: ẋ = η x + Bxy + xy ẏ = η x y. 3.6 In Eq. 3.6 there are two trivial equilibrium points E =0± η η > 0 and two nontrivial equilibrium points E 34 = B B ± B 4η +η + η B ± B 4η. In [3]and[6] wecan find thecompletebifurcationdiagrams ofsystem 3.3. Herewe list some of them. Theorem 3. a If B <0 then the bifurcation diagram of system 3.3 consists of the origin and the following curves: M = { η η :η =0η 0 } { N = η η :η = } B η + ϑ η 3 η 0. Along M and N a saddle-node bifurcation and pitchfork bifurcation occur respectively. System 3.3 has no periodic orbits. Moreover if η η is in the region between M and N then the solution of system 3.3 goes asymptotically to one of the equilibrium points E E and E 3. b If B >0 then the bifurcation diagram of system 3.3 consists of the origin the curves M and N and the following curves: H = { η η :η =0η >0 } { S = η η :η = B 3B + η + ϑ η 3/ } η >0. Along M and N we have exactly the same bifurcation as in a. Along H and S a Hopf bifurcation and a heteroclinic bifurcation occur respectively. If η η lies between the curves H and S then system 3.3 has a unique limit cycle which is unstable and becomes a heteroclinic orbit when η η S. Figures and showaandboftheorem3.respectively.

18 Cai et al. Advances in Difference Equations 08 08:438 Page 8 of Figure When B < 0 the bifurcation diagrams and phase portraits of system 3.3see[8] Figure When B > 0 the bifurcation diagrams and phase portraits of system 3.3see[8] 4 Numerical simulations In this section we give some examples to explain the theoretical results. Set q =8 0 4 h = 3 k =.5andε =4 0 and consider the following system: dx dt = ε dz dt x u x x hz x+u = x z + kzt τ. By calculations we obtain a = i a = a 3 = i a =.9430 a =.453 a 3 = a 4 =.654 b = i b = i b = i b =.8473 c = i c = i c = 0.07 c = d = i d =

19 Cai et al. Advances in Difference Equations 08 08:438 Page 9 of Figure 3 The system is asymptotically stable around the equilibrium point for μ μ = The green line represents x the red line represents z. Waveform diagram for variable of x z left. Phase diagram for variable x zright Figure 4 The system has an asymptotically stable periodic orbit near τ for μ μ = The green line represents x the red line represents z. Waveform diagram for variable of x z left. Phase diagram for variable x zright i d = d = ; the equilibrium point is and τ = For small μweobtaink Conclusions In this article we have discussed the Hopf-zero bifurcation of Oregonator oscillator with delay. We thoroughly analyze the distribution of the eigenvalues of the corresponding characteristic equation and find some specific conditions ensuring that all the eigenvalues have negative real parts. We also can discover the factors that make system. undergo a Hopf-zero bifurcation at equilibrium x + z +. Meanwhile by using the normal form method and the center manifold theorem we have derived the normal form of the reduced system on the center manifold and discussed the Hopf-zero bifurcation with parameters in system.. Besides we have obtained bifurcation diagrams and phase portraits of system 3.3whenB >0andB < 0 respectively. We also note that a saddle-node bifurcation and pitchfork bifurcation occur along M and Nrespectivelyand a Hopf bifurcation and a heteroclinic bifurcation occur along H and S respectively. Finally numerical stimulations see Figure 3 4 and 5 have been given to illustrate the theoretical results. Our work is a further study of the Oregonator oscillator which will be useful in the research of the complex phenomenon caused by high codimensional bifurcation of a delaydifferential equation.

20 Cai et al. Advances in Difference Equations 08 08:438 Page 0 of Figure 5 Waveform diagram for variable of x z. The first two figures show that when τ =.50<τ 0 the system is stable around the equilibrium point. When τ =.80>τ 0 the next two figures show that the system is unstable around the equilibrium point Acknowledgements The authors wish to express their gratitude to the editors and the reviewers for the helpful comments. Funding This research is supported by the Heilongjiang Provincial Natural Science Foundation No. A0506. Competing interests The authors declare that they have no competing interests. Authors contributions The idea of this research was introduced by YC LL and CZ. All authors contributed to the main results and numerical simulations and approved the final manuscript. Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 9 September 08 Accepted: 9 November 08 References. Cao X. Song Y. Zhang T.: Hopf bifurcation and delay-induced Turing instability in a diffusive lac operon model. Int. J. Bifurc. Chaos Chang X. Wei J.: Hopf bifurcation and optimal control in a diffusive predator prey system with time delay and prey harvesting. Nonlinear Anal. Model. Control Chow S.N. Li C. Wang D.: Normal forms and bifurcation of planar vector fields. Cambridge Ding W. Liao X.F. Dong T.: Hopf bifurcation in a love-triangle model with time delays. Neurocomputing Ding Y. Jiang W. Yu P.: Hopf-zero bifurcation in a generalized Gopalsamy neural network model. Nonlinear Dyn Faria T. Magalhaes L.T.: Normal forms for retarded functional differential equation with parameters and applications to Hopf bifurcation. J. Differ. Equ Gazor M. Sadri N.: Bifurcation control and universal unfolding for Hopf-zero singularities with leading solenoidal terms. SIAM J. Appl. Dyn. Syst Guo Y. Jiang W.: Hopf bifurcation in two groups of delay-coupled Kuramoto oscillators. Int. J. Bifurc. Chaos

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