Dynamic properties of the coupled Oregonator model with delay ∗

. This work explores a coupled Oregonator model. By analyzing the associated characteristic equation, linear stability is investigated and Hopf bifurcations are demonstrated, as well as the stability and direction of the Hopf bifurcation are determined by employing the normal form method and the center manifold reduction. We also discussed the Z 2 equivariant property and the existence of multiple periodic solutions. Numerical simulations are presented to illustrate the results in Section 5.


Introduction
Delay in dynamical systems is exhibited whenever the system behavior is dependent at least in part on its history. Many technological and biological systems are known to exhibit such behavior, such as coupled laser systems, high-speed milling, population dynamics and gene expression [1][2][3][4] where X ∈ U , F ∈ C 2 (U ), U ∈ R n is a compact closure of the open set. When two identical oscillators coupling in the way of linear difference, the equation of motion of the system isẊ where K 1 , K 2 are the coupling coefficient matrix (see [5][6][7][8]). Chemical diffusion coupling often described in this form [9]. In 1979, Tyson simplified the three-dimensional oscillator Oregonator to 2-D: here x = [HBrO 2 ], z = Ce (IV). In 1999, Tianshou Zhou and Chunsuo Zhang proposed a coupled Oregonator model [10] ε When electric current is applied, the catalyst Ce (IV) is perturbed and other species are not affected (see [11]). Consequently, in modeling, the perturbation term is introduced only in equation dz/dt = x − z, and we rewrite this equation as the following form: Then the purpose of this paper is to consider coupled Oregonator model with a delay where ε = 4 × 10 −2 , δ = 4 × 10 −4 , u = 8 × 10 −4 , h ∈ (0, 1) is an adjustable parameter. The remainder of this paper organized as follows. In the next section, we shall consider the stability and the local Hopf bifurcation. Base on the symmetric bifurcation theorem of Golubitsky [12], we also discussed the Z 2 equivariant property and the existence of multiple periodic solutions in Section 3. In Section 4, based on the normal form method and the center manifold reduction introduced by Hassard et al. [13], we derive the formulae determining the direction, stability and the period of the bifurcating periodic solution at the critical value of τ , a conclusion is drawn in this section. To verify the theoretic analysis, numerical simulations are given in Section 5. www.mii.lt/NA

Stability and local Hopf bifurcations
Through out the paper, we assume that k < 1 (resulting in equilibrium point x 0 /z 0 = 1 − k > 0).
is the uniform steady state of system (5). If x s 1 = x s 2 , z s 1 = z s 2 , then S is an uniform steady state. Let S(x 1 , z 1 , x 2 , z 2 ) be an equilibrium point of system (5). Obviously, S(x 1 , z 1 , x 2 , z 2 ) satisfying equation group Let Equation (7) have has three roots x = 0, So, system (5) has three steady-state solution, Obviously, there is an unique uniformly positive steady state.
The following is to prove that the uniformly positive steady state is unique. (7) and (8) we conclude there is an unique x + satisfying G(x) = 0, and when 0 < x < x + , we have G(x) > 0; when x > x + , we have G(x) < 0. Further more we obtain G(x + ) = 0,Ġ(x + ) < 0. From (6) we have The following we will study the function Clearly g(x + , D) = 0, that means x is a positive real root of Eq. (10). Notice that 0 < x < x + , fromĠ(x + ) < 0 and G(x) > 0, we can obtain x − G(x)/D < x < x + and On the other hand when x > x + , from G(x) < 0, we conclude that x − G(x)/D > x > x + and G(x − G(x)/D) < 0, further more we have g(x, D) < 0. Then, the following lemma holds.
The following work is expanded around the uniformly positive steady state and we don't consider of other steady state. Let S + (x 10 , z 10 , x 20 , z 20 ) satisfying Eqs. (6), where Then we can rewrite (5) as the following equivalent system: Set x 0 = x 10 , z 0 = z 10 . The linearization of system (11) at (0, 0, 0, 0) is where

Moreover, its corresponding characteristic equation is
where In this section, we will study the distribution of roots of Eq. (12). We first introduce the following important result, which was been proved by Ruan and Wei using Rouche theorem [14]. For τ = 0, the two roots of ∆ 1 = 0 have negative real parts if and Thus, the two roots of ∆ 2 = 0 have negative real parts. We impose the following condition: is satisfied, all the roots of (12) have negative real parts, hence (x 10 , z 10 , x 20 , z 20 ) is asymptotically stable.
Next, we mainly focus on the case of τ > 0.
Separating the real and imaginary parts, we obtain which implies Let z = w 2 1 and denote Then, (15) becomes m 2 + u 1 m + r 1 = 0.
In order to seek a positive solution for Eq. (14), we impose the following condition: Clearly, under the condition (B1), (12) has a unique positive Under the condition (B2), (12) has no positive root.
Under the condition (B3), if there are real positive roots, then |k| is very large, h infinitely close to one, does not match with the actual situation.
Summarizing the above discussions, we obtain the following.
Lemma 3. For the polynomial equation (15), we have the following result: Suppose that Eq. (15) has positive roots. Without loss of generality, we assume that it has a positive root defined by m. Then, Eq. (14) has a positive root ω 1 , moreover ω 1 must satisfies the following equation: By (13), we have Thus, denote . . , then ±iw 1 is a pair of purely imaginary roots of (12) with τ = τ 1j .
If (B1) hold, we have k < 0, then Separating the real and imaginary parts, we obtain which implies Then, Eq. (17) becomes In order to seek a positive solution for (18), we impose the following condition: Clearly, under the condition (C1), (18) has a unique positive Under the condition (C2), (18) has no positive root.
Under the condition (C3), if u 2 2 − 4r 2 > 0, then (18) has a pair of roots Summarizing the above discussions, we obtain the following.
Suppose that Eq. (18) has some positive roots. Without loss of generality, we assume that it has a positive root defined by m. Then, (14) has a positive root ω 2 , moreover ω 2 must satisfies the following equations: By (16), we have Thus, denoting where j = 0, 1, 2, . . . , then ±iw 2 is a pair of purely imaginary roots of (12) with τ = τ 2j . If (C1) hold, we have As the same, for condition (C3) and u 2 2 − 4r 2 > 0 is satisfied, then (17) has at least two roots w . And Because the condition (C3) and 2j }. Here we consider the whole system (12). Note that when τ = 0, Eq. (12) becomes Using Lemmas 1-3, we have the following results.
is satisfied, then all roots with positive real parts of equation ∆ 2 = 0 has the same sum as those of the polynomial equation For convenience, we make some hypotheses as follows: Then denoting τ j = τ 1j .
Lemma 6. Suppose that one the the hypothesis (P2), (P3), (P4) is satisfied, then (d (Re λ(τ ) (12) and differentiating the resulting equation in τ , we obtain we have We first focus on the case ∆ 1 = 0. Under the condition (P2), when τ = τ 1j , we have ∆ 1 = 0. Thus Eq. (20) becomes We can easily obtain . www.mii.lt/NA For a 1 < 0, we have k < 0. In the previous part of this paper we know |k| can't be very large. As mentioned above, it can be obtained that For a 1 > 0, we have 0 < k < 1 − 2f and a 1 < 2. It can be obtained that Then we focus on the case where ∆ 2 = 0. As the same, we have .
In the previous part of this paper we know |k| and w 2 can't be very large, ε is very small, it can be obtained that So under the condition (P3), we have As the same, under the condition (P4), we have In the case of r 1 < 0, r 2 < 0, we have k < 0. Then we have the following results: In addition, the second condition of (P4) is not established. Then condition (P4) modified as r 1 < 0, r 2 < 0. Therefore, the transversality condition holds and Hopfbifurcation occurs at τ = τ j . From Lemmas 4 and 5, we have the following. (5) is asymptotically stable for all τ > 0.

Existence of multiple periodic solutions
In the following, we consider the symmetric properties of Eq. (5). Using the theories of functional differential equation, we know that the system (5) for any U r in R 2 . It is much interesting to consider the spatio-temporal patterns of bifurcating periodic solutions. For this purpose, we give the concepts of some spatiotemporal symmetric periodic solutions. Assume that the state (u 1 (t), v 1 (t), u 2 (t), v 2 (t)) can possess two different types of symmetry: spatial and temporal. The oscillators (u 1 (t), v 1 (t)) and (u 2 (t), v 2 (t)) are in-phase if the state taking the form for all times t. On the other hand, oscillator (u 1 (t), v 1 (t)), is half a period out of phase with (anti-synchronous) oscillator (u 2 (t), v 2 (t)) means the state taking the form Now, we explore the possible (spatial) symmetry of the system (5). Consider the action of where S 1 is the temporal. Let T = 2π/ω 1 or T = 2π/ω 2 , and denote P T the Banach space of all continuous T -periodic function x(t). Denoting SP T the subspace of P T consisting of all T -periodic solution of system (5) with τ = τ kj (k = 1, 2), then for each subgroup Σ ⊂ Z 2 × S 1 , is a subspace.
Theorem 2. The trivial solution of system (5) undergoes a Hopf bifurcation at giving rise to one branch of in-phase (respectively, anti-phase) periodic solutions.
Proof. Let ω 1 satisfies Eq. (15). The corresponding eigenvectors of ∆ 1 = 0 can be chosen as The isotropic subgroup of Z 2 × S 1 is z 2 (ρ), the center space associated to eigenvalues ±iω 1 is spanned by q 1 (θ) andq 1 (θ), and the bifurcated periodic solutions are in-phase, taking the form Similarly, if ω 2 satisfies Eq. (18), then corresponding eigenvectors of ∆ 2 = 0 can be chosen as Z 2 × S 1 has another isotropic subgroup z 2 (ρ, π), the center space associated to eigenvalues ±iω 2 is spanned by q 2 (θ),q 2 (θ) which implies that the bifurcated periodic solutions are anti-phase, i.e., taking the form where T is a period.

Direction and stability of the Hopf bifurcation
In the previous section, we have obtained some conditions to ensure that the system (5) undergoes a single Hopf bifurcation at the origin (x 10 , z 10 , x 20 , z 20 ) when τ = τ j passes through certain critical values. In this section, we shall study the direction, stability, and the period of the bifurcating periodic solutions. The method we used is based on the normal form method and the center manifold theory introduced by Hassard et al. [13]. We first focus on the case ∆ 1 = 0, because the other case can be dealt with analogously. We re-scale the time by t → t/τ , to normalize the delay so that system (11) can be written as Letting τ = τ 1 + a (a ∈ R), then a = 0 is Hopf bifurcation value of (21), Eq. (21) can be rewritten as: By the Riesz representation theorem, there exists a function η(θ, µ) (0 θ 1), whose elements are of bounded variation such that We choose η(θ, a) = (τ 1 + a)Aδ(θ) where δ is defined by  f (a, ϕ), θ = 0.
In the following, we follow the ideas in Hassard et al. [13] and by using the same notations as there to compute the coordinates describing the center manifold C 0 at a = 0. Let u t be the solution of (22) when a = 0. Define z(t) = q * , u t , W (t, θ) = u t (θ) − 2 Re{z(t)q(θ)}.
On the center manifold C 0 we have γ and γ are local coordinates for center manifold C 0 in the direction of q * and q * . Note that W is real if u t is real. We only consider real solutions. For solution u t ∈ C 0 of (15), since a = 0, we have We rewrite this equation as with g(γ,γ) =q * (0)f W (γ,γ, 0) + 2 Re γ(t)q(0) = g 20 γ 2 2 + g 11 γγ + g 02γ It follows from (23) and (24) that Comparing of coefficients we have: Moreover E 1 and E 2 satisfies the following equations, respectively: And Hence we have the following theorem by the result of Hassard et al. [13].

Conclusions
For a coupled Oregonator model with delay, an important issue is how delays change the stability of Oregonator model states, steady or oscillatory, causing further oscillations or significantly altering existing ones and hence inducing delay-controlled periodic behavior. In this paper, experimental and numerical investigations on the effect of electrical feedback in the oscillating Belousov-Zhabotinsky reaction are studied. By analyzing the associated characteristic equation and means of space decomposition, we subtly discuss the distribution of zeros of the characteristic equation, and then derive some sufficient conditions ensuring that all the characteristic roots have negative real parts. By regarding the eigenvalues of the connection matrix of the system as bifurcation parameters, we discuss Hopf bifurcation of the equilibria. Meanwhile, with the help of center manifold reduction and normal form theory, we study Hopf bifurcation of the equilibria, and obtain the detailed information about the bifurcation direction and stability of various bifurcated periodic solutions. Finally, numerical simulations have demonstrated the correctness of the theoretical results. From a chemical viewpoint, both means that time delay could cause a stable equilibrium to become unstable and cause the properties in a coupled Oregonator model to fluctuate: if τ < τ j , the density of various elements reach an equilibrium. If τ increases and crosses the value τ j , then this equilibrium becomes unstable: the density of various elements oscillates around the unstable equilibrium. www.mii.lt/NA