The effect of delayed feedback on the dynamics of an autocatalysis reaction–diffusion system

Abstract. This paper deals with an arbitrary-order autocatalysis model with delayed feedback subject to Neumann boundary conditions. We perform a detailed analysis about the effect of the delayed feedback on the stability of the positive equilibrium of the system. By analyzing the distribution of eigenvalues, the existence of Hopf bifurcation is obtained. Then we derive an algorithm for determining the direction and stability of the bifurcation by computing the normal form on the center manifold. Moreover, some numerical simulations are given to illustrate the analytical results. Our studies show that the delayed feedback not only breaks the stability of the positive equilibrium of the system and results in the occurrence of Hopf bifurcation, but also breaks the stability of the spatial inhomogeneous periodic solutions. In addition, the delayed feedback also makes the unstable equilibrium become stable under certain conditions.


Introduction
Autocatalysis is the process whereby a chemical is involved in its own production.In recent years, the diffusive autocatalysis reaction models, which attracts much attention, have been extensively used in the studies of Turing instability or Turing pattern.For example, see [8,21,29] for the Brusselator model, see [4,20,23] for the Sel'kov model, see [14,15,27] for the Lengyel-Epstein model, see [17,26] for the Schnakenberg model.
When the reaction rates are the same and the reactor is assumed to be closed, we obtain an arbitrary-order autocatalysis model with Neumann boundary conditions, which X. Wei, J. Wei takes the following form: u(x, 0) = u 0 (x) 0, v(x, 0) = v 0 (x) 0, x ∈ ∂Ω. ( Here u and v describe the dimensionless concentrations of the reactant and autocatalyst, respectively, and Ω is an open bounded domain in R N (N 1) with a smooth boundary ∂Ω, ν is the outward unit normal vector on ∂Ω.Moreover, a denotes the initial concentration of the reaction precursor, p is the order of the reaction with respect to autocatalytic species, and d 1 , d 2 are the diffusion coefficients of reactant and autocatalyst, respectively.The parameters a, p, d 1 and d 2 are assumed to be positive constants.The derivation of the model and more details can be found in [6,18].In [11], Guo et al. proved the existence of Hopf bifurcation and steady state bifurcation of system (1) by taking a as a parameter.In addition, the authors also derived the conditions for the occurrence of Turing instability.
In [10], Guo et al. supplemented and improved the results in [11] and further established the Turing instability region determined by diffusion coefficients.In addition, the authors discussed the effect of diffusion coefficients on the existence of Hopf bifurcation.It is well known that time delay is universal in ecological and chemical systems, and its influence on the dynamics of systems is crucial and instrumental.In [19], Ott et al. first proposed delay to control system by utilizing the input signals adjusted to the temporal states of the system, and then delayed feedback and its modifications are widely applied to control chaos and to stabilize unstable oscillations.Motivated by the idea of Ott and Grebogi, many investigators have studied the effect of time delay in ecological and chemical models (see [1,2,7,9,13,16,24,28,30]). Although many researches have been devoted to the experiments about the suppression of the delayed feedback on the chemical turbulent, there is few analysis of the effect of delayed feedback on the dynamics of chemical reaction models theoretically.Based on the previous work, we consider the following autocatalysis model with delayed feedback: where the last term in the first equation g(u(x, t − τ ) − u(x, t)) denotes the local delayed feedback control, g denotes the feedback intensity, and τ the time delay.
In this paper, we perform a detailed analysis of delayed feedback on the dynamics of system (2).The rest of the paper is organized as follows.In Section 2, by analyzing the distribution of the roots of the associated characteristic equation, we study the stability of the positive equilibrium E * .As time delay varies, we study its effect on the stability of the positive equilibrium E * and prove the existence of Hopf bifurcation.In Section 3, by applying the normal form theory and center manifold reduction for partial differential systems, the explicit formulas, which determine the stability and the direction of the bifurcating periodic solutions, are given.In Section 4, some simulations are given to illustrate our theoretical results.The simulations show that the delayed feedback not only breaks the stability of E * and results in the occurrence of the Hopf bifurcation, but also effects the stability of the spatial inhomogeneous periodic solutions.

Stability analysis
It is easy to see that system (2) has a unique positive equilibrium E * (u * , v * ) = (a 1−p , a).In this paper, we consider system (2) on the domain Ω = (0, lπ), l ∈ R + .Now, we devote our attention to the study of the effect of delayed feedback on the stability of ) and U (t) = (u 1 (t), u 2 (t)) T , then system (2) can be written as an abstract differential equation in the phase space C = C([−τ, 0], X) as follows: where .
Linearizing system (3) at the positive equilibrium E * , we have where L : C → X is given by and From Wu [25] the corresponding characteristic equation of ( 4) can be written as where y ∈ dom(∆) \ {0}, dom(∆) ⊂ X, and It is well known that the following eigenvalue problem into the characteristic equation ( 5), we have Therefore, the characteristic equation ( 5) is equivalent to where When τ = 0, equation (6) becomes where From Theorem 2.1 in [11] and Theorem 3.1 in [10] we obtain the following results on system (2) without delayed feedback (τ = 0).
(ii) Assume that p > 1 and a > (p − 1) 2 , then the positive equilibrium E * of system (2) in the absence of diffusion is locally asymptotically stable, and unstable whenever the diffusion is present.Now, we investigate the effect of time delay on the stability of E * when p > 1.From Lemma (1) we can see that E * (u * , v * ) of ( 2) is locally asymptotically stable under the assumption: when τ = 0.In fact, if a < (p − 1) 1/p , then (H1) is false.In this case, we can still use the following techniques to analyze the effect of time delay on the stability of E * .According to Corollary 2.4 in Ruan and Wei [22], we have that the stability of the equilibrium E * of system (2) changes only if the characteristic equation ( 6) has a root appears on or crosses the imaginary axis.It can be verified that 0 is not a root of the characteristic equation ( 6) for any n ∈ N 0 .Therefore, we only need to check whether the characteristic equation ( 6) has purely imaginary roots.Let ±iω (ω > 0) be the roots of ( 6), then we have Separating the real and imaginary parts, it yields to It follows that ω should satisfy with where Q n is defined as in (7).Denote z = ω 2 , then we can rewrite equation ( 9) as follows: Nonlinear Anal.Model.Control, 23(5):749-770 From the first equation of (10) we know that the graph of or Note that Q n > 0 for any n ∈ N 0 when (H1) holds.We know that the sign of is determined by the sign of the following variable: when (H1) holds.Similar to the above analysis, we obtain that the sufficient condition for B 2 n − D 2 n > 0 can be given by or Hence, we have that 12) and ( 14), or ( 12) and ( 15), or ( 13) and ( 14), or ( 13) and ( 15) hold.Recall that z = ω 2 , we deduce that characteristic equation ( 6) does not have purely imaginary roots if equation (11) does not have positive roots.We know that z + n and z − n are negative when either of the following assumptions holds: Hence, we can summarize the above discission as follows: Lemma 2. Suppose that (H1) holds.If (12) and (14), or (12) and (15), or (13) and (14), or (13) and (15), or (H3) holds, then all the roots of the characteristic equation (6) have negative real parts. https://www.mii.vu.lt/NA From the above analysis we know that if equation ( 11) has a positive root, then the characteristic equation ( 6) has a pair of simply imaginary roots.It implies that there exists This means that equation ( 11) has a positive root z n satisfying Then equation ( 6) has a pair of simply imaginary roots ±iω + n as long as τ = τ + n,j for 0 n N 0 , where for j ∈ N 0 , 0 n N 0 .Summarizing the above discussion, we have the following result.
Lemma 3. Suppose that (H1) holds.If g > a p /(2(p − 1)), then there exists a minimal integer N 0 ∈ N 0 such that for 0 n N 0 , the characteristic equation ( 6) has a pair of imaginary roots ±iω + n as long as τ = τ + n,j , and all the other roots of equation (6), except ±iω + , have negative roots.
Assume that g > a p /(2(p − 1)).For n > N 0 , we further assume that then equation ( 11) has two positive roots, which can be given by From the first formula of (10) we obtain that By equation ( 8), we have for j ∈ N 0 , N 0 < n N .From the above analysis and according to Lemma 3, we have the following results.
and (6) only has a pair of simply imaginary roots just for n = 0. (ii) Assume that T 1 < 0 is satisfied.Then there must exist an integer n ∈ N such that (6) has a pair of simply imaginary roots.
holds, then the characteristic equation ( 6) has two pairs of simply imaginary roots.
Proof.Differentiating the two sides of equation ( 6) with respect to τ , it follows that Thus, dλ dτ From ( 6) we obtain that According to ( 8) and ( 9), then we have , then the conclusion is claimed. Denote It is obvious from ( 17) that {τ ± n,j }| ∞ j=0 is increasing on j for the fixed n ∈ D 2 , so, for the fixed n, τ ± n,0 = min j∈N0 {τ ± n,j }.Recall that E * is locally asymptotically stable when τ = 0 under the assumption (H1), then necessarily τ + n,0 τ − n,0 (n ∈ N 0 ) when (H1) holds.Hence, for all n ∈ D 1 ∪ D 2 , we can define the smallest critical value such that the stability of E * (u * , v * ) will change, which is given by From Lemmas (2)-( 4) and Proposition (1) we obtain the following conclusion.
Remark 2. Assume that a < (p−1) 1/p and other conditions in (H1) still hold.According to the discussion in [3], we have that the conclusion in Theorem 1 is still true just as (H4) or (H5) holds.In this case, the stability switches may occur.

Stability and direction of Hopf bifurcation
In the previous section, we have verified that system (2) undergoes Hopf bifurcation at the positive equilibrium E * (u * , v * ) = (a 1−p , a) as τ = τ + n,j (τ − n,j ).In this section, we will investigate the direction and stability of period solutions bifurcating from the positive equilibrium E * by applying the center manifold theorem and normal form theory for the partial differential equations presented in Faria [5] and Wu [25].
Setting u 1 (x, t) = u(x, tτ )−a 1−p , u 2 (x, t) = v(x, tτ )−a, U (t) = (u 1 (x, t), u 2 (x, t)), then system (2) can be written in the following form: where Setting τ = τ * + α, then system (2) can be rewritten in an abstract form in the space where for ϕ ∈ C. From the previous section it is easy to see that system (18) undergoes Hopf bifurcation at the equilibrium (0, 0) when α = 0. We also know that ±iω * τ * are the simply purely imaginary characteristic values of linearised system of ( 18) at (0, 0) as α = 0 and all the other characteristic values have negative real parts as α = 0.
From previous analysis we have that the characteristic values of τ D∆ on X are where the inner product is given by So, on M n , ( 19) is equivalent to the following equation on R 2 : Using the Riesz representation theorem, we know that there exists a bounded variation for ϕ ∈ C([−1, 0], R 2 ).In fact, we can choose Let A denote the infinitesimal generator of the semigroup defined by ( 20) with α = 0 and n = n 0 , and A * denote the formal adjoint of A under the bilinear form ).According to the above discussion, we see that ±iω * τ * are simply imaginary eigenvalues of A and A * .In the following, we let n = n 0 .Let be the eigenvectors of A and A * corresponding to the eigenvalue iω * τ * .By calculation, we get q(0) = (1, q 1 ) T , q * (0) = (1, q 2 ), where .
Then g 21 can be determined.Consequently, we can compute the following quantities: According to the general theory in Hassard [12], we know that µ 2 determines the direction of the Hopf bifurcation: the direction of the Hopf bifurcation is forward(backward) when µ 2 > 0 (< 0), that is, the bifurcating periodic solutions exist for τ > τ * (τ < τ * ); β 2 determines the stability of the bifurcating periodic solutions: the bifurcating periodic solutions are orbitally asymptotically stable(unstable) when β 2 < 0 (> 0); and T 2 determines the period of the bifurcating periodic solutions: the period of the bifurcating periodic solutions increases(decreases) when T 2 > 0 (< 0).

Numerical simulations
In this section, we present some numerical simulations to illustrate the above theoretical results.

Conclusion
We have studied the effect of delayed feedback on the dynamics of an arbitrary-order autocatalysis model.By selecting the appropriate ratio of diffusion coefficients, the delayed feedback can be used to control the stability of the positive equilibrium.When time delay increases and crosses through some critical values, we found that the positive equilibrium E * becomes unstable and induces the occurrence of spatial homogeneous periodic solutions.By computing the normal form on the center manifold, we analyzed the direction and stability of the periodic solutions.In addition, feedback intensity affected the stability of the spatial inhomogeneous periodic solutions, that is, as the feedback intensity reaches and exceeds some critical values, the spatial inhomogeneous periodic solutions become unstable.Our study also shows that delayed feedback control can stabilize the unstable equilibrium E * under certain conditions.