Bifurcation Behaviors of Steady-State Solution to a Discrete General Brusselator Model

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear u in the standard Brusselator model to the nonlinear f(u). Assume that f ∈ C([0,∞), [0,∞)) is a strictly increasing function, and f′(f 1(a)) ∈ (0,∞). Taking b as the bifurcation parameter, we obtain that the solution set of the problem constitutes a constant solution curve and a nonconstant solution curve in a small neighborhood of the bifurcation point


Introduction
In 1968, Prigogine and Lefever [1] introduced first the Brusselator model for a chemical reaction-diffusion of selfcatalysis as follows: where Ω ⊂ R N (N ≥ 1) is a smooth and bounded domain, n denotes the outward unit normal vector on zΩ, u and v represent the concentration of two intermediary reactants having the diffusion rates d 1 , d 2 ∈ (0, ∞) with d 2 > d 1 , and a, b > 0 are the fixed concentrations. is chemical reaction plays an important role due to its similarities with neuronal and biological networks. erefore, (1) has been extensively investigated in the last decades from both analytical and numerical point of view (see [2][3][4][5][6][7][8][9][10][11][12]). Most of them are interested in finding spatially nonconstant solutions of the equilibrium problem From the definition of Strogatz [13], chaos sensitivity depends on initial conditions, which shows that nearby trajectories diverge exponentially. Continuous systems in a 2-dimensional phase space cannot experience such divergence; hence, chaotic behaviors can only be observed in deterministic continuous systems with a phase space of dimension 3, at least. On the contrary, in a discrete map, it is well known that chaos occurs also in one dimension. erefore, discrete chaotic systems exhibit chaos whatever their dimension is.
It is worth to note that discrete models governed by difference equations are more appropriate than the continuous one due to their efficient computational results and rich dynamical behavior (see [14,15]). erefore, the discrete Brusselator model has been studied by several authors, and they got some results ( [16][17][18] and the references therein). In particular, Din [16] applied forward Euler's method to one-dimensional model (1) as follows: where 0 < h < 1 represents the step size for Euler's method. e local dynamical behaviors are obtained for (3). Note that [16][17][18] only studied the dynamical behaviors of the discrete-time Brusselator model. e reason is that the partial difference equation is very difficult for us. Indeed, the discrete-space Brusselator model is also worth studying due to the discontinuity of the space. erefore, we will consider the discrete space, more general form of (2) with N � 1: x ∈ T, where f(u)/u can be regarded as a variable coefficient. It is well known that the linear terms (b + 1)u and bu in (2) cannot withstand any small perturbation. In fact, (5) has an important application value in biology and chemistry. Xu et al. [19] said that model (1) includes a basic assumption: the cells always live in a continuous patch environment. However, this may not be the case in reality, and the motion of individuals of given cells is random and isotropic, i.e., without any preferred direction, the cells are also absolute individuals. e cells or units are also absolute individuals in microscopic sense, and each isolated cell exchanges materials by diffusion with its neighbors. us, it is reasonable to consider a 1D or 2D spatially discrete reaction-diffusion system in order to explain the chemical system.
Kang [20] discussed the dynamics of the local map of a discrete version of the Brusselator model. To discretize system (1), he employed the following discretizations.
For the derivative in time, he used For the space derivative, he used It is important to note that Δ in (6)-(8) is different from Δ in this paper. Our discretization is consistent with Kang's, and we chose the step size to be 1. When f(u) � u, (5) is the steady-state form of the problem studied in [19,20].
On the contrary, the Brusselator system has been investigated from the numerical point of view (see [21] and references therein). Most modern texts on numerical analysis give an introduction to numerical solutions of partial differential equations using the finite-difference approach. Twizell et al. [22] had given a second-order finitedifference scheme for the Brusselator reaction-diffusion system. It is well known that (2) is an important mathematical dynamics model in biology and chemistry. In some ways, (5) is even more practical than (2).

2
Discrete Dynamics in Nature and Society We will study the local and global bifurcation of nonnegative nonconstant solutions of (4) under the following assumptions: (4) is the discrete version of (2) with N � 1. Obviously, discrete Brusselator model (4) is a second-order difference boundary value problem. e rest of the paper is organized as follows: in Section 2, we give a priori estimate and some preliminary results. Section 3 is devoted to studying the local bifurcation of nonnegative nonconstant solutions of (4) under conditions (H1) and (H2). Finally, in Section 4, we add condition (H3) to obtain the global bifurcation of nonnegative nonconstant solutions of (4).

Preliminary Results
At first, let us look for the constant solution of (4). To get it, it suffices to look for the constant solution of the following problem: By (H1), problem (4) has a unique constant solution We can easily obtain the following a priori estimate of the nonnegative nonconstant solutions of (4).

Lemma 1. Let (H1), (H2), and (H3) hold. en, any non-
Proof. Let x 0 ∈ T be the minimum point of u. We have en, Let bf u Combining this with (13), from (H3), we show Now, let x 2 ∈ T be the maximum point of w. Observe that en, from (H1), it is easy to see u( Combining this with (17), we know that, for any x ∈ T, Discrete Dynamics in Nature and Society en, and so Consequently, the proof is completed.
□ Lemma 2 (see [23]). Assume T ≥ 2 is an integer. en, the discrete second-order linear Neumann eigenvalue problem has T real and simple eigenvalues, which can be ordered as follows: Moreover, for j ∈ 1, . . . , T − 1 { }, the eigenfunction φ j corresponding to the eigenvalue μ j has exactly j − 1 simple generalized zeros.
For any fixed T ≥ 2, it is well known that and the corresponding eigenfunctions are Lemma 3 (see [18], eorem 2.5). Let a be a constant. en, for ΔC(i) � 0, Lemma 4 (see [18], eorem 2.7). If z n is an indefinite sum of y n , then where

Local Bifurcation
By the second part, w ≔ (f − 1 (a), ab/[f − 1 (a)] 2 ) is the unique constant solution of (4). Define the mapping P: (0, ∞) × X ⟶ Y: For the fixed b > 0, w � (u, v) is a solution of (4) if and only if (b, w) is a zero-point of P. Note that P(b, w) � 0 0 since w is the constant solution of (4). Let We also have to Taylor expand f at the point f − 1 (a). e purpose of the rest of this section is to solve b 0 and prove that (b 0 , w) is the bifurcation point of P(b, w) � 0 0 .

Discrete Dynamics in Nature and Society
First of all, we substitute (32) and (33) into (4) and let the higher-order term of ε be equal to 0. en, we can get the problem In (34), by using undetermined coefficient method, it follows that Moreover, it is not difficult to prove (34) has a nontrivial solution (u 1 , v 1 ): Next, we substitute (32) and (33) into (4) and let the higher-order term of ε 2 be equal to 0; then, (4) becomes the following system: where In order to solve b 1 from (37), let us consider the following adjoint system of the homogeneous system related to (37): Discrete Dynamics in Nature and Society It is not difficult to verify that (39) has a solution (y 2 , z 2 ): By virtue of the solvability condition for (37), it is obvious that In fact, We know that sin(jπ/T)(x − (1/2)) 2 sin(jπ/2T) (44) en b j 1 ≔ b 1 � 0, and so F 1 will reduce to erefore, a particular solution (u 2 , v 2 ) of (37) can be obtained as follows: Since b 1 � 0, we have to solve b 2 . We substitute (32) and (33) into (4) and let the higher-order term of ε 3 be equal to 0; then, a problem similar to (37) is obtained: Discrete Dynamics in Nature and Society Clearly, (39) is also the adjoint system of the homogeneous system related to (48); then, According to values of u 1 , u 2 , v 1 , and v 2 , we have From Lemmas 3 and 4, for any j ∈ 1, 2, . . . , T − 1 { } and j ≠ (nT/2), n ∈ 1, 2, . . . (52) From the above analysis, we obtain the main result of this section. , v(ε)) of (4) if ε is small enough, where b, u, and v are continuous with respect to ε: e set of zero-points of P constitutes two curves in a neighborhood of bifurcation point (b