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

Abstract

We study the local and global bifurcation of nonnegative nonconstant solutions of a discrete general Brusselator model. We generalize the linear in the standard Brusselator model to the nonlinear . Assume that is a strictly increasing function, and . Taking 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 . Moreover, via the Rabinowitz bifurcation theorem, we obtain the global structure of the set of nonconstant solutions under the condition that is nonincreasing in . In this process, we also make a priori estimation for the nonnegative nonconstant solutions of the problem.

1. Introduction

In 1968, Prigogine and Lefever [1] introduced first the Brusselator model for a chemical reaction-diffusion of self-catalysis as follows:where is a smooth and bounded domain, denotes the outward unit normal vector on , and represent the concentration of two intermediary reactants having the diffusion rates with , and are the fixed concentrations. This chemical reaction plays an important role due to its similarities with neuronal and biological networks. Therefore, (1) has been extensively investigated in the last decades from both analytical and numerical point of view (see [2–12]). 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. Therefore, 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]). Therefore, the discrete Brusselator model has been studied by several authors, and they got some results ([16–18] and the references therein). In particular, Din [16] applied forward Euler’s method to one-dimensional model (1) as follows:where represents the step size for Euler’s method. The local dynamical behaviors are obtained for (3).

Note that [16–18] only studied the dynamical behaviors of the discrete-time Brusselator model. The 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.

Therefore, we will consider the discrete space, more general form of (2) with :where , , is an integer, are fixed parameters and , and is a bifurcation parameter. Clearly, ; then, (4) is seen to be equivalent towhere can be regarded as a variable coefficient. It is well known that the linear terms and 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. The cells or units are also absolute individuals in microscopic sense, and each isolated cell exchanges materials by diffusion with its neighbors. Thus, 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 in (6)–(8) is different from in this paper. Our discretization is consistent with Kang’s, and we chose the step size to be 1. When , (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 finite-difference 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).

We will study the local and global bifurcation of nonnegative nonconstant solutions of (4) under the following assumptions: (H1) is a strictly increasing function (H2)  (H3) is nonincreasing in

Remark 1. If , then (4) is the discrete version of (2) with . Obviously, discrete Brusselator model (4) is a second-order difference boundary value problem.
The 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).

2. 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. Then, any nonnegative nonconstant solution of (4) satisfies

Proof. Let be the minimum point of . We haveThen,Then, by (H1), and soLet be the maximum point of . Similarly, we can get thatThen,Combining this with (13), from (H3), we showLet . Then, it follows from (4) thatNow, let be the maximum point of . Observe thatThen, from (H1), it is easy to see . Combining this with (17), we know that, for any ,Then,If is the minimum point of , thenand soConsequently, the proof is completed.

Lemma 2. (see [23]). Assume is an integer. Then, the discrete second-order linear Neumann eigenvalue problemhas real and simple eigenvalues, which can be ordered as follows:

Moreover, for , the eigenfunction corresponding to the eigenvalue has exactly simple generalized zeros.

For any fixed , it is well known thatand the corresponding eigenfunctions are

Lemma 3. (see [18], Theorem 2.5). Let be a constant. Then, for ,

Lemma 4. (see [18], Theorem 2.7). If is an indefinite sum of , thenLetwhere and .

3. Local Bifurcation

By the second part, is the unique constant solution of (4).

Define the mapping :

For the fixed , is a solution of (4) if and only if is a zero-point of . Note that since is the constant solution of (4).

Let

We also have to Taylor expand at the point .

The purpose of the rest of this section is to solve and prove that is the bifurcation point of .

First of all, we substitute (32) and (33) into (4) and let the higher-order term of be equal to 0. Then, 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 :

Next, we substitute (32) and (33) into (4) and let the higher-order term of be equal to 0; then, (4) becomes the following system:where

In order to solve from (37), let us consider the following adjoint system of the homogeneous system related to (37):

It is not difficult to verify that (39) has a solution :

By virtue of the solvability condition for (37), it is obvious that

In fact,

We know that

From Lemmas 3 and 4, for any and , we obtain

Then , and so will reduce to

Therefore, a particular solution of (37) can be obtained as follows:where

Since , we have to solve . We substitute (32) and (33) into (4) and let the higher-order term of be equal to 0; then, a problem similar to (37) is obtained:where

Clearly, (39) is also the adjoint system of the homogeneous system related to (48); then,

According to values of , and , we have

From Lemmas 3 and 4, for any and , we know that

Thus,

From the above analysis, we obtain the main result of this section.

Theorem 1. Assume that (H1) and (H2) hold. Then, for any positive integer and , is a bifurcation point of . Moreover, there is a nontrivial solution of (4) if is small enough, where , and are continuous with respect to :

The set of zero-points of constitutes two curves in a neighborhood of bifurcation point .

Let be the closure of the nonconstant solution set of and be a connected component of and . In a small neighborhood of bifurcation point , the curve is determined by the eigenfunction , where has exactly simple generalized zeros.

4. Global Bifurcation

Theorem 2. Let (H1), (H2), and (H3) hold. If , then projection of continuum is unbounded on the -axis.

Proof. (4) can be written as follows:whereLet . Then, (55) is equivalent to the following problem:where and are higher-order terms of andIn this way, we convert the constant solution of (4) to the trivial solution of (57).
Let and be the inverse of operators and with Neumann boundary conditions, respectively, where and . Set :It can be verified that (57) is equivalent toin . For any fixed , and are linear compact operators in and . By the Rabinowitz global bifurcation theorem [24], we need to verify(i)1 is an eigenvalue of , and its algebraic multiplicity is 1(ii)The index of changes when crosses Now, we will prove (i). Suppose . Leti.e.,Thus,whereBy computation, if and only if ; taking leads toThen, . This implies that 1 is the eigenvalue of and . The algebraic multiplicity of eigenvalue 1 is the dimension of the generalized null space ; therefore, .
Let be the transposed matrix of :and . Suppose . Then,From the definition of , (67) can also be written asThat is to say,whereSimilarly, if and only if ; taking leads toThen, . According to , we obtainThis suggests that , and so (i) is proved.
Now, we will prove (ii). From (i), for any and belongs to a small neighborhood of , and is a bijection. Fix ; then, is a solution of (60), and is isolated. From Leray–Schauder fixed point theory, we can getwhere is a sufficiently small ball centered at , is the sum of the algebraic multiplicity of the eigenvalues of , and .
We are going to verify that, for is small enough,If is an eigenvalue of and is the corresponding eigenfunction, theni.e.,By virtue of and , we can getThen, the characteristic equation isIf , can be solved from (78):Therefore, by calculating the corresponding eigenvalues of (78), we can obtain that when passes through , the number of eigenvalues of which is greater than 1 is the same, and their algebraic multiplicity are equal. By plugging (79) into (78), we haveThen,and so (80) has two different roots . Thus, two things will happen:(a)If , then .(b)If , then .When scenario (a) occurs, passes through and . From (78), . Therefore, the matrix has exactly one more eigenvalue that is than does, and its algebraic multiplicity is 1. Then, (74) holds. That is to say, the index jumps as goes through .
When scenario (b) occurs, passes through and . From (78), . Similarly, the index jumps as goes through . Therefore, (ii) is true regardless of (a) or (b).
Thus, by the index jump principle and [24], Theorem 1.3, it follows that there exists a connected component of nontrivial solutions of (60), and comes from the bifurcation point . We know that is also the connected component of the nonconstant solution of (4) from . and are both in . By the Rabinowitz global bifurcation theorem, the connected component joins to either or in , where .
We first prove that the latter situation will not happen. According to Theorem 1, the solution on the connected component sent from is related to , and has exactly simple generalized zeros. In the same way, the solution on the connected component sent from is related to , and has simple generalized zeros. If the connected component sent joining to , the solution is related to both and , which is impossible. On the contrary, Lemma 1 shows that if , then the solutions and of (4) are both bounded. So, the connected component will not join to . Therefore, the connected component can only join to either or , where . But, in any case, the projection of continuum is unbounded on the -axis.

Data Availability

Data sharing is not applicable to this article as no datasets were generated.

Conflicts of Interest

All authors declare no conflicts of interest.

Authors’ Contributions

The authors claim that the research was realized in collaboration with the same responsibility. All authors read and approved the last version of the manuscript.

Acknowledgments

This work was supported by the National Natural Science Foundation of China (no. 11671322).