Abstract
The local dynamics, chaos, and bifurcations of a discrete Brusselator system are investigated. It is shown that a discrete Brusselator system has an interior fixed point if . Then, by linear stability theory, local dynamical characteristics are explored at interior fixed point . Furthermore, for the discrete Brusselator system, the existence of periodic points is investigated. The existence of bifurcations around an interior fixed point is also investigated and proved that the discrete Brusselator model undergoes hopf and flip bifurcations if and , respectively. The next feedback control method is utilized to stabilize the chaos that exists in the discrete Brusselator system. Finally, obtained results are verified numerically.
1. Introduction
1.1. Mathematical Formulation of a Continuous-Time Brusselator System
The Brusselator is a theoretical system for a specific form of (cubic) autocatalytic chemical reaction found in a variety of chemical and biological systems. At least one product of an autocatalytic process is also a reactant. Glycolysis, clock reactions (product and reactant concentrations change on a regular basis), ozone formation from atomic oxygen (via triple oxygen atom collisions and enzymatic reactions), and the more esoteric but widely quoted BZ-reaction generate autocatalysis from a mixture of potassium bromate, malonic acid, and manganese sulphate in a solution of heated sulfuric acid. In chemistry and biology, the full Brusselator system is a reaction-diffusion system. Reaction-diffusion models show how the concentration of one or more substances distributed in space (e.g., in a cellular space) changes as a result of local chemical reactions in which the substances are transformed into one another as well as diffusion of the substances spread out over the surface. The reaction components of the Brusselator system consist of a pair of intermediate state variables and with time varying concentrations and reacting with product chemicals , , and whose constraints , , and are kept constants. Therefore, reaction models are established by straightforward application of mass action to the following Brusselator system reactions:
Moreover, the full Brusselator system is obtained by adding diffusion terms and with a parameter proportional to the diffusion ratio of the two species and . Therefore, the full continuous-time Brusselator system representing systems of differential equations takes the form as follows:
Furthermore, without loss of generality, one considered the rate constants to 1, and moreover continuous-time Brusselator system (2) takes the following form under the assumption of negligible diffusion, i.e., , and [1]:
Finally, if , then continuous-time Brusselator system becomes the following form (see page 230 in [2]):with .
1.2. Review of Literature and Statement of the Problem
In recent years, many papers have appeared on investigating the dynamical characteristics and numerical solution of continuous-time chemical models. For example, the dynamical characteristics of the following chemical system were studied by Zafar et al. [3]:where is the substrate concentration and is the intermediate complex. The parameters , , and are all dimensionless. Edeki et al. [4] used a hybrid approach to investigate the numerical solution of the following biochemical system:where represent the substrate concentration at a given time and represent dimensionless parameters. Brown and Davidson [5] have studied the global bifurcation of a Brusselator system. Guo et al. [6] have investigated the hopf bifurcation of a general Brusselator model with diffusion. Li and Wang [7] have explored the hopf bifurcation and diffusion-driven instability of a Brusselator system. Li and Zhang [8] have studied stochastic bifurcation and stability of a continuous Brusselator system with multiplicative white noise. More precisely, Li and Zhang [8] have transformed the original continuous Brusselator system to an Itô averaging system by polar coordinates transformation and the stochastic averaging method. Furthermore, the local and global stabilities are analyzed by the singular boundary theory and largest Lyapunov exponent, and phenomenological bifurcation is also studies by examining the associated Fokker–Planck equation. Finally, numerical simulations are given to show the effectiveness of the obtained results. Zuo and Wei [9] have studied bifurcations of a diffusive Brusselator system with delayed feedback control. Zhao and Ma [10] have studied global and local bifurcations of a general Brusselator model. Ji and Shen [11] have studied the turing instability of a Brusselator in the reaction-diffusion network. Luo and Guo [12] have explored period-1 evolutions to chaos in a Brusselator. Ma [13] has studied the stochastic hopf bifurcation of a Brusselator system. Fu et al. [14] have investigated bifurcation in a diffusive Brusselator model. On the other hand, due to their efficient computing results and rich dynamical behavior, discrete models governed by difference equations are more appropriate than continuous models [15]. For instance, Kang and Pesin [16] explored the local dynamics of a discrete Brusselator model. Khan [17] recently investigated bifurcations of a discrete chemical model:which is the discrete version of the following continuous glycolytic oscillator model:
by Euler’s forward formula, where are positive constants, and , respectively, represent fructose-6-phosphate and adenosine diphosphate. Naik et al. [18] have explored the bifurcations of the following discrete chemical model:where and are the substrate concentrations at time , and dimensionless parameters are denoted by . Motivated by the aforementioned studies, the purpose of this paper is to explore the chaos and bifurcations of the following discrete Brusselator system:which is the discrete form of continuous-time Brusselator system (4) by Euler’s forward formula.
1.3. Main Contribution
The purpose of this paper is to investigate local dynamics, chaos, and bifurcations of a discrete Brusselator system (10). More precisely, our main contribution in this paper includes(1)Local dynamics at a fixed point of the discrete Brusselator system (10)(2)Existence of periodic point of discrete Brusselator system (10)(3)Existence of possible bifurcations around (4)Controlling the chaos(5)Verification of theoretical results numerically
1.4. Paper Layout
We study fixed points with the linearized form of the discrete Brusselator system (10) in Section 2. Topological classifications at are explored in Section 3. In Section 4, periodic points of the Brusselator system are investigated. The bifurcation of a Brusselator system (10) around a fixed point is explored in Section 5. Section 6 is about the controlling of chaos. The main findings are numerically validated in Section 7. The closing remarks are given in Section 8.
2. Fixed Point with Linearized Form
In this section, we study the existence of equilibria with the linearized form of the discrete Brusselator system (10). We first summarize the result regarding the existence of a fixed point of the discrete Brusselator system (10) in the following Lemma.
Lemma 1. For all, Brusselator system (10) has an interior fixed point P.
Proof. If is an equilibrium of the discrete Brusselator system (10), thenFrom (11), the computation yieldsUsing equation of system (12) into equation, one gets . Now, if , then from equation of system (12), one gets . Therefore, for all , is interior fixed point of (10).
Now, one has the map in order to write the linearized form of (10) at :whereFinally, variational matrix at of the discrete Brusselator system (10) under the map (13) is as follows:
3. Topological Classifications at
This section investigates the topological classifications of system (10) at . at is
In addition, the corresponding characteristic equation iswhere
Finally, roots of (17) arewhere
Now, topological classifications of the Brusselator system about are given based on the sign of .
Lemma 2. If, then the following topological classifications hold aroundof the Brusselator system (10):(i) is stable focus if(ii) is unstable focus if(iii) is nonhyperbolic if
Lemma 3. If , then the following topological classifications hold aroundof the Brusselator system(10):(i) is stable node if(ii) is unstable node if(iii) is nonhyperbolic if or
4. Periodic Points of the Brusselator System
Theorem 1. of the Brusselator system(10) is a periodic point with prime period-1.
Proof. From (10), definewhere and are represented in (14). Now, the computation yieldsFrom (29), we can deduce that of the Brusselator system (10) is a periodic point with prime period-1.
Theorem 2. of Brusselator system (10) is a periodic point of period-.
Proof. From (28), one has the required statement
5. Bifurcation
In the present section, we explore bifurcation analysis of the Brusselator system (10) around a fixed point .
5.1. Hopf Bifurcation at
If (23) holds, then from (19), one gets . Hence, from this, one can conclude that at , the Brusselator system (10) may undergoes hopf bifurcation if respective parameters located in the set as follows:
However, the following theorem guarantees the fact that for the Bruesslator system (10), there must exist a hopf bifurcation at .
Theorem 3. At, the Bruesslator system (10) undergoes the hopf bifurcation if.
Proof. It is noted that if is considered as a bifurcation parameter, then the Bruesslator system (10) becomeswhere . Further, for -dependence the Bruesslator system (32), complex eigenvalues are as follows:whereFrom (33) and (34), one hasFrom (35), the computation yields and . Now, for the occurrence of hopf bifurcation, it is required that which is equivalent to . However, if (23) holds, and thus . Therefore, its only requires that , i.e., . Now, the following transformations are utilized in order to shift the fixed point into :In view of (32) and (36), one getsIf , normal form of (37) is explored. By Taylor series expansion about , (37) givesNow, the following invertible transformation is easily constructed, which converts the linear part of (38) into canonical form:From (38) and (39), one getswhereFrom (41), one hasNow, the nondegeneracy condition for hopf bifurcation is given by [19–27]whereUsing values from (42) in (44), one getsFinally, utilizing (45) in (43) and after manipulation, one getsAs , which shows that around , Bruesslator system (10) undergoes hopf bifurcation. Also, supercritical (resp., subcritical) hopf bifurcation occurs if (resp. ).
5.2. Flip Bifurcation around
If (26) holds, then from (19), one gets but , which further implies that Brusselator system (10) may undergoe flip bifurcation if respective parameters are located in the set:
However, the following result guarantees that around , the Bruesslator system (10) undergo the flip bifurcation.
Theorem 4. At , the Bruesslator system (10) undergoes flip bifurcation if .
Proof. If considered as a bifurcation parameter, then Bruesslator system (10) takes the form that is already presented in (32). Furthermore, Bruesslator system (10) becomesby using (36). Now, by utilizing the transformation,equation (48) further takes the form aswhereNow, the center manifold isAfter calculation, one getsSo, map (50) restricted to iswhereIn order to show that the Brusselator system undergoes flip bifurcation, the following discriminant must be nonzero [19, 20]:From (56), it is noted that if as . Therefore, the Bruesslator system (10) undergoes flip bifurcation. Further, if , respectively, , then period-2 points bifurcating from are stable (respectively, unstable).
6. Chaos Control
Several methods can be used for obtaining chaos control in discrete-time models. A few of these methods are the pole-placement technique, the hybrid control method, and the state feedback method [28, 29]. We give a feedback control method to stabilize chaotic orbits at of the discrete Bruesslator system (10), with the following controlled form:where with denote feedback gains. For controlled system (57), is
Moreover, auxiliary equation of is
If are roots of (59), thenand
The lines of marginal stability are calculated by the solution and , and these conditions confirm that . If , then from (61), one has
If , then from (60) and (61), one has
Finally, if , then from (60) and (61), one gets
Hence, , and in -plane define a triangular region that gives .
7. Numerical Simulations
The following three cases are to be considered for the completeness of this section.
Case 1. If one chooses , then from (23), one gets , which is the value of the bifurcation parameter. From Lemma 2, interior fixed point of the discrete Brusselator model (10) is stable (respectively unstable) focus if (respectively, ). So, if , then Figure 1(a) shows that of the Brusselator model (10) is stable focus, and moreover Figures 1(b)–1(h) also show the same qualitative behavior for indicated bifurcation value . On the other hand, if , then Figure 2(a) shows that of the Brusselator system (10) changes behavior, and as a consequence, stable curves appear. Now, if , then from (46), one gets , which shows that model (10) undergoes supercritical hopf bifurcation. , Figures 2(b)–2(h) also show that model (10) also undergoes supercritical hopf bifurcation for indicated bifurcation value . The hopf bifurcation diagrams are drawn in Figure 3. Moreover, the maximum Lyapunov exponent corresponding to Figure 3 is drawn in Figure 4. Finally, 3D bifurcation diagram is plotted in Figure 5.
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Case 2. Hereafter, obtained results in subsection 5.2 have been verified numerically if . If one choose , then from (26), one gets , which is the value of the bifurcation parameter. From Lemma 3, is a stable node if . So, if , then Figure 6(a) shows that of the Brusselator system (10) is a stable node and moreover, Figures 6(b)–6(d) also show the same qualitative behavior for the indicated bifurcation value . On the other hand, if , then of the Brusselator system (10) changes behavior, and as a consequence, flip bifurcation occurs, i.e., if , then from (56), one gets and . Since , period-2 points bifurcate from . So, the flip bifurcation diagrams are drawn in Figure 7. Moreover, the maximum Lyapunov exponent corresponding to Figure 7 is drawn in Figure 8. Finally, the 3D bifurcation diagram is plotted in Figure 9.
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Case 3. Now, numerical simulations are provided to verify obtained results in Section 6. For instance, if , then from (62)–(64), one getsandHence, the lines that are presented in (65), (66), and (67) determine triangular region that gives (see Figure 10).
Remark 1. It is noted here that our simulation revels that no flip bifurcation occurs if .
8. Conclusion
This work includes the study of topological classifications around a fixed point, periodic points, chaos, and bifurcations of a discrete Buesslator system (10). We proved that for all parametric values , the Bruesslator system (10) has an interior fixed point . Furthermore, we studied the local stability with different topological classifications around a fixed points and proved that it is a stable focus if where , unstable focus if , nonhyperbolic if ; stable node if , unstable node if , and nonhyperbolic if or . Next for the Bruesslator system (10), we also studied the existence of periodic points by existing theory. The existence of possible bifurcations around of the Bruesslator system (10) is explored. We also investigated that around both hopf and flip bifurcations if and , respectively. Further, the state feedback control method is utilized in order to stabilize the chaos that exists in the discrete Bruesslator system. Finally, the obtained results are numerically verified.
Data Availability
All the data utilized in this article have been included, and the sources from where they were adopted were cited accordingly.
Conflicts of Interest
The authors declare that they have no conflicts of interest regarding the publication of this paper.