Abstract

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of . It is proved that, for all parametric values, is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium have been investigated by employing the method of linearization. Existence of prime period and periodic points of the model under consideration are also investigated. It is proved that, about the fixed point , the discrete-time glycolytic oscillator model undergoes no bifurcation, except Neimark–Sacker bifurcation. A further hybrid control strategy is applied to control Neimark–Sacker bifurcation in the discrete-time model. Finally, theoretical results are verified numerically.

1. Introduction

In glycolysis, glucose decomposes in the presence of various enzymes including ten steps in which five are termed the preparatory phase or phosphorylation, while the remaining steps are called the pay-off phase. Phosphofructokinase is one of the enzymes which is responsible for the occurrence of glycolytic oscillation [1–4]. This step is considered the control unit of glycolysis due to the presence of enzyme phosphofructokinase. A biochemical reaction that occurs in metabolic systems has the following sequence of steps [5]: where the quantities involved in the above biochemical reaction are depicted in Table 1.

Generally, it is assumed that the enzyme PFK consists of two steps as given below. Among these, in one step, adenosine diphosphate simulates the allosteric regulatory enzyme and yields a more active form. Therefore, the product formed during the reaction mediated by phosphofructokinase increases the rate of reaction. The systematic description of the kinetics is as follows: and we have where and denote FGP and ADP, respectively, and are positive constants. Using the nonstandard finite difference scheme, the discrete analogue of the glycolytic oscillator model, which is depicted in (3), is

Here, our purpose is to explore the local dynamics, N-S bifurcation, and hybrid control in a glycolytic oscillator model (4) in the interior of . Specifically, our key findings are as follows:(1)About the unique fixed point , we explored the dynamical properties along with different topological classifications.(2)We explored the existence of bifurcation about .(3)Prime period and periodic points of the glycolytic oscillator model (4) are investigated.(4)To prove, the glycolytic oscillator model (4) undergoes an N-S bifurcation when parameters vary in a small neighborhood of , and no other bifurcation occurs except the N-S bifurcation.(5)The N-S bifurcation is controlled by the hybrid control strategy.(6)Theoretical results are verified numerically.

The flow pattern of the remaining of this paper is as follows: Section 2 is about the existence of the unique equilibrium point and corresponding linearized form of the glycolytic oscillator model (4). In Section 3, we study the local dynamical properties with topological classification about of the glycolytic oscillator model (4), whereas existence of prime period and periodic points of model (4) is explored in Section 4. The comprehensive N-S bifurcation analysis about is given in Section 5. In Section 6, simulations are presented to verify theoretical results. In Section 7, a hybrid control strategy is applied to control N-S bifurcation. Concluding remarks are given in Section 8.

2. Existence of the Unique Equilibrium Point and Corresponding Linearized Form of the Glycolytic Oscillator Model (4)

The existence of the fixed point in the interior of and corresponding linearized form of the glycolytic oscillator model (4) are explored in this section. More specifically, the existence result about the equilibrium point can be summarized as follows.

Lemma 1. parametric values , the discrete-time glycolytic oscillator model (4) has a unique equilibrium point .
Now, the linearized form of the glycolytic oscillator model (4) about under the map becomes where

3. Local Dynamical Properties of Model (4) about

The local dynamics about of the glycolytic oscillator model (4) is explored in this section. about is

The auxiliary equation of about is where

And eigenvalues of about are where

Hereafter in the following, we will explore the necessary and sufficient condition under which of the glycolytic oscillator model (4) is a sink, a repeller, a saddle, and nonhyperbolic.

Lemma 2. For of the glycolytic oscillator model (4), the following holds:(i) is a sink iff (ii) is a repeller iff (iii) is a saddle iff (iv) is nonhyperbolic iff

Proof. Follows from Theorem 1.1.1 of [6].
Hereafter in the next lemma, we will explore local dynamics along with topological classification about if of the glycolytic oscillator model (4).

Lemma 3. If then for , the following holds:(i) is a locally asymptotically stable focus if (ii) is an unstable focus if (iii) is nonhyperbolic if Hereafter, we will establish that of (4) is the periodic point of prime period . Moreover, equilibrium is the periodic point having period .

4. Existence of Prime Period and Periodic Points of (4)

Theorem 1. of the model (4) is a periodic point of prime period .

Proof. From (4), we define where After manipulation, from (19), one gets From (21), one can conclude that of (4) is a periodic point of prime period .

Theorem 2. of the glycolytic oscillator model (4) is a periodic point of period .

Proof. From (19), one has From (22), of (4) is a periodic point of period .

5. Bifurcation Analysis about of the Glycolytic Oscillator Model (4)

Recall that if (18) holds, then , which implies that the glycolytic oscillator model (4) undergoes N-S bifurcation if parameters go through the curve:

Hereafter, we will give detailed N-S bifurcation about when parameters go through the curve, which is depicted in (23), by using bifurcation theory [7, 8]. For given parameters , let

After simplifying (24), one gets

From (25), one gets . So, by implicit function theorem, one can obtain such that and hence choose β as a bifurcation parameter. Now, consider β in a small neighborhood of , , , where and hence the glycolytic oscillator model (4) becomes where be its unique positive fixed point. Moreover, about the fixed point of (26) is

The auxiliary equation of about is where

The zeroes of (28) are where

Additionally, it is required that , which is equivalent to , and hence by manipulation, it is true. If , then of the glycolytic oscillator model (26) transforms into . So where . Hereafter, the normal form of (33) is studied if . By Taylor series expansion about , from (33), one gets where

Hereafter, one contracts the following invertible matrix T that puts the linear part of (34) into the canonical form: where and is depicted in (31). Hence, (34) then implies where in which by

From (39), one gets

In order to ensure the N-S bifurcation for (38), it is required that the following discriminatory quantity should be nonzero ([7–14]): where

After manipulation, one gets

From the analysis, one can conclude the following.

Theorem 3. If , then the glycolytic oscillator model (4) undergoes N-S bifurcation about the fixed point: as parameters go through the curve, which is depicted in (23). Additionally, attracting (resp., repelling) closed curve bifurcates appear from if (resp., ).
According to bifurcation theory, the bifurcation is supercritical (resp., subcritical) N-S bifurcation if (resp., ). In the subsequent section, simulations guarantee that the glycolytic oscillator model (4) undergoes the supercritical N-S bifurcation if go through (23).
Remark. The discrete-time glycolytic oscillator model (4) cannot undergo any bifurcation, except N-S bifurcation, about the positive fixed point .

6. Numerical Simulations

Theoretical results are verified numerically in this section. For instance, if , then from (18), one gets . Theoretically, equilibrium of the glycolytic oscillator model (4) is a locally asymptotically stable focus if , , condition (16) holds. To see this, if , then the unique positive equilibrium point of (4) is a locally asymptotically stable focus (see Figure 1(a)). Similarly for other values of β, if , then of (4) is a locally asymptotically stable focus (see Figures 1(b)–1(l)). But if , then positive equilibrium becomes an unstable focus, , condition (17) holds. Precisely, if , then positive equilibrium loses its stability and meanwhile an attracting closed curve appears. The appearance of this closed curve indicates that the glycolytic oscillator model (4) undergoes a supercritical N-S bifurcation if vary in a neighborhood of . To see this, if , then eigenvalues of about are

After some manipulation, from (43), one gets

In view of (46) and (47), from (43), one gets . Hence, if , then model (4) undergoes a supercritical N-S bifurcation and so a stable curve appears, which is depicted in Figure 2(a). Also for different choices of the bifurcation parameter, if , then one gets (see Table 2), and corresponding closed curves are depicted in Figures 2(b)–2(l). Moreover, the bifurcation diagrams along with the maximum Lyapunov exponent are plotted and drawn in Figure 3. Finally, bifurcation diagrams are plotted and drawn in Figure 4.

7. Control of N-S Bifurcation about of the Glycolytic Oscillator Model (4)

The hybrid control strategy is applied to control N-S bifurcation about of the glycolytic oscillator model (4) motivated from the work of [15–18]. The control system corresponding to the glycolytic oscillator model (4) is

about of the controlled system, which is depicted in (48), is

The rest of this section deals with the study of N-S bifurcation of the controlled system (48) at . In this case, (49) becomes whose auxiliary equation is where . In view of Lemma 1 of [15], one gets

Let , then from (52), one gets which hold true. Also, in terms of the bifurcation parameter β becomes whose roots are where

If , then from (32), one gets

From (57), the nondegenerate condition holds, and hence, (48) undergoes N-S bifurcation at . In terms of above chosen parametric values: , and , the controlled system (48) becomes whose fixed point is . Now, we transform into using the following transformation: where (58) becomes

Since the linear part of (60) is just as with eigenvalues,

Now, (61) takes the following form: where by

In view of (63), from (45), one gets

In view of (61) and (65), from (43), one gets . Finally, the controlled results of N-S bifurcation are plotted and drawn in Figure 5. For more interesting results for solving nonlinear equations using the multipoint iterative method, we refer the reader to [19] and references cited therein.

8. Conclusion

In the proposed work, we have explored the local dynamics, N-S bifurcation, and hybrid control in a discrete-time glycolytic oscillator model, which is depicted in (4), in the interior of . It is proved that , the glycolytic oscillator model (4) has a unique equilibrium point . By the method of linearization, we have explored the local dynamics along with topological classification about of the discrete-time glycolytic oscillator model (4), and conclusions are presented in Table 3. We have also investigated the prime period and periodic points for the model (4). Furthermore, we have explored the existence of bifurcation about and proved that the glycolytic oscillator model (4) undergoes no bifurcation, except N-S bifurcation, when go through the curve . We have performed the N-S bifurcation about by bifurcation theory. Moreover, the hybrid control strategy is applied to control the N-S bifurcation in the glycolytic oscillator model (4). Finally, theoretical results are verified numerically.

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.

Acknowledgments

A. Q. Khan and E. Abdullah research was partially supported by the Higher Education Commission of Pakistan, while the research of Tarek F. Ibrahim was funded by a Deanship of Scientific Research in King Khalid University under grant number GRP-324-40.