# Neimark–Sacker Bifurcation of a Two-Dimensional Discrete-Time Chemical Model

Authors: A. Q. Khan
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

In this paper, the local dynamics and Neimark–Sacker bifurcation of a two-dimensional glycolytic oscillator model in the interior ofℝ+2 are explored. It is investigated that for all αandβ, the model has a unique equilibrium point: Pxy+α/β+α2,α. Further about Pxy+α/β+α2,α, local dynamics and the existence of bifurcation are explored. It is investigated about Pxy+α/β+α2,α that the glycolytic oscillator model undergoes no bifurcation except the Neimark–Sacker bifurcation. Some simulations are given to verify the obtained results. Finally, bifurcation diagrams and the corresponding maximum Lyapunov exponent are presented for the glycolytic oscillator model.


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## Body

## 1. Introduction

Many chemical models are governed by difference as well as differential equations. As compared to the continuous model, discrete models designated by difference equations are better explored in recent years. Mathematical models of chemistry, physics, physiology, psychology, ecology, engineering, and social sciences have given birth to major areas of research during the last few decades. For instance, Edeki et al. [1] have explored the numerical solution of the following nonlinear biochemical model by using the hybrid technique:(1)ηdxdt=y−αx−xy,dydt=−y+α−β,where xandy are the substrate concentrations at time t and η,α,andβ are the dimensionless parameters. Zafar et al. [2] have investigated the equilibria and convergence analysis of the following nonlinear biochemical reaction networks:(2)dxdt=x+β−α+xy,dydt=1σx−βy−xy,where x is the concentration of the substrate, y is the intermediate complex, and the parameters σ,α,andβ are the dimensionless parameters. Inspired from the aforementioned studies, the goal of this paper is to investigate the bifurcation analysis of a glycolytic oscillator model:(3)xn+1=xn+hα−βxn−xnyn2,yn+1=yn+hβxn+xnyn2−yn,which is the discrete analogue of the following continuous-time model, by Euler’s forward formula:(4)dxdt=α−βx−xy2,dydt=βx+xy2−y,where x and y, respectively, denote fructose-6-phosphate and adenosine diphosphate and αandβ are the positive constants. For more detailed background and mathematical modelling of the glycolytic oscillator model (4), the reader is referred to [3–7]. More specifically, the main finding in this article is as follows:(1)
Study of the dynamics aboutPxy+α/β+α2,α of model (3)(2)
Existence of possible bifurcation aboutPxy+α/β+α2,α(3)
To investigate, for the model under consideration, that no other bifurcation exists except the Neimark–Sacker bifurcation(4)
Verification of theoretical results numericallyThe rest of the article is organized as follows: Section2 is about the existence of a positive fixed point in ℝ+2 and the corresponding linearized form of the glycolytic oscillator model (3). Local dynamics about Pxy+α/β+α2,α of (3) is investigated in Section 3. Existence of bifurcation about Pxy+α/β+α2,α is investigated in Section 4, whereas detailed Neimark–Sacker bifurcation analysis is given in Section 5. Simulations are given in Section 6. The conclusion of the paper is given in Section 7.


## 2. Existence of Positive Equilibrium Point and Linearized Form of Model (3)

The existence of a positive fixed point inℝ+2 and the corresponding linearized form of the model are explored in this section. Specifically, the existence result about the positive fixed point can be stated as the following lemma.Lemma 1.
For allαandβ, Pxy+α/β+α2,α is the unique positive equilibrium point of model (3).
Hereafter, aboutPxyx,y, the linearized form of model (3) is constructed. For the corresponding linearized form of (3), one has the following map:(5)Ψ1,Ψ2↦xn+1,yn+1,where(6)Ψ1=xn+hα−βxn−xnyn2,Ψ2=yn+hβxn+xnyn2−yn.
JPxyx,y about Pxyx,y under (5) is(7)JPxyx,y=1−hβ+y2−2xyhhβ+y21−h+2hxy.


## 3. Local Dynamics aboutPxy+α/β+α2,α of Model (3)

The local dynamics of the glycolytic oscillator model (3) is explored by utilizing the linearization method. JPxy+α/β+α2,α about Pxy+α/β+α2,α is(8)JPxy+α/β+α2,α=1−hβ+α2−2α2hα2+βhβ+α2α2+β1−h+2hα2α2+β.The characteristic equation ofJPxy+α/β+α2,α about Pxy+α/β+α2,α is(9)λ2−pλ+q=0,where(10)p=2−hβ+α2+1+2hα2α2+β,q=1−h1−hα2+β+2hα2α2+β.And, the eigenvalues ofJPxy+α/β+α2,α about Pxy+α/β+α2,α are(11)λ1,2=p±Δ2,where(12)Δ=p2−4q=2−hβ+α2+1+2hα2α2+β2−41−h1−hβ+α2+2hα2α2+β.Now, in the following two lemmas, local dynamics aboutPxy+α/β+α2,α for the model under consideration is studied.Lemma 2.
IfΔ=2−hβ+α2+1+2hα2/α2+β2−41−h1−hα2+β+2hα2/α2+β<0, then for Pxy+α/β+α2,α, the following holds:(i)
Pxy+α/β+α2,α is a stable focus if(13)h<1+β−α2β+α22;(ii)
Pxy+α/β+α2,α is an unstable focus if(14)h>1+β−α2β+α22;(iii)
Pxy+α/β+α2,α is a nonhyperbolic if(15)h=1+β−α2β+α22.Lemma 3.
IfΔ=2−hβ+α2+1+2hα2/α2+β2−41−h1−hα2+β+2hα2/α2+β/≥0, then for Pxy+α/β+α2,α, the following holds:(i)
Pxy+α/β+α2,α is a stable node if(16)h>max21+β+α2α2+β−4α2−4α2−21+β+α2β+α22−16β+α232α2+β2,21+β+α2α2+β−4α2+4α2−21+β+α2β+α22−16β+α232β+α22;(ii)
Pxy+α/β+α2,α is an unstable node if(17)h<min21+β+α2α2+β−4α2−4α2−21+β+α2β+α22−16β+α232α2+β2,21+β+α2α2+β−4α2+4α2−21+β+α2β+α22−16β+α232α2+β2;(iii)
Pxy+α/β+α2,α is a nonhyperbolic if(18)h=21+β+α2α2+β−4α2−4α2−21+β+α2β+α22−16β+α232β+α22,or(19)h=21+β+α2α2+β−4α2+4α2−21+β+α2α2+β2−16α2+β32α2+β2.
Now, the existence of bifurcation aboutPxy+α/β+α2,α is explored based on the above theoretical results.


## 4. Existence of Bifurcations aboutPxy+α/β+α2,α

(i)
From Lemma2, one can obtain that if (15) holds, then(20)λ1,2=2−hβ+α2+1+2hα2/α2+β±ι41−h1−hα2+β+2hα2/α2+β−2−hβ+α2+1+2hα2/α2+β22=1.Equation (20) implies that model (3) undergoes Neimark–Sacker bifurcation if α,β,h are in the following set:(21)HBPxy+α/β+α2,α=α,β,h:h=1+β−α2β+α22.(ii)
There does not exist period-doubling bifurcation as eigenvalues ofJPxy+α/β+α2,α about Pxy+α/β+α2,α are neither −1 nor 1 if (18) or (19) holds.


## 5. Neimark–Sacker Bifurcation aboutPxy+α/β+α2,α

Hereafter, by using bifurcation theory [8, 9], the detailed Neimark–Sacker bifurcation about Pxy+α/β+α2,α is explored if h goes through HBPxy+α/β+α2,α. Now, if h varies in a small nbhd of h∗, that is, h=h∗+ε with ε≪1, then model (3) becomes(22)xn+1=xn+h⋆+εα−βxn−xnyn2,yn+1=yn+h⋆+εβxn+xnyn2−yn.JPxy+α/β+α2,α about Pxy+α/β+α2,α of (22) is(23)JPxy+α/β+α2,α=1−h∗+εβ+α2−2h∗+εα2β+α2h∗+εβ+α2β+α21−h∗+ε+2h∗+εα2α2+β.The characteristic equation ofJPxy+α/β+α2,α about Pxy+α/β+α2,α is(24)λ2−pελ+qε=0,where(25)pε=2−h∗+εβ+α2+1+2h∗+εα2α2+β,qε=1−h∗+ε1−h∗+εβ+α2+2h∗+εα2α2+β.From (24), one gets(26)λ1,2=pε±ι4qε−p2ε2=2−h∗+εβ+α2+1+2h∗+εα2/α2+β2±ι2Δ∗,where(27)Δ∗=41−h∗+ε1−h∗+εα2+β+2h∗+εα2α2+β−2−h∗+εβ+α2+1+2h∗+εα2α2+β2,(28)dλ1,2dεε=0=β−α2+α2+β22β+α2>0.Additionally, it is required thatλ1,2ℓ≠1,ℓ=1,…,4, which corresponds to p0≠1,2,0,−1 and so it is true by calculation. Now, if un=xn−x∗,vn=yn−y∗, then Pxy+α/β+α2,α of the glycolytic oscillator model (3) transforms into O0,0. So,(29)un+1=un+h∗+εα−βun+x∗−un+x∗vn+y∗2,vn+1=vn+h∗+εβun+x∗+un+x∗vn+y∗2−vn+y∗,where x∗=α/β+α2andy∗=α. Hereafter, the normal form of (29) is studied if ε=0. From (29), one gets(30)un+1=a11un+a12vn+a13unvn+a14vn2+a15unvn2,vn+1=a21un+a22vn+a23unvn+a24vn2+a25unvn2,where(31)a11=1−h∗β+y∗2,a12=−2h∗x∗y∗,a13=−2h∗y∗,a14=−h∗x∗,a15=−h∗,a21=h∗β+y∗2,a22=1+h∗2x∗y∗−1,a23=2h∗y∗,a24=h∗x∗,a25=h∗.Now, the matrixT is obtained that puts the linear part of (30) into a conoidal form:(32)T≔a120η−a11−ζ,where(33)η=2−h∗β+α2+1+2h∗α2/α2+β2,ζ=12Δ∗,and Δ∗ is depicted in (27). Hence, (30) then implies that(34)Xn+1=ηXn−ζYn+P¯,Yn+1=ζXn+ηYn+Q¯,where(35)P¯Xn,Yn=l11Xn2+l12XnYn+l13Yn2+l14Xn3+l15XnYn2+l16YnXn2,Q¯Xn,Yn=l21Xn2+l22XnYn+l23Yn2+l24Xn3+l25XnYn2+l26YnXn2,(36)l11=η−a11a12a12a13+a14η−a11,l12=−ζa12a12a13+2a14η−a11,l13=a14a12ζ2,l14=a15η−a112,l15=a15ζ2,l16=−2a15ζη−a11,l21=η−a11ζa12η−a11a12a13−a24+a14η−a11−a122a23,l22=η−a11a12η−a112a12a24−2a14−a12a13−a12a23,l23=ζa12a14η−a11−a12a24,l24=η−a112ζa15η−a11−a12a25,l25=ζa15η−a11−a12a25,l26=η−a112a12a25−2a15η−a112,by(37)unvn≔a120η−a11−ζXnYn.Also,(38)P¯XnXnO=2l11,P¯XnYnO=l12,P¯YnYnO=2l13,P¯XnXnXnO=6l14,P¯XnXnYnO=2l16,P¯XnYnYnO=2l15,P¯YnYnYnO=0,Q¯XnXnO=2l21,Q¯XnYnO=l22,Q¯YnYnO=2l23,Q¯XnXnXnO=6l24,Q¯XnXnYnO=2l26,Q¯XnYnYnO=2l25,Q¯YnYnYnO=0.It is noted that the following relation should be nonzero in order for (34) to undergo the Neimark–Sacker bifurcation (see [8–16]):(39)Ω=−Real1−2λ¯λ¯21−λξ11ξ20−12ξ112−ξ022+Realλ¯ξ21,where(40)ξ02=18P¯XnXn−P¯YnYn+2Q¯XnYn+ιQ¯XnXn−Q¯YnYn+2P¯XnYnO,ξ11=14P¯XnXn+P¯YnYn+ιQ¯XnXn+Q¯YnYnO,ξ20=18P¯XnXn−P¯YnYn+2Q¯XnYn+ιQ¯XnXn−Q¯YnYn−2P¯XnYnO,ξ21=116P¯XnXnXn+P¯XnYnYn+Q¯XnXnYn+Q¯YnYnYn+ιQ¯XnXnXn+Q¯XnYnYn−P¯XnXnYn−PYnYnYnO.After manipulation, one gets(41)ξ02=14l11−l13+l22+ιl21+l12−l23,ξ11=12l11+l13+ιl21+l23,ξ20=14l11−l13+l22+ιl21−l23−l12,ξ21=183l14+l15+l26+ι3l24+l25−l16.From the analysis and Neimark–Sacker bifurcation conditions discussed in [8, 9], one has the following results.Theorem 1.
If (39) holds, then the glycolytic oscillator model (3) undergoes Neimark–Sacker bifurcation about Pxy+α/β+α2,α as α,β,h pass through HBPxy+α/β+α2,α. Furthermore, attracting (respectively, repelling) the closed curve bifurcates from Pxy+α/β+α2,α if Ω<0respectively,Ω>0.Remark 1.
It is noted here that bifurcation is supercritical (respectively, subcritical) Neimark–Sacker bifurcation ifΩ<0respectively,Ω>0. In the next section, simulations guarantee that (3) undergoes supercritical Neimark–Sacker bifurcation if α,β,h vary in an nbhd of Pxy+α/β+α2,α.


## 6. Numerical Simulations

Some simulations will be presented for the correctness of the obtained results in this section. For instance, ifα=0.7andβ=0.6, then from (15), one gets h=1.0925847992593216. Theoretically, equilibrium Pxy+α/β+α2,α of (3) is a stable focus if h<1.0925847992593216. To see this, if h=0.6<1.0925847992593216, then Figure 1(a) implies that Pxy+α/β+α2,α is a stable focus. Similarly, for other values of h, if h<1.0925847992593216, then Pxy+α/β+α2,α of (3) is a stable focus (see Figures 1(b)–1(l)). But, if h goes through 1.0925847992593216, then Pxy+α/β+α2,α becomes unstable, and as a consequence, an attracting closed curve appears. This closed curve indicates that model (3) undergoes a supercritical Neimark–Sacker bifurcation if α,β,h go through the curve, which is depicted in (21). To see this, if h=1.1>1.0925847992593216, then eigenvalues of Pxy+0.6422018348623854,0.7 about 0.6422018348623854,0.7 are(42)λ1,2=0.3449954128440368±0.9433286759155826ι,and the nongenerate for the existence of Neimark–Sacker bifurcation holds, i.e., β−α2+α2+β2/2α2+β=0.5954587155963303>0. Moreover, after some manipulation from (41), one gets(43)ξ02=0.2735375582937717+0.3287376065788703ι,ξ11=0.004623532110091866−0.0021810537860846857ι,ξ20=0.2735375582937717−0.03107799351733123ι,ξ21=−0.3481094018033289−0.025816775834865524ι.Figure 1
Trajectories of model (3). (a) h = 0.6 with (0.47; 0.42), (b) h = 0.67 with (0.8; 0.8), (c) h = 0.7 with (0.38; 0.38), (d) h = 0.79 with (0.38; 0.38), (e) h = 0.7965 with (0.1; 0.2), (f) h = 0.8 with (0.61; 0.62), (g) h = 0.89 with (0.61; 0.62), (h) h = 0.894 with (0.61; 0.62), (i) h = 0.9 with (0.81; 0.62), (j) h = 0.97 with (0.81; 0.62), (k) h = 1.0 with (0.81; 0.62), and (l) h = 1.08 with (0.81; 0.62).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Using (42) and (43) in (39), one gets Ω=−0.32927462274555946<0. So, if h=1.1, model (3) undergoes supercritical Neimark–Sacker bifurcation, and hence a stable curve appears (see Figure 2(a)). In particular, the occurrence of closed curves indicates that model (3) undergoes a supercritical Neimark–Sacker bifurcation, i.e., FGP and ADP coexist with a long time. Also, for the rest of the values of parameters, the values of Ω<0 (see Table 1) and corresponding closed curves are presented in Figures 2(b)–2(l). Moreover, bifurcation diagrams along with the maximum Lyapunov exponent are plotted in Figure 3. Also, 3D bifurcation diagrams are plotted and drawn in Figure 4. Finally, the topological classification about Pxy+α/β+α2,α of model (3) is presented in Figure 5.Figure 2
Supercritical Neimark–Sacker bifurcation of model (3). (a) h = 1.1 with (0.81; 0.62), (b) h = 1.11 with (0.81; 0.62), (c) h = 1.113 with (0.081; 0.062), (d) h = 1.11345 with (0.01; 0.02), (e) h = 1.114 with (1:4; 0.2), (f) h = 1.115 with (0.68; 0.7), (g) h = 1.1156 with (0.68; 0.67), (h) h = 1:22 with (0.08; 0.07), (i) h = 1.235 with (0.1; 0.8), (j) h = 1.245 with (0.7; 0.8), (k) h = 1.247 with (0.7; 0.9), and (l) h = 1.2479 with (0.6; 0.7).
(a)(b)(c)(d)(e)(f)(g)(h)(i)(j)(k)(l)Table 1
Values ofΩ for h>1.0925847992593216.
Value ofh if h>1.0925847992593216Numerical value ofΩ1.1−0.32927462274555946<01.11−0.33856187233019613<01.113−0.34138558178129375<01.11345−0.3418106432332544<01.114−0.3423306971220448<01.115−0.343277757130833<01.1156−0.3438469278876272<01.22−0.4541948828452207<01.235−0.472021931523424<01.245−0.48420168645269523<01.247−0.4866663794705075<01.2479−0.48777863891987355<0Figure 3
(a-b) Bifurcation diagram of the model ifh∈0.5,1.5 with the initial condition 0.47,0.42. (c) Maximum Lyapunov exponent corresponding to (a-b).
(a)(b)(c)Figure 4
3D bifurcation diagrams of model (3).
(a)(b)(c)(d)(e)Figure 5
Topological classification aboutPxy+α/β+α2,α of model (3).


## 7. Conclusion

The dynamics and Neimark–Sacker bifurcation of the glycolytic oscillator model inℝ+2 have been investigated. It has been proved that for all αandβ, model (3) has a positive equilibrium point: Pxy+α/β+α2,α. The local dynamics about Pxy+α/β+α2,α has been studied by the method of linearization. It is proved that Pxy+α/β+α2,α is a stable focus if h<1+β−α2/β+α22, unstable focus if h>1+β−α2/β+α22, and nonhyperbolic if h=1+β−α2/β+α22. Further, it is investigated that if h=1+β−α2/β+α22, then (20) holds which implies that (3) undergoes Neimark–Sacker bifurcation when α,β,h are located in the set: HBPxy+α/β+α2,α=h,α,β:h=1+β−α2/β+α22. Then, Neimark–Sacker bifurcation about Pxy+α/β+α2,α is studied by using bifurcation theory. It is also proved that under certain parametric conditions, Pxy+α/β+α2,α is a stable node, unstable node, and nonhyperbolic. It is also explored that for the model under consideration, no other bifurcation occurs except the Neimark–Sacker bifurcation. Finally, theoretical results are verified numerically.



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