# Supercritical Neimark–Sacker Bifurcation and Hybrid Control in a Discrete-Time Glycolytic Oscillator Model

Authors: A. Q. Khan; E. Abdullah; Tarek F. Ibrahim
Journal: Mathematical Problems in Engineering
 (2020)

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## Abstract

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior ofℝ+2. It is proved that, for all parametric values, Pxy+α/β+α2,α is the unique positive equilibrium point of the glycolytic oscillator model. Further local dynamical properties along with different topological classifications about the equilibrium Pxy+α/β+α2,α 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 Pxy+α/β+α2,α, 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.


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## Body

## 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]:(1)where the quantities involved in the above biochemical reaction are depicted in Table 1.Table 1
Quantities along their chemical interpretation.
Quantities
Corresponding chemical interpretation
GGP
Glucose-6-phosphate
FGP
Fructose-6-phosphate
FDP
Fructose-1,6-diphosphate
PFK
Phosphofructokinase
ATP
Adenosine triphosphate
ADP
Adenosine diphosphateGenerally, 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:(2)and we have(3)dxdt=α−βx−xy2,dydt=βx+xy2−y,where xn and yn denote FGP and ADP, respectively, and αandβ are positive constants. Using the nonstandard finite difference scheme, the discrete analogue of the glycolytic oscillator model, which is depicted in (3), is(4)xn+1=αh+xn1+hβ+hyn2,yn+1=yn+hβxn+hxnyn21+h.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 ℝ+2. Specifically, our key findings are as follows:(1)
About the unique+ve fixed point Pxy+α/β+α2,α, we explored the dynamical properties along with different topological classifications.(2)
We explored the existence of bifurcation aboutPxy+α/β+α2,α.(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 Pxy+α/β+α2,α, 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: Section2 is about the existence of the unique +ve 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 Pxy+α/β+α2,α 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 Pxy+α/β+α2,α 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+ve Equilibrium Point and Corresponding Linearized Form of the Glycolytic Oscillator Model (4)

The existence of the+ve fixed point in the interior of ℝ+2 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 αandβ, the discrete-time glycolytic oscillator model (4) has a unique +ve equilibrium point Pxy+α/β+α2,α.
Now, the linearized form of the glycolytic oscillator model (4) about Pxyx,y under the map Ψ1,Ψ2⟶xn+1,yn+1 becomes(5)Λn+1=JPxyx,yΛn,where(6)JPxyx,y=11+hβ+y2−2yhx+αh1+hβ+y22hβ+y21+h1+2hxy1+h,Ψ1=αh+xn1+hβ+hyn2,Ψ2=yn+hβxn+hxnyn21+h.


## 3. Local Dynamical Properties of Model (4) about Pxy+α/β+α2,α

The local dynamics aboutPxy+α/β+α2,α of the glycolytic oscillator model (4) is explored in this section. JPxy+α/β+α2,α about Pxy+α/β+α2,α is(7)JPxy+α/β+α2,α=11+hβ+α2−2α2hα2+β1+hβ+α2hβ+α21+hα2+β+2hα2β+α21+h.The auxiliary equation ofJPxy+α/β+α2,α about Pxy+α/β+α2,α is(8)λ2−pαβ+α2,αλ+qαβ+α2,α=0,where(9)pαβ+α2,α=1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α2,qαβ+α2,α=α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2.And eigenvalues ofJPxy+α/β+α2,α about Pxy+α/β+α2,α are(10)λ1,2=pα/β+α2,α±Δ2,where(11)Δ=pαβ+α2,α2−4qαβ+α2,α=1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α22−4α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2.Hereafter in the following, we will explore the necessary and sufficient condition under whichPxy+α/β+α2,α of the glycolytic oscillator model (4) is a sink, a repeller, a saddle, and nonhyperbolic.Lemma 2.
ForPxy+α/β+α2,α of the glycolytic oscillator model (4), the following holds:(i)
Pxy+α/β+α2,α is a sink iff(12)1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α2<1−α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2<2.(ii)
Pxy+α/β+α2,α is a repeller iff(13)α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2>1,1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α2<1−α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2.(iii)
Pxy+α/β+α2,α is a saddle iff(14)1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α22+41−α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2>0,1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α2>1−α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2.(iv)
Pxy+α/β+α2,α is nonhyperbolic iff(15)1+hβ+α2+α2+β+2hα21+hα2+β1+hα2+β1+hβ+α2=1−α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2.Proof.
Follows from Theorem 1.1.1 of [6].
Hereafter in the next lemma, we will explore local dynamics along with topological classification aboutPxy+α/β+α2,α if Δ=1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α22−4α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2<0, of the glycolytic oscillator model (4).Lemma 3.
IfΔ=1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α22−4α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2<0, then for Pxy+α/β+α2,α, the following holds:(i)
Pxy+α/β+α2,α is a locally asymptotically stable focus if(16)α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2<1.(ii)
Pxy+α/β+α2,α is an unstable focus if(17)α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2>1.(iii)
Pxy+α/β+α2,α is nonhyperbolic if(18)α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2=1.
Hereafter, we will establish thatPxy+α/β+α2,α of (4) is the periodic point of prime period −1. Moreover, equilibrium Pxy+α/β+α2,α is the periodic point having period −2,3,…,n.


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

Theorem 1.
P
x
y
+
α
/
β
+
α
2
,
α of the model (4) is a periodic point of prime period −1.Proof.
From (4), we define(19)Fx,y=fx,y,gx,y,where(20)fx,y=αh+x1+hβ+hy2,gx,y=y+hβx+hxy21+h.
After manipulation, from (19), one gets(21)FPxy+α/β+α2,α=Pxy+αβ+α2,α.
From (21), one can conclude that Pxy+α/β+α2,α of (4) is a periodic point of prime period −1.Theorem 2.
P
x
y
+
α
/
β
+
α
2
,
α of the glycolytic oscillator model (4) is a periodic point of period −2,3,…,n.Proof.
From (19), one has(22)F2x,y=αh+fx,y1+hβ+hgx,y2,gx,y+hβfx,y1+h+hfx,ygx,y21+h⟹FPxy+α/β+α2,α2=Pxy+αβ+α2,α,F3x,y=αh+f2x,y1+hβ+hg2x,y2,g2x,y2+hβf2x,y1+h+hf2x,yg2x,y21+h⟹FPxy+α/β+α2,α3=Pxy+αβ+α2,α,⋮Fix,y=αh+fi−1x,y1+hβ+hgi−1x,y2,gi−1x,y+hβfi−1x,y1+h+hfi−1x,ygi−1x,y21+h⟹FPxy+α/β+α2,αi=Pxy+αβ+α2,α.
From (22), Pxy+α/β+α2,α of (4) is a periodic point of period −2,3,…,n.


## 5. Bifurcation Analysis aboutPxy+α/β+α2,α of the Glycolytic Oscillator Model (4)

Recall that if (18) holds, then λ1,218=1, which implies that the glycolytic oscillator model (4) undergoes N-S bifurcation if parameters h,α,andβ go through the curve:(23)N−SBPxy+α/β+α2,α=h,α,β:α2+β+2hα2+2h2α2α2+β1+hα2+β1+hβ+α2=1,h,α,β>0.Hereafter, we will give detailed N-S bifurcation aboutPxy+α/β+α2,α when parameters h,α,andβ go through the curve, which is depicted in (23), by using bifurcation theory [7, 8]. For given parameters h,α,andβ, let (24)Fh,α,β=α2+β+2hα2+2h2α2α2+β−1+hα2+β1+hβ+α2.After simplifying (24), one gets(25)Fh,α,β=α2+hα4−β2−2α2β−β−β2h−α4.From (25), one gets ∂Fh,α,β/∂β<0. So, by implicit function theorem, one can obtain β=βh,α such that Fh,α,βh,α=0 and hence choose β as a bifurcation parameter. Now, consider β in a small neighborhood of β∗, i.e., β=β∗+ε, where ε≪1 and hence the glycolytic oscillator model (4) becomes(26)xn+1=αh+xn1+hβ∗+ε+hyn2,yn+1=yn+hβ∗+εxn+hxnyn21+h,where Pxy+α/β∗+ε+α2,α be its unique positive fixed point. Moreover, JPxy+α/β∗+ε+α2,α about the fixed point Pxy+α/β∗+ε+α2,α of (26) is(27)JPxy+α/β∗+ε+α2,α=11+hβ∗+ε+α2−2α2hα2+β∗+ε1+hβ∗+ε+α2hβ∗+ε+α21+hα2+β∗+ε+2hα2β∗+ε+α21+h.The auxiliary equation ofJPxy+α/β∗+ε+α2,α about Pxy+α/β∗+ε+α2,α is(28)λ2−pελ+qε=0,where(29)pαβ∗+ε+α2,α=1+hβ∗+ε+α2+α2+β∗+ε+2hα21+hα2+β∗+ε1+hα2+β∗+ε1+hβ∗+ε+α2,qαβ∗+ε+α2,α=α2+β∗+ε+2hα2+2h2α2α2+β∗+ε1+hα2+β∗+ε1+hβ∗+ε+α2.The zeroes of (28) are(30)λ1,2=pε±ι4qε−p2ε2=1+hβ∗+ε+α2+α2+β∗+ε+2hα21+hα2+β∗+ε/1+hα2+β∗+ε1+hβ∗+ε+α22±ι2Δ∗,where(31)Δ∗=4α2+β∗+ε+2hα2+2h2α2α2+β∗+ε1+hα2+β∗+ε1+hβ∗+ε+α2−1+hβ∗+ε+α2+α2+β∗+ε+2hα21+hα2+β∗+ε1+hα2+β∗+ε1+hβ∗+ε+α22,(32)λ1,2=qε1/2,dλ1,2dεε=0≠0.Additionally, it is required thatλ1,2m≠1,m=1,…,4, which is equivalent to p0≠−2,0,1,2, and hence by manipulation, it is true. If un=xn−x∗,vn=yn−y∗, then Pxy+α/β∗+ε+α2,α of the glycolytic oscillator model (26) transforms into P000,0. So(33)un+1=αh+un+x∗1+hβ∗+ε+hvn+y∗2−x∗,vn+1=vn+y∗+hun+x∗β∗+ε+vn+y∗21+h−y∗,where x∗=α/β+α2andy∗=α. Hereafter, the normal form of (33) is studied if ε=0. By Taylor series expansion about un,vn=0,0, from (33), one gets(34)un+1=a11un+a12vn+a13unvn+a14vn2+oun+vn2,vn+1=a21un+a22vn+a23unvn+a24vn2+oun+vn2,where(35)a11=11+hβ∗+y∗2,a12=−2hy∗x∗+hα1+hβ∗+y∗2,a13=−2hy∗1+hβ∗+y∗22,a14=hx∗+hα−1+hβ∗+hy∗2+4y∗2h1+hβ∗+y∗23,a21=hβ∗+y∗21+h,a22=1+2hx∗y∗1+h,a23=2hy∗1+h,a24=hx∗1+h.Hereafter, one contracts the following invertible matrixT that puts the linear part of (34) into the canonical form:(36)T≔a120η−a11−ζ,where(37)η=1+hβ+α2+α2+β+2hα21+hα2+β21+hα2+β1+hβ+α2,ζ=12Δ∗,and Δ∗ is depicted in (31). Hence, (34) then implies(38)Xn+1=ηXn−ζYn+P¯,Yn+1=ζXn+ηYn+Q¯,where(39)P¯Xn,Yn=l11Xn2+l12XnYn+l13yn2+oXn+Yn2,Q¯Xn,Yn=l21Xn2+l22XnYn+l23yn2+oXn+Yn2,in which(40)l11=η−a11a12a12a13+a14η−a11,l12=−ζa12a12a13+2a14η−a11,l13=a14a12ζ2,l21=η−a11ζa12η−a11a12a13−a24−a14a24−a122a23,l22=−η−a11a122a14η−a11+a12a13+2a24+a23a122,l23=ζa12a14ζη−a11−a12a24,by(41)unvn≔a120η−a11−ζXnYn.From (39), one gets(42)P¯XnXnP000,0=2l11,P¯XnYnP000,0=l12,P¯YnYnP000,0=2l13,Q¯XnXnP000,0=2l21,Q¯XnYnP000,0=l22,Q¯YnYnP00+0,0=2l23.In order to ensure the N-S bifurcation for (38), it is required that the following discriminatory quantity should be nonzero ([7–14]):(43)Ω=−Re1−2λ¯λ¯21−λξ11ξ20−12ξ112−ξ022+Reλ¯ξ21,where(44)ξ02=18P¯XnXn−P¯YnYn+2Q¯XnYn+ιQ¯XnXn−Q¯YnYn+2P¯XnYnP000,0,ξ11=14P¯XnXn+P¯YnYn+ιQ¯XnXn+Q¯YnYnP000,0,ξ20=18P¯XnXn−P¯YnYn+2Q¯XnYn+ιQ¯XnXn−Q¯YnYn−2P¯XnYnP000,0,ξ21=116P¯XnXnXn+P¯XnYnYn+Q¯XnXnYn+Q¯YnYnYn+ιQ¯XnXnXn+Q¯XnYnYn−P¯XnXnYn−PYnYnYnP000,0.After manipulation, one gets(45)ξ02=182l11−l12+2l22+2ιl21+l12−l23,ξ11=12l11+l13+ιl21+l23,ξ20=14l11−l13+l22+ιl21−l23−l12,ξ21=0.From the analysis, one can conclude the following.Theorem 3.
IfΩ≠0, then the glycolytic oscillator model (4) undergoes N-S bifurcation about the fixed point: Pxy+α/β+α2,α as parameters h,α,andβ go through the curve, which is depicted in (23). Additionally, attracting (resp., repelling) closed curve bifurcates appear from Pxy+α/β+α2,α if Ω<0 (resp., Ω>0).
According to bifurcation theory, the bifurcation is supercritical (resp., subcritical) N-S bifurcation ifΩ<0 (resp., Ω>0). In the subsequent section, simulations guarantee that the glycolytic oscillator model (4) undergoes the supercritical N-S bifurcation if h,α,andβ go through (23).
Remark. The discrete-time glycolytic oscillator model (4) cannot undergo any bifurcation, except N-S bifurcation, about the positive fixed point Pxy+α/β+α2,α.


## 6. Numerical Simulations

Theoretical results are verified numerically in this section. For instance, ifα=0.6,h=0.7, then from (18), one gets β=0.16105922820694768. Theoretically, equilibrium Pxy+α/β+α2,α of the glycolytic oscillator model (4) is a locally asymptotically stable focus if β>0.16105922820694768, i.e., condition (16) holds. To see this, if β=0.4>0.16105922820694768, then the unique positive equilibrium point Pxy+0.7894736842105263,0.6 of (4) is a locally asymptotically stable focus (see Figure 1(a)). Similarly for other values of β, if β>0.16105922820694768, then Pxy+α/β+α2,α of (4) is a locally asymptotically stable focus (see Figures 1(b)–1(l)). But if β<0.16105922820694768, then positive equilibrium Pxy+α/β+α2,α becomes an unstable focus, i.e., condition (17) holds. Precisely, if β<0.16105922820694768, then positive equilibrium Pxy+α/β+α2,α 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 α,β,andh vary in a neighborhood of Pxy+α/β+α2,α. To see this, if β=0.16<0.16105922820694768, then eigenvalues of JPxy+1.1538461538461537,0.6 about 1.1538461538461537,0.6 are(46)λ1,2=0.615799684186781±0.3270196350217114ι.Figure 1
Trajectories of the discrete-time glycolytic oscillator model (4) about Pxy+α/β+α2,α: (a) β = 0.4 with (0.35, 0.39); (b) β = 0.37 with (0.99, 0.80); (c) β = 0.35 with (0.56, 0.49); (d) β = 0.2965 with (0.1, 0.2); (e) β = 0.27 with (0.9, 0.8); (f) β = 0.24 with (0.66, 0.67); (g) β = 0.22 with (0.5, 0.4); (h) β = 0.2 with (0.4, 0.3); (i) β = 0.19 with (0.59, 0.39); (j) β = 0.199 with (0.9, 0.7); (k) β = 0.18 with (0.9, 0.7); (l) β = 0.17 with (0.8, 0.7).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)After some manipulation, from (43), one gets(47)ξ02=−0.007611881686547611+0.024203687145576566ι,ξ11=0.03740896019401933−0.1966375637334364ι,ξ20=0.009013748156314753−0.0593513169940678ι,ξ21=0.In view of (46) and (47), from (43), one gets Ω=−0.029495873864781315<0. Hence, if β=0.16<0.16105922820694768, 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 β<0.16105922820694768, then one gets Ω<0 (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, 3D bifurcation diagrams are plotted and drawn in Figure 4.Figure 2
Supercritical N-S bifurcation of the discrete-time glycolytic oscillator model (4) about Pxy+α/β+α2,α: (a) β = 0.16 with (0.64, 0.62); (b) β = 0.15 with (0.64, 0.62); (c) β = 0.14 with (0.69, 0.6); (d) β = 0.149 with (0.081, 0.072); (e) h = 0.1495 with (0.9, 0.8); (f) β = 0.13 with (0.98, 0.19); (g) β = 0.139 with (0.99, 0.67); (h) β = 0.12 with (0.91, 0.72); (i) β = 0.1239 with (0.59, 0.35); (j) β = 0.11 with (0.7, 0.5); (k) β = 0.1135 with (1.5, 0.5); (l) β = 0.114 with (0.02, 0.03).
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
(l)Table 2
Numerical values ofΩ if β<0.16105922820694768.
Value ofβ if q>1
Numerical value ofΩ
0.16
−
0.029495873864781315
<
0
0.15
−
0.024558009135300485
<
0
0.14
−
0.025661100529415302
<
0
0.149
−
0.02516944528172212
<
0
0.1495
−
0.02461170028243595
<
0
0.13
−
0.026828812841863014
<
0
0.139
−
0.025774885406930285
<
0
0.12
−
0.028066167307452507
<
0
0.1239
−
0.027574969477341914
<
0
0.11
−
0.029378827673349428
<
0
0.1135
−
0.028910444972707013
<
0
0.114
−
0.028844333111783357
<
0Figure 3
Bifurcation diagram and its corresponding maximum Lyapunov exponent of the model (4) about Pxy+α/β+α2,α. (a, b) Bifurcation diagram of the model if β∈0.1,0.29 and initial condition 0.74,0.62. (c) Maximum Lyapunov exponent corresponding to (a) and (b).
(a)
(b)
(c)Figure 4
3
D bifurcation diagrams of the model (4).
(a)
(b)
(c)
(d)
(e)


## 7. Control of N-S Bifurcation aboutPxy+α/β+α2,α of the Glycolytic Oscillator Model (4)

The hybrid control strategy is applied to control N-S bifurcation aboutPxy+α/β+α2,α of the glycolytic oscillator model (4) motivated from the work of [15–18]. The control system corresponding to the glycolytic oscillator model (4) is(48)xn+1=cαh+xn1+hβ+hyn2+1−cxn,yn+1=cyn+hβxn+hxnyn21+h+1−cyn.J
P
x
y
+
α
/
β
+
α
2
,
α about Pxy+α/β+α2,α of the controlled system, which is depicted in (48), is(49)JPxy+α/β+α2,α=c1+hβ+hα2+1−c−2hcα2α2+β1+hβ+y22hcα2+β1+hcα2+β+2hα21+hβ+α2+1−c.The rest of this section deals with the study of N-S bifurcation of the controlled system (48) at h=0.7,β=0.1601<0.16105922820694768,andα=0.6. In this case, (49) becomes(50)JPxy+1.15362,0.6=1−0.2669c−0.710407c0.214159c1+0.158261c,whose auxiliary equation is(51)λ2−pλ+q=0,where p=2−0.108492candq=1−1.150755c+1.152135c2. In view of Lemma 1 of [15], one gets(52)1−2−0.108492c+1−1.150755c+1.152135c2>0,1−1.150755c+1.152135c2=1,2−0.108492c2−41−1.150755c+1.152135c2<0.Letc=0.998, then from (52), one gets(53)1−2−0.108492c+1−1.150755c+1.152135c2=0.107354>0,1−1.150755c+1.152135c2=1,2−0.108492c2−41−1.150755c+1.152135c2=−0.417687<0,which hold true. Also, JPxy+1.15362,0.6=1.15362,0.6 in terms of the bifurcation parameter β becomes(54)JPxy+α/β+α2,α=0.002+0.9980.252+0.7β−0.838320.6+0.420.36+β1.252+0.7β20.36+β0.4109410.36+β0.002+0.5870590.864+β0.36+β,whose roots are(55)λ1,2=1.69341+3.55991β+0.70336β2−Δ∗∗20.766224+2.5568β+1.19β2,where(56)Δ∗∗=−1.69341−3.55991β−0.70336β22−40.224208−1.2023β−1.1886β20.766224+2.5568β+1.19β2.Ifβ=0.1601, then from (32), one gets(57)dλ1,2dεε=0=3.82461>0.From (57), the nondegenerate condition holds, and hence, (48) undergoes N-S bifurcation at β=0.1601. In terms of above chosen parametric values: β=0.1601,α=0.6,h=0.7, and c=0.998, the controlled system (48) becomes(58)xn+1=0.002xn+0.41916+0.998xn1.11207+0.7yn2,yn+1=0.002yn+0.5882350.6986xn+0.998yn+0.7xnyn2,whose fixed point is Pxy+1.15362,0.6. Now, we transform Pxy+1.15362,0.6 into P000,0 using the following transformation:(59)un=xn−1.15362,vn=yn−0.6,where (58) becomes(60)un+1=0.733634un−0.967106vn−0.450543unvn−0.154225vn2+oun+vn2,vn+1=0.213731un+1.15794vn+0.493129unvn+0.474072vn2+oun+vn2.Since the linear part of (60) is just as JPxy+α/β+α2,α=1.15362,0.6 with eigenvalues,(61)λ1,2=0.6185299315830337±0.4140823777730747ι.Now, (61) takes the following form:(62)Xn+1Yn+1=0.6185299315830337−0.41408237777307470.4140823777730747−0.6185299315830337XnYn+P¯Xn,YnQ¯Xn,Yn,where(63)P¯Xn,Yn=0.0539722Xn2+0.201764XnYn+0.0273435yn2+oXn+Yn2,Q¯Xn,Yn=−0.17732Xn−0.00184395XnYn−0.203906yn2+oXn+Yn2,by(64)unvn≔−0.96710632570659470−0.11510407546939039−0.4140823777730747XnYn.In view of (63), from (45), one gets(65)ξ02=−0.012188390118933261+0.05708718392199485ι,ξ11=0.04065784974095253−0.190612959926066ι,ξ20=0.006196180814291302−0.04379461343161802ι,ξ21=0.In view of (61) and (65), from (43), one gets Ω=−0.028844333111783357<0. 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.Figure 5
Control results of N-S bifurcation of the glycolytic oscillator model (4) about Pxy+α/β+α2,α if α=0.6,h=0.7,c=0.998, and β∈0.1,0.29.
(a)
(b)
(c)


## 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 ℝ+2. It is proved that ∀ αandβ, the glycolytic oscillator model (4) has a unique +ve equilibrium point Pxy+α/β+α2,α. By the method of linearization, we have explored the local dynamics along with topological classification about Pxy+α/β+α2,α 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 Pxy+α/β+α2,α and proved that the glycolytic oscillator model (4) undergoes no bifurcation, except N-S bifurcation, when h,α,andβ go through the curve N−SBPxy+α/β+α2,α=h,α,β:α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2=1. We have performed the N-S bifurcation about Pxy+α/β+α2,α 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.Table 3
Fixed pointPxy+α/β+α2,α along with qualitative behavior of the glycolytic oscillator model (4).
Equilibrium point
Corresponding behavior
P
x
y
+
α
/
β
+
α
2
,
α
Sink iff1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α2<1−α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2<2
Repeller iffα2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2>1 and 1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α2<1−α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2
Saddle iff1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α22+41−α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2>0 and 1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α2>1−α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2
Nonhyperbolic iff1+hβ+α2+α2+β+2hα21+hα2+β/1+hα2+β1+hβ+α2=1−α2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2
Locally asymptotically stable focus ifα2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2<1
Unstable focus ifα2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2>1
Nonhyperbolic (for complex eigenvalues) ifα2+β+2hα2+2h2α2α2+β/1+hα2+β1+hβ+α2=1



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*Source: 7834076-2020-01-22.xml*