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

A Q Khan,Tarek F Ibrahim,E Abdullah

doi:10.1155/2020/7834076

Abstract

We study the local dynamical properties, Neimark–Sacker bifurcation, and hybrid control in a glycolytic oscillator model in the interior of ℝ+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.

Highlights

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 [1,2,3,4]. is 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]: Glucose

Summary

Introduction

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. Our purpose is to explore the local dynamics, N-S bifurcation, and hybrid control in a glycolytic oscillator model (4) in the interior of R2+. E flow pattern of the remaining of this paper is as follows: Section 2 is about the existence of the unique +ve equilibrium point and corresponding linearized form of the glycolytic oscillator model (4).

Results

Conclusion