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

A Q Khan

doi:10.1155/2020/3936242

Abstract

In this paper, the local dynamics and Neimark–Sacker bifurcation of a two-dimensional glycolytic oscillator model in the interior of ℝ+2 are explored. It is investigated that for all α and β, 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.

Highlights

Many chemical models are governed by difference as well as differential equations
Edeki et al [1] have explored the numerical solution of the following nonlinear biochemical model by using the hybrid technique: η dx y − αx − xy, dt (1) dy −y +(α − β), dt where x and y are the substrate concentrations at time t and η, α, and β are the dimensionless parameters
Zafar et al [2] have investigated the equilibria and convergence analysis of the following nonlinear biochemical reaction networks: dx x +(β − α) + xy, dt 1 σ (x xy), where x is the concentration of the substrate, y is the intermediate complex, and the parameters σ, α, and β are the dimensionless parameters

Summary

Research Article

Received 27 February 2020; Revised 18 June 2020; Accepted 22 June 2020; Published 6 August 2020. The local dynamics and Neimark–Sacker bifurcation of a two-dimensional glycolytic oscillator model in the interior of R2+ are explored. It is investigated that for all α and β, the model has a unique equilibrium point: P+xy((α/β + α2), α). Further about P+xy((α/β + α2), α), local dynamics and the existence of bifurcation are explored. It is investigated about P+xy((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. Bifurcation diagrams and the corresponding maximum Lyapunov exponent are presented for the glycolytic oscillator model

Introduction

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Numerical value of Ω

Unstable focus