Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

Qiuyan Zhang; Lingling Liu; Weinian Zhang; Qiuyan Zhang; Lingling Liu; Weinian Zhang

doi:10.3934/mbe.2017078

Mathematical Biosciences and Engineering

2017, Volume 14, Issue 5&6: 1499-1514. doi: 10.3934/mbe.2017078

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Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network

1.
Department of Mathematics, Sichuan University, Chengdu, Sichuan 610064, China
2.
College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, Sichuan 610225, China
3.
School of Sciences, Southwest Petroleum University, Chengdu, Sichuan 610500, China

Received: 02 July 2016 Accepted: 01 January 2017 Published: 01 October 2017
MSC : 34C23, 92C45

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.
- Enzyme-catalyzed reaction,
- cusp,
- Bogdanov-Takens bifurcation,
- normal form
Citation: Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network[J]. Mathematical Biosciences and Engineering, 2017, 14(5&6): 1499-1514. doi: 10.3934/mbe.2017078

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Abstract

There have been some results on bifurcations of codimension one (such as saddle-node, transcritical, pitchfork) and degenerate Hopf bifurcations for an enzyme-catalyzed reaction system comprising a branched network but no further discussion for bifurcations at its cusp. In this paper we give conditions for the existence of a cusp and compute the parameter curves for the Bogdanov-Takens bifurcation, which induces the appearance of homoclinic orbits and periodic orbits, indicating the tendency to steady-states or a rise of periodic oscillations for the concentrations of the substrate and the product.

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