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

Qiuyan Zhang; Lingling Liu; Weinian Zhang

doi:10.3934/mbe.2017078

Article Contents

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

* Corresponding author: Weinian Zhang
* Corresponding author: Weinian Zhang

Received: July 02, 2016

Accepted: January 2017

Published: December 2017

Supported by NSFC # 11221101, # 11231001, # 11501475 and SPDEF 16ZB0080.

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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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Mathematics Subject Classification: 34C23, 92C45.

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Figure 1. Reaction scheme

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Figure 2. Bifurcation surfaces projection on the $(a, \kappa)$-plane

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Figure 3. Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$

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Figure 4. An attracting limit cycle

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Table 4. Dynamics of system (4) in various cases of parameter $(a, \kappa)$

Parameters $(a, \kappa)$	Equilibria				Limit cycles and homoclinic orbits	Region in bifurcation diagram
Parameters $(a, \kappa)$	$E_0$	$E_1$	$E_2$	$E_*$	Limit cycles and homoclinic orbits	Region in bifurcation diagram
$ \mathcal{D}_{I}$	saddle	unstable focus	saddle			$\mathcal{D}_{I}$
$ \mathcal{L}$	saddle	unstable focus	saddle		one homoclinic rrbit	$\mathcal{L}$
$\mathcal{D}_{II}$	saddle	unstable focus	saddle		one limit cycle	$\mathcal{D}_{II}$
$\mathcal{H}$	saddle	stable focus	saddle			$\mathcal{H}$
$\mathcal{D}_{III}$	saddle	stable focus	saddle			$\mathcal{D}_{III}$
$ \mathcal{SN}^+$				saddle-node		$\mathcal{SN}^+$
$ \mathcal{D}_{IV}$						$\mathcal{D}_{IV}$
$(a_, \kappa_)$				cusp		$(a_, \kappa_)$
$ \mathcal{SN}^-$				saddle-node		$\mathcal{SN}^-$

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