# American Institute of Mathematical Sciences

October  2017, 14(5&6): 1499-1514. doi: 10.3934/mbe.2017078

## 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

Received  July 02, 2016 Accepted  January 2017 Published  May 2017

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

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.

Citation: Qiuyan Zhang, Lingling Liu, Weinian Zhang. Bogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1499-1514. doi: 10.3934/mbe.2017078
##### References:

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##### References:
Reaction scheme
Bifurcation surfaces projection on the $(a, \kappa)$-plane
Bifurcation diagrams of system (4) for the case that $c>1$ and $b=(c+1)^2/4$
An attracting limit cycle
Dynamics of system (4) in various cases of parameter $(a, \kappa)$
 Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram $E_0$ $E_1$ $E_2$ $E_*$ $\mathcal{D}_{I}$ 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 $\mathcal{SN}^-$
 Parameters $(a, \kappa)$ Equilibria Limit cycles and homoclinic orbits Region in bifurcation diagram $E_0$ $E_1$ $E_2$ $E_*$ $\mathcal{D}_{I}$ 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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