Open AccessBogdanov-Takens bifurcations in the enzyme-catalyzed reaction comprising a branched network
Author(s) -
Qiuyan Zhang,
Lingling Liu,
Weinian Zhang
Publication year - 2017
Publication title -
mathematical biosciences and engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.451
H-Index - 45
eISSN - 1551-0018
pISSN - 1547-1063
DOI - 10.3934/mbe.2017078
Subject(s) - homoclinic orbit , pitchfork bifurcation , bogdanov–takens bifurcation , homoclinic bifurcation , cusp (singularity) , mathematics , degenerate energy levels , saddle node bifurcation , hopf bifurcation , bifurcation , mathematical analysis , transcritical bifurcation , pure mathematics , physics , geometry , nonlinear system , quantum mechanics
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.