Resumen de Bogdanov–Takens and Hopf Bifurcations Analysis of a Genetic Regulatory Network

Ming Liu, Fanwei Meng, Dongpo Hu

• The stability and two kinds of bifurcations of a genetic regulatory network are considered in this paper. We have given a complete stability analysis involved in mentioned model. The Hopf bifurcation of codimension 1 and Bogdanov–Takens bifurcation of codimension 2 for the nonhyperbolic equilibria of the model is characterized analytically. In order to determine the stability of limit cycle of Hopf bifurcation, the first Lyapunov number is calculated and a numerical example is given to illustrate graphically. Three bifurcation curves related to Bogdanov–Takens bifurcation, namely saddle-node, Hopf and homoclinic bifurcation curves, are given explicitly by calculating a universal unfolding near the cusp. Moreover, the numerical continuation results show that the model has other bifurcation types, including saddle-node and cusp bifurcations. The bifurcation diagram and phase portraits are also given to verify the validity of the theoretical results. These results show that there exists rich bifurcation behavior in the genetic regulatory network.