Transcritical and zero-Hopf bifurcations in the Genesio system

Abstract

In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation $${\dddot{x}} + a {\ddot{x}} + b {\dot{x}} + c x - x^2 = 0$$ , called the Genesio equation, which has a unique quadratic nonlinear term and three real parameters. More precisely, writing this differential equation as a first-order differential system in $$\mathbb {R}^3$$ we prove: first that the system exhibits a transcritical bifurcation at the equilibrium point located at the origin of coordinates when $$c=0$$ and the parameters (a, b) are in the set $$\{(a,b) \in \mathbb {R}^2 : b \ne 0\} {\setminus } \{(0,b) \in \mathbb {R}^2 : b > 0\}$$ , and second that the system has a zero-Hopf bifurcation also at the equilibrium point located at the origin when $$a=c=0$$ and $$b>0$$ .