Transcritical and zero-Hopf bifurcations in the Genesio system

© 2016, Springer Science+Business Media Dordrecht. In this paper we study the existence of transcritical and zero-Hopf bifurcations of the third-order ordinary differential equation x⃛ + ax¨ + bx˙ + cx- x2= 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 R3 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) ∈ R2: b≠ 0 } \ { (0 , b) ∈ R2: 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.