Hopf bifurcation and Si’lnikov chaos of Genesio system

Genesio system, which is a three-dimensional system with only one quadratic nonlinear term, is considered. It has two equilibrium points for some parameters. The Hopf bifurcation is discussed, and the existence of the homoclinic orbit for this system has been proven by using the undetermined coefficient method. As a result, the Si’lnikov criterion guarantees that the Genesio system has Smale horseshoe chaos.