Hopf bifurcation formulae and applications to the Genesio-Tesi system

Abstract

The purpose of this paper is to propose some formulae for Hopf bifurcation analysis, and investigate applications to a chaotic system. We perform a substantial simplification for the classical Hopf bifurcation formulae. Our results can be extended to multi-dimensional quadratic systems. As an application, we consider the Genesio-Tesi system. Finally, the dynamics of the system are analyzed by bifurcation diagrams, Lyapunov exponents, phase portraits and Poincaré maps. We show that the system can generate chaos via a Hopf bifurcation and period doubling cascade as the control parameter varies. Some other bifurcations can be observed, which includes saddle-node bifurcations, interior crises and boundary crises.