Periodically Perturbed Hopf Bifurcation

Abstract

In this paper the influence of small periodic perturbations on systems exhibiting Hopf bifurcation is studied in detail. In two-dimensional systems that ordinarily exhibit Hopf bifurcation, the addition of small periodic parametric excitation gives rise to interesting “secondary phenomena.” Explicit results for various primary and secondary bifurcations, along with their stabilities, are obtained. In this work, the ideas related to method of averaging, Poincare–Birkhoff normal forms, and center manifold theorem are used appropriately at different stages. It is found that various results obtained using these techniques agree in their common regions of validity.