Hopf Bifurcation Analysis of the Halvorsen System

This paper investigates local bifurcations in the Halvorsen system, focusing specifically on transcritical and Hopf bifurcations. The behavior of equilibrium points during bifurcations is studied using Sotomayor's theorem for transcritical bifurcation and normal form theory, which is based on Hassard's formulas, for Hopf bifurcation. When the bifurcation parameter exceeds a critical value, a Hopf bifurcation emerges. By applying normal form theory, we establish the conditions under which a Hopf bifurcation occurs. Furthermore, we discuss the direction of the Hopf bifurcation and the stability of the resulting periodic orbits. Finally, numerical simulations are provided to support the theoretical findings.