Qualitative and numerical analysis of the Rössler model: Bifurcations of equilibria

In this paper, we show the combined use of analytical and numerical techniques in the study of bifurcations of equilibria of low-dimensional chaotic problems. We study in detail different aspects of the paradigmatic Rössler model. We provide analytical formulas for the stability of the equilibria as well as some of their codimension one, two, and three bifurcations. In particular, we carry out a complete study of the Andronov–Hopf bifurcation, establishing explicit formulas for its location and studying its character numerically, determining a curve of generalized-Hopf bifurcation, where the Hopf bifurcation changes from subcritical to supercritical. We also briefly study some routes among the different Andronov–Hopf bifurcation curves and how these routes are influenced by the local and global bifurcations of limit cycles. Finally, we show the U-shape of the homoclinic bifurcation curve at the studied parameter values.