Chapter 4: Degenerate Hopf bifurcations

A methodology for computation of the local bifurcation diagrams obtained near the Hopf bifurcation degeneracies has not yet been presented. As the bifurcation curves become more complex when more defining conditions are satisfied, as will be seen in this chapter, the parameter space needs to be divided appropriately to reflect the distinct behaviors in different regions. After that, the corresponding bifurcation diagrams can be constructed, at least locally. Moreover, intersections of the degenerate bifurcations can be obtained, which generates other more complex singularities. The importance of detecting such higher co-dimensional degeneracies is crucial for analyzing all the distinctive features of the local dynamics that are preserved under relatively large variation of system parameters. In this chapter, a methodology for computing the local bifurcation diagrams obtained near the simplest Hopf bifurcation degeneracies is presented.