Bifurcation theory

Summary

Bifurcation theory studies how the qualitative behaviour of a family of dynamical systems changes as parameters vary. A bifurcation happens at a parameter value where the phase portrait changes its topological type, for example when equilibria appear, disappear or change stability. The theory classifies the typical bifurcations, such as the saddle-node and Hopf bifurcations, and gives normal forms for them. It explains sudden transitions in models from physics, biology and engineering.