Bifurcation theory is the
mathematical study of changes in the qualitative or
topological structure of a given family, such as the
integral curves of a family of
vector fields, and the solutions of a family of
differential equations. Most commonly applied to the
mathematical study of
dynamical systems, a
bifurcation occurs when a small smooth change made to the parameter values (the bifurcation parameters) of a system causes a sudden 'qualitative' or topological change in its behaviour. Bifurcations occur in both continuous systems (described by
ODEs,
DDEs or
PDEs), and discrete systems (described by maps). The name "bifurcation" was first introduced by
Henri Poincaré in 1885 in the first paper in mathematics showing such a behavior.
Henri Poincaré also later named various types of
stationary points and classified them.