Bifurcation Theory

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields or the solutions of a family of differential equations. Most commonly applied to dynamical systems, a bifurcation occurs when a small, smooth change to a system's parameter values causes a sudden qualitative or topological change in its behavior, and bifurcations occur in both continuous systems, described by ordinary, delay or partial differential equations, and discrete systems, described by maps. Henri Poincaré first introduced the name bifurcation in 1885, in the first paper to show such behavior mathematically.