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Bifurcation theory
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, L'Équilibre d'une masse fluide animée d'un mouvement de rotation, Acta Mathematica, t.7, pp. 259-380, sept 1885.〕 Henri Poincaré also later named various types of stationary points and classified them.
==Bifurcation types==
It is useful to divide bifurcations into two principal classes:
* Local bifurcations, which can be analysed entirely through changes in the local stability properties of equilibria, periodic orbits or other invariant sets as parameters cross through critical thresholds; and
* Global bifurcations, which often occur when larger invariant sets of the system 'collide' with each other, or with equilibria of the system. They cannot be detected purely by a stability analysis of the equilibria (fixed points).
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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