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[[File:Saddlenode.gif|thumb|right|300px|Phase portrait showing saddle-node bifurcation]]

[[File:Saddlenode.gif|thumb|right|300px|Phase portrait showing saddle-node bifurcation]]

显示鞍结分岔的相位图

显示鞍结分岔的相位图

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 behavior. 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.

'''分岔理论'''是[[数学]]中研究给定[[族]]的定性或拓扑结构的改变，例如[[向量场]]中的一族[[积分曲线]]以及[[微分方程]]的一族解。'''分岔'''常用于动力系统的[[数学]]研究中，是指当系统的参数值(分岔参数)发生微小平滑的变化时，系统发生突然的“定性”或拓扑变化。<ref>{{Cite book |first=P. |last=Blanchard |first2=R. L. |last2=Devaney|author2-link= Robert L. Devaney |first3=G. R. |last3=Hall |title=Differential Equations |location=London |publisher=Thompson |year=2006 |pages=96–111 |isbn=978-0-495-01265-8 }}</ref> 分岔在连续系统(由[[常微分方程]]、[[微分方程]]或[[偏微分方程]]描述)和离散系统(由映射描述)中均存在。1885年，[[亨利 · 庞加莱]]首次在论文中提到“分岔”一词，这也是数学中揭示该行为的第一篇论文。<ref>Henri Poincaré. "''L'Équilibre d'une masse fluide animée d'un mouvement de rotation''". ''Acta Mathematica'', vol.7, pp. 259-380, Sept 1885.</ref>后来[[亨利 · 庞加莱]]也对不同的驻点进行了命名和分类。

==分岔类型==

==分岔类型==

主要将分岔划分为以下两个类别：

主要将分岔划分为以下两个类别：