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第7行： − Phase portrait showing saddle-node bifurcation − '''Bifurcation theory''' is the [[Mathematics|mathematical]] study of changes in the qualitative or [[topological structure]] of a given [[Family of curves|family]], such as the [[integral curve]]s of a family of [[vector field]]s, and the solutions of a family of [[differential equation]]s. Most commonly applied to the [[mathematics|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.<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> Bifurcations occur in both continuous systems (described by [[Ordinary differential equation|ODE]]s, [[Delay differential equation|DDE]]s or [[Partial differential equation|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.<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> [[Henri Poincaré]] also later named various types of [[stationary points]] and classified them. − 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。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，后来亨利 · 庞加莱也对不同的驻点进行了命名和分类。 − − − − ==Bifurcation types== − − ==Bifurcation types== − It is useful to divide bifurcations into two principal classes: − It is useful to divide bifurcations into two principal classes:
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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
'''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：