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第5行：
第5行： − + − − − 显示鞍结分岔的相位图 第20行：
第17行： − + 第26行：
第23行： − − A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (those described by maps rather than ODEs), this corresponds to a fixed point having a [[Floquet multiplier]] with modulus equal to one. In both cases, the equilibrium is ''non-hyperbolic'' at the bifurcation point.+ − − A local bifurcation occurs when a parameter change causes the stability of an equilibrium (or fixed point) to change. In continuous systems, this corresponds to the real part of an eigenvalue of an equilibrium passing through zero. In discrete systems (those described by maps rather than ODEs), this corresponds to a fixed point having a Floquet multiplier with modulus equal to one. In both cases, the equilibrium is non-hyperbolic at the bifurcation point. − − 当参数变化引起平衡点(或不动点)的稳定性变化时，就会发生局部分岔。在连续系统中，这相当于一个平衡特征值通过零点的实部。在离散系统中(用映射而不是常微分方程描述的系统) ，这相当于一个具有模等于1的 Floquet 乘数的不动点。在这两种情况下，平衡点在分岔点都是非双曲的。 − − The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local'). − − The topological changes in the phase portrait of the system can be confined to arbitrarily small neighbourhoods of the bifurcating fixed points by moving the bifurcation parameter close to the bifurcation point (hence 'local'). − − 通过将分岔参数移动到分岔点附近，系统相图的拓扑变化可以局限于分岔不动点的任意小邻域内。 − − − − More technically, consider the continuous dynamical system described by the ODE − 更严格地说，考虑常微分方程描述的连续动力系统+ − + − + − 数学点 x f (x，λ) f 冒号 mathbb { r } ^ n times mathbb { r } right tarrow mathbb { r } ^ n / math − − A local bifurcation occurs at <math>(x_0,\lambda_0)</math> if the [[Jacobian matrix and determinant|Jacobian]] matrix − − A local bifurcation occurs at <math>(x_0,\lambda_0)</math> if the Jacobian matrix − − 如果雅可比矩阵的数学(x0， lambda 0) / − − <math> \textrm{d}f_{x_0,\lambda_0}</math> − − <math> \textrm{d}f_{x_0,\lambda_0}</math> − − 0，lambda 0} / math − − has an [[Eigenvalue, eigenvector and eigenspace|eigenvalue]] with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely imaginary, this is a [[Hopf bifurcation]]. − − has an eigenvalue with zero real part. If the eigenvalue is equal to zero, the bifurcation is a steady state bifurcation, but if the eigenvalue is non-zero but purely imaginary, this is a Hopf bifurcation. − − 有一个实部为零的特征值。如果特征值等于零，则分岔为稳态分岔，但如果特征值非零而纯为虚数，则为霍普夫分岔。 − − − − For discrete dynamical systems, consider the system − For discrete dynamical systems, consider the system 第85行：
第43行： − <math>x_{n+1}=f(x_n,\lambda)\,.</math>+ − + − Math x { n + 1} f (xn， lambda) ，. / math − − Then a local bifurcation occurs at <math>(x_0,\lambda_0)</math> if the matrix − − Then a local bifurcation occurs at <math>(x_0,\lambda_0)</math> if the matrix − − 然后在 math (x0， lambda 0) / math if the matrix − − − − − − 0，lambda 0} / math − − has an eigenvalue with modulus equal to one. If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation. − − has an eigenvalue with modulus equal to one. If the eigenvalue is equal to one, the bifurcation is either a saddle-node (often called fold bifurcation in maps), transcritical or pitchfork bifurcation. If the eigenvalue is equal to −1, it is a period-doubling (or flip) bifurcation, and otherwise, it is a Hopf bifurcation. − − 有一个模等于1的特征值。如果特征值等于1，则分岔可以是跨临界的鞍结点(在映射中通常称为折叠分岔) ，也可以是叉式分岔分岔。如果特征值等于-1，则为倍周期分叉，否则为霍普夫分岔分叉。 − − − − Examples of local bifurcations include: − − Examples of local bifurcations include: − − 局部分岔的例子包括: + − + − + − + − + − + − + − +
无编辑摘要
{{Technical|date=June 2019}}
{{Technical|date=June 2019}}
[[File:Saddlenode.gif|thumb|right|300px|Phase portrait showing saddle-node bifurcation]]
[[File:Saddlenode.gif|thumb|right|300px|显示鞍结分岔的相位图]]
*局部分岔Local bifurcations是指可被[[平衡点]]的局部稳定性、周期轨道或其他不变集作为参数穿过临界阈值完全分析的分岔；
*局部分岔Local bifurcations是指可用[[平衡点]]的局部稳定性、周期轨道或其他不变集作为参数穿过临界阈值完全分析的分岔；
===局部分岔Local bifurcations===
===局部分岔Local bifurcations===
[[File:Chaosorderchaos.png|300px|right|thumb|周期减半分岔(L)导致有序，周期倍增分岔(R)导致混沌.]]
[[File:Chaosorderchaos.png|300px|right|thumb|周期减半分岔(L)导致有序，周期倍增分岔(R)导致混沌.]]
当参数的改变引起平衡点(或不动点)的稳定性改变时，就会发生局部分岔。在连续系统中，这相当于平衡点的特征值实部通过零点。在离散系统(用映射而不是常微分方程描述的系统) 中，这相当于不动点有一个模数等于1的[[Floquet乘数]]。在这两种情况下，平衡点在分岔点处都是''非双曲''的。通过将分岔参数移动到分岔点附近，可将系统相图的拓扑变化局限于分岔不动点的任意小邻域内，因此称为局部分岔。
More technically, consider the continuous dynamical system described by the ODE
More technically, consider the continuous dynamical system described by the ODE
更严格地说，考虑由常微分方程描述的连续动力系统
:<math>\dot x=f(x,\lambda)\quad f\colon\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.</math>
:<math>\dot x=f(x,\lambda)\quad f\colon\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.</math>
<math>\dot x=f(x,\lambda)\quad f\colon\mathbb{R}^n\times\mathbb{R}\rightarrow\mathbb{R}^n.</math>
如果雅可比矩阵<math> \textrm{d}f_{x_0,\lambda_0}</math>
具有实部为零的[[特征值]]，则在<math>(x_0,\lambda_0)</math>处发生局部分岔。若特征值为零，则分岔为稳态分岔，若特征值非零而为纯虚数，则分岔为[[霍普夫分岔]]。
对于离散动力系统，考虑系统
对于离散动力系统，考虑系统
:<math>x_{n+1}=f(x_n,\lambda)\,.</math>
:<math>x_{n+1}=f(x_n,\lambda)\,.</math>
如果矩阵<math> \textrm{d}f_{x_0,\lambda_0}</math>具有模数等于1的特征值，则在<math>(x_0,\lambda_0)</math>处发生局部分岔。若特征值为1，则分岔为鞍结分岔（在映射中常称为折叠分岔）、跨临界分岔、叉式分岔。若特征值为-1，则分岔为周期倍增（或翻转）分岔，否则为
霍普夫分岔。
<math> \textrm{d}f_{x_0,\lambda_0}</math>
<math> \textrm{d}f_{x_0,\lambda_0}</math>
局部分岔的例子有:
* [[Saddle-node bifurcation|Saddle-node]] (fold) bifurcation
* [[鞍结]] (折叠) 分岔
* [[Transcritical bifurcation]]
* [[跨临界分岔]]
* [[Pitchfork bifurcation]]
* [[叉式分岔]]
* [[Period-doubling bifurcation|Period-doubling]] (flip) bifurcation
* [[周期倍增]] (翻转) 分岔
* [[Hopf bifurcation]]
* [[霍普夫分岔]]
* [[Neimark–Sacker bifurcation|Neimark–Sacker]] (secondary Hopf) bifurcation
* [[Neimark–Sacker]] (次级霍普夫)分岔
===Global bifurcations===
===全局分岔===
[[Image:homoclinic_bif.png|frame|right|A phase portrait before, at, and after a homoclinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a [[homoclinic orbit]]. After the bifurcation there is no longer a periodic orbit. '''Left panel''': For small parameter values, there is a [[saddle point]] at the origin and a [[limit cycle]] in the first quadrant. '''Middle panel''': As the bifurcation parameter increases, the limit cycle grows until it exactly intersects the saddle point, yielding an orbit of infinite duration. '''Right panel''': When the bifurcation parameter increases further, the limit cycle disappears completely.]]
[[Image:homoclinic_bif.png|frame|right|A phase portrait before, at, and after a homoclinic bifurcation in 2D. The periodic orbit grows until it collides with the saddle point. At the bifurcation point the period of the periodic orbit has grown to infinity and it has become a [[homoclinic orbit]]. After the bifurcation there is no longer a periodic orbit. '''Left panel''': For small parameter values, there is a [[saddle point]] at the origin and a [[limit cycle]] in the first quadrant. '''Middle panel''': As the bifurcation parameter increases, the limit cycle grows until it exactly intersects the saddle point, yielding an orbit of infinite duration. '''Right panel''': When the bifurcation parameter increases further, the limit cycle disappears completely.]]