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| | * [[Neimark–Sacker(次级霍普夫)分岔]] | | * [[Neimark–Sacker(次级霍普夫)分岔]] |
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| − | ===全局分岔=== | + | ===全局分岔Global bifurcations=== |
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| − | [[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|2维同宿分岔前后的相位图 |
| | + | 周期轨道逐渐增长，直到它与鞍点重合。在分岔点处，周期轨道的周期已经增长到无穷大，并成为一个[[同宿轨道]]。分岔之后不再存在周期轨道。 |
| | + | '''左侧'''：对于小参数值，在原点处有一个[[鞍点]]，在第一象限有一个[[极限环]]。 |
| | + | '''中间'''：随着分岔参数的增加，极限环逐渐增大，直到与鞍点完全相交，形成一个无限长的轨道。 |
| | + | '''右侧'''：当分岔参数进一步增加时，极限环完全消失。]] |
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| − | 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.]]
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| − | 一个相位的肖像前，在，和之后的一个同宿分岔在2 d。周期轨道不断增长，直到与鞍点相撞。在分岔点，周期轨道的周期已经增长到无穷大，它已经成为一个[同宿轨道]。分岔后不再存在周期轨道。左面板: 对于小参数值，在原点有一个鞍点，在第一象限有一个极限环。中面板: 随着分岔参数的增加，极限环逐渐增大，直到与鞍点完全相交，形成一个无限长的轨道。右面板: 当分支参数进一步增加时，极限环完全消失
| + | 当'较大的'不变集(如周期轨道)与平衡点重合时，就会出现全局分岔。这导致相空间中轨迹的拓扑结构发生变化，而且这种变化不能像局部分叉那样局限于一个小的邻域内。事实上，拓扑结构的变化可以延伸到任意大的距离，因此称为全局分岔。 |
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| | + | 全局分岔的例子有： |
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| − | Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance (hence 'global').
| + | *'''同宿分岔'''是指极限环与鞍点相重合。<ref>{{cite book |last=Strogatz |first=Steven H. |date=1994 |title=Nonlinear Dynamics and Chaos |publisher=[[Addison-Wesley]] |page=262 |isbn=0-201-54344-3 |author-link=Steven Strogatz}}</ref> 同宿分岔出现在超临界或亚临界状态下。上面的变体是“小”或者“I型”同宿分岔。 二维情况下，在同宿轨道“捕获”鞍的不稳定和稳定流形的另一端存在“大”或“II型”同宿分岔。在三维或多维情况下，可能会出现高共维分岔，产生复杂性系统，可能是[[混沌]]动力学。 |
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| − | Global bifurcations occur when 'larger' invariant sets, such as periodic orbits, collide with equilibria. This causes changes in the topology of the trajectories in the phase space which cannot be confined to a small neighbourhood, as is the case with local bifurcations. In fact, the changes in topology extend out to an arbitrarily large distance (hence 'global').
| + | *'''异宿分岔'''是指极限环与两个或多个鞍点重合，这涉及到[[异宿环]]。<ref>{{cite book |title=Bifurcation Theory and Methods of Dynamical Systems|last=Luo|first=Dingjun|authorlink= |publisher=World Scientific|year=1997|isbn=981-02-2094-4|page=26}}</ref> 异宿分岔有两种类型：共振分岔和横向分岔，两种类型的分岔都会导致异宿环稳定性的改变。 在共振分岔处，当环的平衡点的[[特征值]]和特征向量的代数条件满足时，环的稳定性改变。这通常伴随着[[周期轨道]]的出现和消失。当一个异宿环中某个平衡点的横向特征值的实部通过零时，就会引起该环的横向分岔，同时也会引起异宿环稳定性的变化。 |
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| − | 当“较大的”不变集(如周期轨道)与平衡碰撞时，就会出现全局分岔。这导致相空间中轨迹的拓扑结构发生变化，不能像局部分叉那样局限于一个小的邻域。事实上，拓扑的变化延伸到一个任意大的距离(因此是“全局”)。
| + | * '''无限周期分岔'''是指在极限环上同时出现稳定点和鞍点。<ref>James P. Keener, "Infinite Period Bifurcation and Global Bifurcation Branches", ''SIAM Journal on Applied Mathematics'', Vol. 41, No. 1 (August 1981), pp. 127–144.</ref>当参数的[[极限]]接近某个临界值时，振荡速度变慢，周期接近无穷大。无限周期分岔发生在此临界值处。 在临界值外，极限环上相继出现两个不动点，破坏振荡，形成了两个[[鞍点]]。 |
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| | + | * [[蓝天突变]]是指极限环与非双曲环相重合。 |
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| − | Examples of global bifurcations include:
| + | 全局分岔还涉及到更复杂的集合，例如[[混沌吸引子]](如[[危机]])。 |
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| − | Examples of global bifurcations include:
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| − | 全球分歧的例子包括:
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| − | *'''Homoclinic bifurcation''' in which a [[limit cycle]] collides with a [[saddle point]].<ref>{{cite book |last=Strogatz |first=Steven H. |date=1994 |title=Nonlinear Dynamics and Chaos |publisher=[[Addison-Wesley]] |page=262 |isbn=0-201-54344-3 |author-link=Steven Strogatz}}</ref> Homoclinic bifurcations can occur supercritically or subcritically. The variant above is the "small" or "type I" homoclinic bifurcation. In 2D there is also the "big" or "type II" homoclinic bifurcation in which the homoclinic orbit "traps" the other ends of the unstable and stable manifolds of the saddle. In three or more dimensions, higher codimension bifurcations can occur, producing complicated, possibly [[chaos theory|chaotic]] dynamics.
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| − | *'''Heteroclinic bifurcation''' in which a limit cycle collides with two or more saddle points; they involve a [[heteroclinic cycle]].<ref>{{cite book |title=Bifurcation Theory and Methods of Dynamical Systems|last=Luo|first=Dingjun|authorlink= |publisher=World Scientific|year=1997|isbn=981-02-2094-4|page=26}}</ref> Heteroclinic bifurcations are of two types: resonance bifurcations and transverse bifurcations. Both types of bifurcation will result in the change of stability of the heteroclinic cycle. At a resonance bifurcation, the stability of the cycle changes when an algebraic condition on the [[Eigenvalues_and_eigenvectors|eigenvalues]] of the equilibria in the cycle is satisfied. This is usually accompanied by the birth or death of a [[periodic orbit]]. A transverse bifurcation of a heteroclinic cycle is caused when the real part of a transverse eigenvalue of one of the equilibria in the cycle passes through zero. This will also cause a change in stability of the heteroclinic cycle.
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| − | * '''Infinite-period bifurcation''' in which a stable node and saddle point simultaneously occur on a limit cycle.<ref>James P. Keener, "Infinite Period Bifurcation and Global Bifurcation Branches", ''SIAM Journal on Applied Mathematics'', Vol. 41, No. 1 (August, 1981), pp. 127–144.</ref> As the [[Limit (mathematics)|limit]] of a parameter approaches a certain critical value, the speed of the oscillation slows down and the period approaches infinity. The infinite-period bifurcation occurs at this critical value. Beyond the critical value, the two fixed points emerge continuously from each other on the limit cycle to disrupt the oscillation and form two [[saddle point]]s.
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| − | * [[Blue sky catastrophe]] in which a limit cycle collides with a nonhyperbolic cycle.
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| − | Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g. [[Crisis (dynamical systems)|crises]]).
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| − | Global bifurcations can also involve more complicated sets such as chaotic attractors (e.g. crises).
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| − | 全局分岔还可以涉及更复杂的集合，如混沌吸引子(例如混沌吸引子)。危机)。
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| | ==Codimension of a bifurcation== | | ==Codimension of a bifurcation== |