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添加134字节 、 2020年11月18日 (三) 23:27
无编辑摘要
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==Definition==
 
==Definition==
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定义
    
We consider a two-dimensional dynamical system of the form
 
We consider a two-dimensional dynamical system of the form
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==Properties==
 
==Properties==
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属性
    
By the [[Jordan curve theorem]], every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
 
By the [[Jordan curve theorem]], every closed trajectory divides the plane into two regions, the interior and the exterior of the curve.
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==Stable, unstable and semi-stable limit cycles==
 
==Stable, unstable and semi-stable limit cycles==
 
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稳定、不稳定和半稳定极限环
 
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a ''[[stable manifold|stable]]'' or ''attractive'' limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an ''unstable'' limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a ''semi-stable'' limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
 
In the case where all the neighboring trajectories approach the limit cycle as time approaches infinity, it is called a ''[[stable manifold|stable]]'' or ''attractive'' limit cycle (ω-limit cycle). If instead, all neighboring trajectories approach it as time approaches negative infinity, then it is an ''unstable'' limit cycle (α-limit cycle). If there is a neighboring trajectory which spirals into the limit cycle as time approaches infinity, and another one which spirals into it as time approaches negative infinity, then it is a ''semi-stable'' limit cycle. There are also limit cycles that are neither stable, unstable nor semi-stable: for instance, a neighboring trajectory may approach the limit cycle from the outside, but the inside of the limit cycle is approached by a family of other cycles (which wouldn't be limit cycles).
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==Finding limit cycles==
 
==Finding limit cycles==
 
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寻找极限环
 
Every closed trajectory contains within its interior a [[stationary point]] of the system, i.e. a point <math>p</math> where <math>V(p)=0</math>. The [[Bendixson–Dulac theorem]] and the [[Poincaré–Bendixson theorem]] predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
 
Every closed trajectory contains within its interior a [[stationary point]] of the system, i.e. a point <math>p</math> where <math>V(p)=0</math>. The [[Bendixson–Dulac theorem]] and the [[Poincaré–Bendixson theorem]] predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.
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==Open problems==
 
==Open problems==
 
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未解决的问题
 
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of [[Hilbert's sixteenth problem]]. It is unknown, for instance, whether there is any system <math>x'=V(x)</math> in the plane where both components of <math>V</math> are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
 
Finding limit cycles, in general, is a very difficult problem. The number of limit cycles of a polynomial differential equation in the plane is the main object of the second part of [[Hilbert's sixteenth problem]]. It is unknown, for instance, whether there is any system <math>x'=V(x)</math> in the plane where both components of <math>V</math> are quadratic polynomials of the two variables, such that the system has more than 4 limit cycles.
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== Applications ==
 
== Applications ==
 
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Applications
 
[[File:Hopfbifurcation.png|thumb|400px|Examples of limit cycles branching from fixed points near [[Hopf bifurcation]]. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.]]
 
[[File:Hopfbifurcation.png|thumb|400px|Examples of limit cycles branching from fixed points near [[Hopf bifurcation]]. Trajectories in red, stable structures in dark blue, unstable structures in light blue. The parameter choice determines the occurrence and stability of limit cycles.]]
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== See also ==
 
== See also ==
 
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另请参见
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==References==
 
==References==
 
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参考资料
 
{{Reflist}}
 
{{Reflist}}
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==Further reading==
 
==Further reading==
 
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延伸阅读
     
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