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==Finding limit cycles==

==Finding limit cycles==

寻找极限环

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.

==Open problems==

==Open problems==

未解决的问题

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.

== Applications ==

== 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.]]