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==Definition==

==Definition==

We consider a two-dimensional dynamical system of the form

We consider a two-dimensional dynamical system of the form

==Properties==

==Properties==

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.

==Stable, unstable and semi-stable limit cycles==

==Stable, unstable and semi-stable 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).

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).

==Finding limit cycles==

==Finding limit cycles==

 

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==

 

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.]]

== See also ==

== See also ==

 

==References==

==References==

 

{{Reflist}}

{{Reflist}}

==Further reading==

==Further reading==