Editing Limit cycle (section)

== 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.]]

Limit cycles are important in many scientific applications where systems with self-sustained oscillations are modelled. Some examples include:

* Aerodynamic limit-cycle oscillations<ref>{{citation | last1 =Thomas| first1 =Jeffrey P.| last2 =Dowell| first2 =Earl H.| last3 =Hall| first3 =Kenneth C.| title =Nonlinear Inviscid Aerodynamic Effects on Transonic Divergence, Flutter, and Limit-Cycle Oscillations| journal =AIAA Journal| volume =40| issue =4| pages =638| publisher =American Institute of Aeronautics and Astronautics| url=https://mems.duke.edu/files/mems/thomas/downloads/hblco.pdf| access-date = December 9, 2019 | bibcode =2002AIAAJ..40..638T | year =2002| doi =10.2514/2.1720}}</ref>
* The [[Hodgkin–Huxley model]] for [[action potential]]s in [[neuron]]s.
* The Sel'kov model of [[glycolysis]].<ref>{{Cite journal|last=Sel'kov|first=E. E.|date=1968|title=Self-Oscillations in Glycolysis 1. A Simple Kinetic Model|journal=European Journal of Biochemistry | language=en | volume=4|issue=1 |pages=79–86 |doi=10.1111/j.1432-1033.1968.tb00175.x | pmid=4230812|issn=1432-1033|doi-access=free}}</ref>
* The daily oscillations in gene expression, hormone levels and body temperature of animals, which are part of the [[circadian rhythm]],<ref>{{Cite journal|last1=Leloup|first1=Jean-Christophe|last2=Gonze|first2=Didier|last3=Goldbeter|first3=Albert| date=1999-12-01|title=Limit Cycle Models for Circadian Rhythms Based on Transcriptional Regulation in Drosophila and Neurospora| journal=Journal of Biological Rhythms|language=en|volume=14|issue=6|pages=433–448| doi=10.1177/074873099129000948| pmid=10643740|s2cid=15074869|issn=0748-7304}}</ref><ref>{{Cite journal|last1=Roenneberg|first1=Till|last2=Chua|first2=Elaine Jane|last3=Bernardo|first3=Ric| last4=Mendoza|first4=Eduardo|date=2008-09-09|title=Modelling Biological Rhythms|journal=Current Biology|volume=18|issue=17| pages=R826–R835| doi=10.1016/j.cub.2008.07.017|pmid=18786388|s2cid=2798371|issn=0960-9822|doi-access=free|bibcode=2008CBio...18.R826R }}</ref> although this is contradicted by more recent evidence.<ref>{{cite journal |last1=Meijer |first1=JH |last2=Michel |first2=S |last3=Vanderleest |first3=HT |last4=Rohling |first4=JH |title=Daily and seasonal adaptation of the circadian clock requires plasticity of the SCN neuronal network. |journal=The European Journal of Neuroscience |date=December 2010 |volume=32 |issue=12 |pages=2143–51 |doi=10.1111/j.1460-9568.2010.07522.x |pmid=21143668|s2cid=12754517 }}</ref>
* The [[Cell migration|migration]] of [[cancer cell]]s in confining micro-environments follows limit cycle oscillations.<ref>{{Cite journal|last1=Brückner|first1=David B.|last2=Fink|first2=Alexandra|last3=Schreiber|first3=Christoph| last4=Röttgermann|first4=Peter J. F.|last5=Rädler|first5=Joachim| last6=Broedersz|first6=Chase P.|date=2019|title=Stochastic nonlinear dynamics of confined cell migration in two-state systems|journal=Nature Physics|language=en|volume=15| issue=6| pages=595–601| doi=10.1038/s41567-019-0445-4| issn=1745-2481|bibcode=2019NatPh..15..595B|s2cid=126819906}}</ref>
* Some non-linear [[Electrical Circuit|electrical circuits]] exhibit limit cycle oscillations,<ref>{{Cite journal| last1=Ginoux| first1=Jean-Marc| last2=Letellier|first2=Christophe|date=2012-04-30|title=Van der Pol and the history of relaxation oscillations: Toward the emergence of a concept|journal=Chaos: An Interdisciplinary Journal of Nonlinear Science|volume=22|issue=2| pages=023120| doi=10.1063/1.3670008|pmid=22757527|issn=1054-1500|arxiv=1408.4890|bibcode=2012Chaos..22b3120G|s2cid=293369}}</ref> which inspired the original [[Van der Pol oscillator|Van der Pol model]].
*The control of respiration and hematopoiesis, as appearing in the [[Mackey-Glass equations|Mackey-Glass]] equations.<ref>{{Cite journal|last1=Mackey|first1=M.|last2=Glass|first2=L|date=1977-07-15|title=Oscillation and chaos in physiological control systems |url=https://www.science.org/doi/10.1126/science.267326|journal=Science|language=en|volume=197 |issue=4300 |pages=287–289|doi=10.1126/science.267326|pmid=267326|bibcode=1977Sci...197..287M|issn=0036-8075|url-access=subscription}}</ref>