### About

This simulation illustrates the concept of a **limit cycle** by using
the following mathematical model:

*dx/dt = y + [K*x*(1 - x^2 - y^2)]/sqrt(x^2 + y^2) *

*dy/dt = -x + [K*y*(1 - x^2 - y^2)]/sqrt(x^2 + y^2)*

The initial conditions are as follows:

*x(0) = x0 *

*y(0) = y0*

The **limit cycle** of this function is a circle centered at the
origin with radius 1 (the unit circle), which can be expressed in the
following statement.

*For all x0,y0 (where x0, y0 are non-zero), x^2 + y^2 approaches 1
as t tends to infinity.*

**Initialize**", then "

**Play**" button to start the simulation. The model trajectory is displayed.

Dpto. de Informática y Automática

E.T.S. Ingeniería Informática, UNED

Juan del Rosal 16, 28040 Madrid, Spain

### Translations

Code | Language | Translator | Run | |
---|---|---|---|---|

### Software Requirements

Android | iOS | Windows | MacOS | |

with best with | Chrome | Chrome | Chrome | Chrome |

support full-screen? | Yes. Chrome/Opera No. Firefox/ Samsung Internet | Not yet | Yes | Yes |

cannot work on | some mobile browser that don't understand JavaScript such as..... | cannot work on Internet Explorer 9 and below |

### Credits

Alfonso Urquía; Carla Martín; Tan Wei Chiong; Loo Kang Wee

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### Sample Learning Goals

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### For Teachers

This simulation illustrates the concept of a limit cycle by using the following mathematical model:

dx/dt = y + [K*x*(1 - x^2 - y^2)]/sqrt(x^2 + y^2)

dy/dt = -x + [K*y*(1 - x^2 - y^2)]/sqrt(x^2 + y^2)

The initial conditions are as follows:

x(0) = x0

y(0) = y0

The limit cycle of this function is a circle centered at the origin with radius 1 (the unit circle), which can be expressed in the following statement.

For all x0,y0 (where x0, y0 are non-zero), x^2 + y^2 approaches 1 as t tends to infinity.

In essence, for any point on the Cartesian plane, it will eventually approach the limit of the unit circle, no matter where the point is.

The graph on the left illustrates the path of the point, while the graph on the right illustrates the graph of y against t (blue) and x against t (red).

Research

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### Video

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### Version:

- http://weelookang.blogspot.com/2018/05/limit-cycle-javascript-simulation.html
- http://www.euclides.dia.uned.es/simulab-pfp/curso_online/cap7_caseStudies/leccion.htm by Alfonso Urquia and Carla Martin-Villalba

### Other Resources

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