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.

The initial value conditions in the XY-plane can be set by using two sliders and fields. Press the "Initialize", then "Play" button to start the simulation. The model trajectory is displayed. Alfonso Urquía and Carla Martín
Dpto. de Informática y Automática
E.T.S. Ingeniería Informática, UNED
Juan del Rosal 16, 28040 Madrid, Spain  

Translations

Code	Language		Translator	Run

Credits

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

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:

1. http://weelookang.blogspot.com/2018/05/limit-cycle-javascript-simulation.html 
   
2. 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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