Limit Cycle JavaScript Simulation Applet HTML5
This simulation (link here) 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.
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).
