Universität Hamburg - Fachbereiche - Fachbereich Mathematik

Explanation of the Applet Period Doubling

It is a well known fact that the scalar logistic map f(x)=a*x*(1-x) undergoes a period doubling bifurcation if parameter a crosses a0=3.

After having started the applet, a visualization of the fixed point iteration is shown which is called "Spinnwebeniteration" with nIt iterations, initiated in (x,x).

[xMin,xMax] x [yMin,yMax] represents the window in the (x,y)-plane. It can be interactively changed.

The user can either change the parameter a directly in the textfield or by using the scrollbar. A click on the button draw again initiates an iteration. A repeated click on the button PD-scenario gnerates different parameters a before and after the period doubling bifurcation at a0=3. Alternatively, you may also use the scrollbar.

In any situation one can choose the starting point x just by a mouse click in the plane immediately followed by a simulation.

Additionally the fixed point and the slope of the function at this point are shown. For a<3 the fixed point is attractive due to the contraction of f nearby. But for a>3 f is expansive, the fixed point is repulsive, and a period 2 orbit is borne.

The user may try to coose a=3.9 with chaotic behavior. Can you find a such that 3-periodic orbits show up?