9  Parabola

The iteration of parabola equation
\(x_{n+1} = r \cdot x_n \cdot (1 \;– \; x_n )\)
converges, shows cycles or chaos depending on the parameter \(r\).

The Parabola Widget

9.1 Calculate

Shows the location of the limit points depending on the parameter \(r\).
Clicking with the left mouse button into the picture shows the parabola of the selected value of \(r\):
\(f(x) = r \cdot x \cdot (1 – x)\)

9.2 Parabola of a single parameter \(r\)

A single parabola with its iterations

A left mouse click starts the iteration within the parabola (initial value).
With middle mouse clicks the iteration can be continued step by step until the selected number of iterations is reached.
A right mouse click performs all iteration steps at once depending on the selected number of iterations.
If [cut first values] is switched on the initialization is cut off: only the limit point resp. cycle resp. chaos is shown.
If [time plot] is switched on the development of the iteration is shown in a separate window.

9.3 PS print/Print

A Postscript output is generated and stored in <Path to Store Directory>/plots.
The PS-files are named automatically.

9.4 Miscellaneous

The tool allows also the visualization of
\(x_{n+1} = x_n + r \cdot x_n \cdot (1 \; – \; x_n)\)
which is the discretization of the logistic differential equation
\(\dot x = x \cdot (1 \; –\; x)\)