Examples

Here you find a simple tutorial on how to create some simple dynamical system's and iterate them on pulga.

Henon

1. The Henon's Map is the following R^2 --> R^2 application:

           f(x1, x2) = (1 - 1.4 * x1 * x1 + 0.3 * x2, x1)

2. Which can be translated to a simple iterated function:

           x1_k+1 = 1 - 1.4 * x1_k * x1_k + 0.3 * x2_k
           
           x2_k+1 = x1_k

3. Set the number of trash points to 100 and generated points to 20000.

Different initial conditions (x1,x2) will have different orbits. For example, try studying the starting point (0.3, 0.1).

What happens with this initial condition?

Try looking at x1 and x2 inside [-1.5,1.5]

Finding fixed points

1. Setup the Henon example.
2. Activate the plugin called ConnectPoints, it connects each two points between iterations.
3. Make the number of iterated points something smaller, like 50.
4. Activate the initial condition plugin, it displays the Initial Condition tool on the tool bar.
5. Whenever you select this tool you are able to pick diferent initial conditions which you want to iterate.
6. Pick this tool and choose some random point, do you see the iteration "path"?
7. Now try picking points closer to (-1.131354477,-1.131354477).
8. Do you see that it might tend to a fixed point?

You cannot be sure that there will be a fixed point somewhere in this region just by using this idea, but with some math intuition you will be able to isolate some areas which might have fixed points.