Basin of Attraction example

Let's go for an advanced example of Pulga's basin of attraction feature.

Twice iterated henon

First we configure Henon's Map as a twice iterated function:

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

           f(x1, x2) = (1 - 1.2 * x1 * x1 + 0.2 * x2, x1)

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

           x1_k+1 = 1 - 1.2 * x1_k * x1_k + 0.2 * x2_k
           
           x2_k+1 = x1_k

   1. So configure the iteration to

              x1 = 1 - 1.2 * x1 * x1 + 0.2 * x2
              
              x2 = x1

   2. Set the iteration power to 2 (twice iterated), this tells Pulga to create iterate a f^2 instead of f.
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]

Basin of Attraction plugin

Activate the basin of attraction plugin

Average Functions

Now we configure the basic plugin options:

1. We pick two average functions that will be executed for different random conditions. The average result is placed in R^2 and differente values will be displayed to us later on.

           first_average(x1,x2) := Math.abs(x1)*Math.abs(x2)
           
           second_average(x1,x2) :=        x1 * x1 + x2 * x2

2. Let's configure the escape time: 0 means that it will run the whole iteration before deciding what that average means.

   Let's change it to 50 so that every 50 iterations it checks what that average means and if there is some significant meaning, it stops.

   (As we shall see, this speeds up the process of drawing basin of attractions but might lose much details in the resulting graphic).

Get the average samples

1. Now open the average sampler in the basin of attraction menu.
2. Start/restart the sampling process.
3. You will see four small 'clouds' forming on the screen.
4. Change the scale values so you can see them more clearly.
5. Stop the sampling process but do not close the sampling window

Create some clouds

Now let's define those four clouds as basins.

1. Click new and select (clockwise) four points around one of those clouds.
2. Right click to finish defining this cloud.
3. Select the created cloud in the cloud selection list (at the left side of the screen) and click edit.
4. Change its name to something meaningfull and leave its color as blue.

   Repeat the above process once for each other 'cloud', picking red, orange and green colors.

   If the average 'stays' at one of those clouds, it has been 'captured' by this basin and will be painted accordingly.

   Now let's create an attractor at infinity:

5. Create a huge cloud that covers all four clouds...

   Change it to reverse mode and pick a light blue color: if the average stays outside the delimited area, it is defined to be at this basin.

Executing

Now close the average sampler and execute the basin of attraction plugin!