Basin of attraction

The animation in this page shows the changes in shape of the basin of attraction for the dynamical system :

   x = x*y+a*x-y     
   y = x+y

as the parameter a varies from 0.4 to 0.75.

The basin of attraction is the set of all initial points in phase space that are drawn to the attractor. The basin boundaries are sometime smooth and sometime they are fractals. By coloring the points just outside the basin according to the number of iterations required for them to leave some region surrounding the attractor (in this case a square) you can produce the so called escape-time fractals.