Set the gain to between 3 and 3.4. Here instead of homing in on a point, we home in on a box where the iterations to the left of the intersection and those to the right of the intersection produce equal changes in y. This gives rise to a double valued attractor. Next push the gain up to 3.48 and look for the attractor.
Notice that as the gain increases beyond about 3.45, the box representing stable iterations splits into two boxes. This results in an four-fold attractor. Continuing to increase the gain will split each of these boxes into two, resulting in an eight fold attractor and so on. This splitting occurs at a gain where the slope of higher iterates becomes steeper than -1 at the intersections with the y=x line. Try the "Action" button starting with higher base iterations than the zero case we have been working with, to explore this idea.