Once the gain exceeds 2, the intersection of the function curve with the y=x line moves out to the right beyond the peak of the function curve. This open up the possibility that iterating with x starting in the interval below the intersection may generate a y which will take the next iteration out beyond the intersection. Once you have escaped from the loop between the function curve and the y=x line, other possibilities appear. The relative slopes of the function curve and the y=x line become important.
The slope of a line is defined as a small change in y along the line divided by the corresponding change in x. The slope of the y=x line then is 1, since everywhere along that straight line a small change in y is equal to the corresponding change in x. The slope of the function curve in the vicinity of the intersection with the y=x line is going to be negative for gains above 2 since y decreases with increasing x. The steepness of that negative slope is the critical issue as far as the nature of the attractor at our chosen gain is concerned.
As long as change in y is less in each successive iteration the function still closes in on the y value at the intersection. This condition is met as long as the magnitude of the negative slope of the function curve is less than that of the y=x line, that is with slopes of absolute value less than 1. The slope of the function curve at the intersection with the y=x line becomes steeper as the intersection moves out to the right, as it does with increasing gain. Once this slope becomes steeper than -1, the change in y near the intersection increases.
The attractor for the function then is single valued for gains such that the slope of the function curve is less steep than -1 at the intersection of the curve with the y=x line. You may calculate the value of gain where the slope becomes more steep than -1 with the application of a little calculus. Even better for our purposes is to experiment with the graphics display, exploring the gains between about 2.70 and 3.25 where the effects we have been discussing are illustrated. Notice that with gain less than 3, iterations home in on a single valued attractor.