For a=0.5 the map is no longer integrable:
Start from various values of y as initial conditions to see what happens to the tori of demonstration 3 as a function of winding number. (The values of the y-coordinates are expressed as a fraction of the range set by ymin and ymax.) The tori corresponding to winding numbers near simple rationals such as 0, 1/2, 1/3, 1/4 ... break down to a ellipses about discrete points corresponding to the frequency locked states. If you look very carefully in the hyperbolic regions between two ellipses you should find chaotic orbits. (Expand these regions as necessary.)