At the same time as the phenomenon of torus breakdown, there is also the possibility of frequency locking i.e. the internal frequency of the dynamics becomes locked to some rational ratio of the strobe frequency of the map. For example for c=0.62 and returning to a=0.8:
the frequency becomes locked to 0.625 i.e. 5/8.
Although the attractor consists just of discrete points, the underlying continuous curve still has significance since it is invariant under the evolution i.e. we can construct a smooth curve such that for any point on the curve as an initial condition for the dynamics, all iterated points also lie on the curve. This corresponds to the "invariant torus" in the 3-dimensional flow. We can approximately construct the invariant curve by studying the transient evolution to the discrete points, since the collapse onto the curve is more rapid than the collapse to the discrete points. Here is the same evolution where only 10 points of the transient are eliminated before plotting: click on the running plot to start new initial conditions, and the invariant curve will become clear.
Increasing a to a=1.2:
and the frequency becomes locked to a ratio of smaller numbers, 2/3.