For nonzero a, e.g. a=0.8 the map becomes nonlinear. The frequency of the orbit is perturbed away from the value c as a changes. It turns out the behavior is highly sensitive to the value of the frequency, and so we keep track of this through the "winding number" - the average rate of rotation around the circle represented by the interval 0<x<1. As a is increased, we adjust the value of c to maintain the winding number at a fixed value around 0.618 (the Golden Mean). We will also accumulate the points more rapidly, focussing on the nature of the orbit rather than its development from individual points.
Although the orbit is distorted geometrically from a circle, it is still smooth and topologically equivalent to a circle, i.e. the motion on the map remains a limit cycle corresponding to quasiperiodic motion of the flow. You can check that one of the Lyapunov exponents is zero, and the other negative.