We found that the single-humped map of the interval, 
, was a good model for some aspects of the dynamics
of the bouncing ball system. Similarly, insight about motion near
a torus attractor can be gained
by studying a circle map ,
where the mapping g most often studied is a two-parameter map of the form
This map has a linear term 
,
a constant bias term 
, and a nonlinear term whose strength
is determined by the constant K.
The frequency of the circle map is monitored by the winding number
If the nonlinear term K equals zero, then 
.
The winding number measures the average increase in the angle
per unit time (Fig. 3.20).
Figure 3.20: The winding number measures the average increase in the
angle of the circle map.
An orbit of the circle map is periodic if, after q iterations,
, for integers p and q. The
winding number for a periodic orbit is W = p/q. A quasiperiodic
orbit has an irrational winding number [22]. Circle maps
have a devilish dynamical structure, which is
explored in reference [22].
