Chapter 19

where we can collapse values of x to the range 0 ≤ x ≤ 1, is sufficiently simple that much is known rigorously [1], yet displays the interplay between frequency locking and the onset of chaos from quasiperiodic motion. There are also a number of excellent reviews e.g. references [2] and [3]. The map is monotonic and invertible for K < 1, but develops an inflection point at x = 0 for K = 1 and then is nonmonotonic and noninvertible for K > 1 (demonstration 1). An important quantity is the winding number W defined as