Trajectory scaling functions for the circle map and the quasiperiodic route to chaos

There is a certain amount of arbitrariness in defining a trajectory scaling function for the circle map for the various quasiperiodic routes to chaos. We show how one may exploit this freedom to select trajectory scaling functions to suit a particular purpose. For comparison with experiment, one might seek a trajectory scaling function that is relatively flat, is smooth, clearly distinguishes between the subcritical and critical regimes of the quasiperiodic route to chaos, and is quite resilient to noise. To allow for experimental difficulties in obtaining (consecutive) exact cycles, we have also developed a trajectory scaling function that is constructed from subcycles within some higher-order cycle. We consider the quasiperiodic routes to chaos with golden-mean and silver-mean (2 -1) rotation winding numbers.