Supercritical Behavior of Disordered Orbits of a Circle Map

Supercritical behavior of the circle map Xn+l =Xn+ A sin(2JZ"xn)+ D is investigated. The windows show the similarity in the parameter space (A, D). The critical phenomena of the width of the windows are characterized by the exponent II, which represents the speed of the collapse of a torus for a given irrational rotation number. Its value is well explained by the RG theory which was originally invented by Feigenbaum et al. and Rand et al. for the subcritical behavior. Next, the notion of "disordering" is introduced to characterize chaotic orbits. The distribution of disordering times is calculated with the use of the induced maps. The distribution shows an exponential decay. The ratio of the decay is related to the instability of unstable cycles. The scaling of the decay is also represented by the exponent II. A conjecture is proposed that the golden mean torus is the first KAM to collapse. Lastly, the period· adding sequence .near the crisis and its scaling behavior are studied in the Appendix.