Supercritical Behavior of Disordered Orbits of a Circle Map

Abstract

Supercritical behavior of the circle map x_n+1=x_n+A sin (2πx_n)+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 ν, 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 ν. 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.

Citing Article(s):

1. Journal of the Physical Society of Japan 60 (1991) pp. 718–719:Quasiperiodic Route to Chaosin a Coupled Logistic Map Kazuhiro Satoh

2. Progress of Theoretical Physics Vol. 76 No. 1 (1986) pp. 302–304:Oscillations of Torus and Collision Torus-Chaos in a Delayed Circle Map Claudio Franciosi

3. Progress of Theoretical Physics Vol. 78 No. 1 (1987) pp. 51–71:Memory Dynamics in Asynchronous Neural Networks Ichiro Tsuda, Edger Koerner and Hiroshi Shimizu