Symmetry structure of the period-doubling bifurcation of the period-2, step-1 accelerator mode in the standard map

The period-a, step-1 accelerator mode in the standard map undergoes a perio d-doubling bifurcation when the stochastic parameter A exceeds the value 0. 6512. The structural transition observed in the boundary islands is studied thoroughly using higher-order symmetry analysis. After the formation of do minant Ei-cycles around the central island at A = 0.6500, the boundary regi on displays 7-cycles inside 12-cycles at A = 0.6506. Then, the principal is land undergoes a period-doubling bifurcation around A = 0.6515, although th e surrounding boundary islands retain the characteristic structure of the o riginal period-2 mode. After the completion of the period-doubling bifurcat ion around A = 0.6520, the principal island splits into a period-4 mode, an d the surrounding higher-order periodic islands become satellites of the pe riod-doubled mode. Analysis of the higher-order symmetry structure reveals that such intricate processes are associated with the splitting of the prin cipal island with accompanying boundary islands formation.