Symmetry Structure of the Period-Doubling Bifurcation of the Period-2, Step-1 Accelerator Mode in the Standard Map

The period-2, step-1 accelerator mode in the standard map undergoes a period-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 dominant 5-cycles around the central island at A = 0.6500, the boundary region displays 7-cycles inside 12-cycles at A = 0.6506. Then, the principal island undergoes a period-doubling bifurcation around A = 0.6515, although the surrounding boundary islands retain the characteristic structure of the original period-2 mode. After the completion of the period-doubling bifurcation around A = 0.6520, the principal island splits into a period-4 mode, and the surrounding higher-order periodic islands become satellites of the period-doubled mode. Analysis of the higher-order symmetry structure reveals that such intricate processes are associated with the splitting of the principal island with accompanying boundary islands formation.