双频标准映射不变环面的破裂

Breakup of Last KAM Torus in Double-Frequency Standard Map

利用旋转数方法以及周期轨道方法确定了双频标准映射中最后一个不变环面破裂的频率为(73+13(5^(1/2)))/(309+55 (5^(1/2))),属于黄金分割频率族,其在连分数展开下具有[4,4,3,3,1,…,1,…]的形式.该环面破裂的临界摄动参数约为0.4476.与单频标准映射相比,增加激发频率可以导致最后一个环面的临界摄动参数减小,使得系统的全局不稳定性增加,一些轨道更容易扩散到相空间作用量较大的地方.

The last KAM torus in the double-frequency standard map under perturbation is investigated with the methods of rotation number and periodic orbits. It has frequency （73 ＋ 13√5）/（309 ＋55√5） , which belongs to the Golden mean group, and has the form of [ 4,4,3,3,1, ..., 1, ... ] under continuation faction expansion. The critical perturbation parameter is roughly 0. 447 6 for this tori. As compared with the last KAM torus in the standard map, the additional excitation frequency decreases the critical perturbation parameter, which increases the instability of the system, thus some orbits can be easier to diffuse away in the phase space.