我們研究了振盪磁場中的阻尼磁針系統，振盪磁場由垂直交錯的兩部分合成，一部分是方向大小固定的磁場B1（可以是地球磁場），另一部分是以正弦形式、振幅為B2的外加振盪磁場。磁針的擺動是複雜的非線性振盪，它會在外磁場振幅增加時，經由週期倍增路徑變成混沌的振盪。系統的運動方程式具有「角度反轉，同時時間平移二分之奇數倍個磁場振盪週期」的對稱性。因為這個對稱性，磁針振盪運動有對稱的週期和混沌吸引子共存。我們用數值方法解微分方程組和畫出相空間圖，檢查當系統的參數改變時，吸引子的性質如何隨參數改變，例如：成對對稱的吸引子是如何出現和合併。有趣的是，我們不只發現了具有相同週期的吸引子成對對稱地出現。在某些參數區間，彼此不對稱的兩個單數週期吸引子也可以共存，以及一對成對對稱的週期二吸引子和一個混沌吸引子的共存也被我們發現。;We consider a magnetic dipole (compass needle) under a constant magnetic (Earth′s) field and an external sinusoidally oscillating magnetic field (of magnitude B2) that is perpendicular to the former. The angular motion displays complex nonlinear oscillations and undergoes a period-doubling route to chaos. The equation of motion of the system possesses a special symmetry when angle inversion together with time translation of half of the driving period is applied. Due to this symmetry, coexistence of attractors, including symmetric periodic states and symmetric chaotic strange attractors, occurs. The properties of these attractors, such as how the symmetric attractor pairs appear and merge, as revealed by numerical solution of the differential equations and phase portraits, are examined in detail as the parameters of the system change. Interestingly, it is found that in addition to the coexistence of symmetric limit cycle attractor pair (both having the same period state), two different odd-periodic states not related by symmetry, can coexist. In addition, a pair of symmetric period-2 limit cycles and a chaotic attractor can coexist in certain parameter regimes.
