发表机构
Facultad de Ciencias, Universidad de la República; Departamento de Física, Universidade Estadual Paulista (UNESP)（共和国大学理学院; 圣保罗州立大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究周期驱动的杜芬-霍姆斯振子中反周期轨道的起源与分布，结合解析与数值方法，建立反周期轨道存在条件，揭示其与周期轨道的关系及宇称选择规则，表明反周期性是驱动系统离散对称性的轨道层面体现。
AI 中文摘要
我们结合解析论证与广泛的数值探索，研究了周期驱动的杜芬-霍姆斯振子中反周期性（满足\(x(t + T)= -x(t)\)的振荡）的起源和分布。首先建立了非平凡反周期轨道存在所需的关于非线性和对称性的最小条件，并描绘了反周期、周期和混沌区域在相空间和参数空间中的组织方式。反周期轨道恰好是在运动方程的半周期移位对称\(S:(x,\dot{x},t)\mapsto(-x,-\dot{x},t + T_d/2)\)（\(T_d\)为驱动周期）下保持不变的周期轨道。这种不变性强加了一个宇称选择规则，在参数扫描中无一例外得到验证：反周期轨道仅在强迫周期的奇数倍处锁定到驱动。缺乏反对称性的周期轨道以通过\(S\)相关的共轭对形式出现，每个轨道是其孪生轨道的点反射；在潜在分岔附近发生的自发对称性破缺选择了每对中的一个成员，而这一对整体恢复了每个轨道单独失去的对称性。因此，反周期性并非特定波形的偶然属性，而是驱动系统离散对称性在轨道层面的体现。
英文摘要
We investigate the origin and distribution of antiperiodicity --- oscillations satisfying $x(t+T)=-x(t)$ --- in the periodically driven Duffing--Holmes oscillator, combining analytical arguments with extensive numerical exploration. Antiperiodic orbits are precisely the periodic orbits invariant under the half-period shift symmetry $S:(x,\dot{x},t)\mapsto(-x,-\dot{x},\,t+T_d/2)$ of the equations of motion, with $T_d$ the driving period. We map the antiperiodic regions across the plane spanned by the amplitude and the frequency of the forcing, together with the periodic and chaotic domains and the potential wells visited by each orbit. The invariance under $S$ imposes a parity selection rule, verified without exception across our parameter sweeps: antiperiodic orbits lock to the drive only at odd multiples of the forcing period. Periodic orbits that lack the antisymmetry occur instead as conjugate pairs related by $S$, each orbit being the point reflection of its twin. We further show that antiperiodic orbits cannot bifurcate through a direct period doubling: the symmetry must break first, in a supercritical pitchfork in which the antiperiodic orbit splits into two conjugate, symmetry-broken orbits; alternatively, the antiperiodic orbit disappears with its symmetry intact, in a saddle-node bifurcation with an antiperiodic saddle. Antiperiodicity thus emerges as the orbit-level manifestation of a discrete symmetry of the driven system.
Comments16 pages
