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
本研究通过随机采样和延拓方法，量化了Duffing--Holmes振子中反周期、周期和混沌吸引子的吸引域相对体积，揭示了全局反周期主导窗口与共存区间，并解释了扫描分支对初始条件的依赖。
AI 中文摘要
周期驱动的Duffing--Holmes振子具有强烈的多稳定性：在相同的强迫振幅和频率下，周期、混沌和反周期吸引子共存，每个吸引子都有其自身的吸引域。反周期轨道是周期轨道，在平移半个驱动周期并同时将坐标取反（这是运动方程的一个对称性）后保持不变。在参数平面的延拓图的引导下（该图定位了各个区域，但无法揭示其共存性），我们沿着两条单参数截线（一条固定强迫频率，另一条固定强迫振幅）探索了初始条件空间。对完整空间进行随机采样，可以得到每个吸引域相对体积作为控制参数函数的无偏估计，采样所用的参数网格与通过延拓计算单轨迹分岔图时所用的网格相同。二维截面上的吸引域图，辅以从每个初始条件到达的吸引子阶数图，解析了吸引子之间竞争最强烈的共存吸引域的几何结构。相对体积揭示了全局反周期主导的窗口，在这些窗口中，反周期轨道几乎吸引所有初始条件，这些窗口被两个或三个性质不同的吸引子以相当比例共享空间的窄区间隔开，同时也揭示了持续共存的广泛范围；在大的强迫振幅下，一个单一的一阶反周期轨道捕获了整个空间。与分岔图相比，吸引域体积的突变是扫描过程中多稳定性引起的间断性的系综层面上的对应物，而反周期吸引域的逐渐缩小解释了为什么扫描所跟随的分支取决于初始状态和驱动相位。
英文摘要
The periodically driven Duffing--Holmes oscillator is strongly multistable: for the same forcing amplitude and frequency, periodic, chaotic, and antiperiodic attractors coexist, each with its own basin of attraction. Antiperiodic orbits are periodic orbits invariant under a shift of half a driving period combined with a sign reversal of the coordinates, a symmetry of the equations of motion. Guided by a continuation map of the parameter plane, which locates the regimes but is blind to their coexistence, we explore the space of initial conditions along two one-parameter cuts, at fixed forcing frequency and at fixed forcing amplitude. Stochastic sampling of the full space yields an unbiased estimate of the relative volume of each basin as a function of the control parameter, on the same parameter grids on which single-trajectory bifurcation diagrams are computed by continuation. Basin maps on two-dimensional sections, complemented by maps of the order of the attractor reached from each initial condition, resolve the geometry of the coexisting basins where the competition between attractors is strongest. The relative volumes reveal windows of global antiperiodic dominance, in which the antiperiodic orbit attracts essentially every initial condition, separated by narrow intervals in which two or three qualitatively different attractors share the space in comparable proportions, as well as extended ranges of persistent coexistence; at large forcing amplitude, a single antiperiodic orbit of order one captures the whole space. Compared with the bifurcation diagrams, the abrupt changes of basin volume are the ensemble-level counterpart of the multistability-induced discontinuities of the sweeps, and the gradual shrinking of the antiperiodic basin explains why the branch followed by a sweep depends on the initial state and on the drive phase.
Comments16 pages
