AI 中文总结
本文对一类对称零散度三维分段线性Filippov系统，在可见-可见双折点情形下，给出了振荡动力学的全局分类，证明了简单周期穿越环的唯一性、稳定性及扰动下的持续性。
AI 中文摘要
本文针对一类对称零散度的三维分段线性Filippov系统，建立了渐近振荡动力学的全局分类，分类结果在除去零Lebesgue测度的初始条件集合后成立。仿射向量场通过一个对合变换相互关联，切换平面包含一个可见-可见双折点。在规范坐标下，特征值为\\(\mu\pm i\\)和\\(-2\mu\\)，而\\(H\\)度量焦点线的倾斜程度。对于每个\\(\mu>0\\)，简单周期穿越环存在当且仅当\\(H\in\mathcal I_\mu\\)，且该环是唯一的、对称的、双曲的，并且轨道渐近稳定。其半周期参数化\\(\mathcal I_\mu\\)，并决定了穿越点、周期和Floquet乘子。全局耗散性排除了具有更高穿越次数的穿越环，并使得所分类的环成为每条仅穿越轨道的\\(\omega\\)-极限集。若吸引滑动没有内部伪平衡点，则滑动是瞬态的，除非轨迹到达双折点。导致到达双折点的初始条件位于可数个解析曲面和曲线的并集中。在Whitney \\(C^1\\)拓扑下的足够小扰动下，该环作为唯一的简单周期穿越极限环持续存在，并保持双曲性和轨道渐近稳定性。
英文摘要
A global classification of the asymptotic oscillatory dynamics is established for a symmetric zero-divergence class of three-dimensional piecewise-linear Filippov systems, up to a set of initial conditions of zero Lebesgue measure. The affine fields are related by an involution, and the switching plane contains a visible--visible two-fold. In canonical coordinates, the eigenvalues are \(μ\pm i\) and \(-2μ\), while \(H\) measures the focal-line inclination. For every \(μ>0\), a simple-period crossing cycle exists if and only if \(H\in\mathcal I_μ\), and is unique, symmetric, hyperbolic, and orbitally asymptotically stable. Its half-period parametrizes \(\mathcal I_μ\) and determines the crossing points, period, and Floquet multipliers. Global dissipation excludes crossing cycles with any higher number of crossings and makes the classified cycle the \(ω\)-limit set of every crossing-only trajectory. If attractive sliding has no interior pseudo-equilibria, sliding is transient unless the trajectory reaches the two-fold. The initial conditions leading to the two-fold lie in a countable union of analytic surfaces and curves. Under sufficiently small perturbations in the Whitney \(C^1\) topology, the cycle persists as the unique simple-period crossing limit cycle and remains hyperbolic and orbitally asymptotically stable.
