具有大滞回的平面控制模型中的极限集与全局分岔结构
Limit Sets and Global Bifurcation Structure in Planar Control Models with Large Hysteresis
- IBILCE–UNESP（伊比拉塞特学院 - 圣保罗州立大学）
- UFSCar（联邦圣卡洛斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过显式解与几何方法，完整刻画了具有大滞回切换的平面控制模型的极限集，并揭示了其全局分岔结构，包括周期轨道演化及动力学状态转变。
AI中文摘要：
本文研究了一个可能引起广泛受众相当兴趣的问题，因为这里所考虑的系统根据一种涉及两个不同动力学状态的切换协议运行。从初始条件出发，演化遵循第一个向量场，直到选定的状态变量$y$达到下阈值$C_1$。此时，动力学切换到第二个向量场。第二个状态保持活跃，直到同一变量达到上阈值$C_2>C_1$，此时第一个向量场被恢复。这种交替过程随后无限重复，产生一个分段光滑向量场。对于每个允许的参数组合和所有初始条件，获得了$\u03c9$-极限集的完整刻画。该分析通过将向量场的显式解与几何论证和首次返回映射相结合来进行。除了极限集的分类之外，本文还描述了该族的全局分岔结构。随着参数变化，系统在不同渐近状态之间经历定性转变，包括周期轨道的产生和消失、其稳定性的变化、在退化情形中周期轨迹连续统的出现，以及有界动力学被单调锯齿运动或无界轨迹所取代。相应的分岔图提供了模型渐近动力学的完整定性描述。
英文摘要：
The present paper addresses a problem that may be of considerable interest to a broad audience since the systems considered here operate according to a switching protocol involving two distinct dynamical regimes. Starting from an initial condition, the evolution follows a first vector field until a selected state variable $y$ reaches a lower threshold $C_1$. At this moment, the dynamics switches to a second vector field. The second regime remains active until the same variable attains an upper threshold $C_2>C_1$, when the first vector field is restored. This alternating procedure is then repeated indefinitely giving rise to a piecewise smooth vector field. A complete characterization of the $ω$-limit sets is obtained for every admissible combination of parameters and all initial condition. The analysis is carried out by combining explicit solutions of the vector fields with geometric arguments and the first return map. Beyond the classification of limit sets, the paper describes the global bifurcation structure of the family. As the parameters vary, the system undergoes qualitative transitions between distinct asymptotic regimes, including the birth and disappearance of periodic orbits, changes in their stability, the occurrence of continuum of periodic trajectories in degenerate situations, and the replacement of bounded dynamics by monotone zig-zag motions or unbounded trajectories. The corresponding bifurcation diagrams provide a complete qualitative description of the asymptotic dynamics of the model.
