分段线性双折叠正则化中滑动环的环数
Cyclicity of sliding cycles in regularizations of piecewise linear two-folds
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中文总结 AI 辅助
本文研究分段线性双折叠正则化中滑动环的环数，通过慢散度积分零点与辅助系统极限环的关联，证明环数以二为界，并给出可见折叠存在两个极限环的充要条件。
中文摘要 AI 辅助
我们研究了具有一般双折叠奇点的分段线性向量场的Sotomayor-Teixeira正则化所产生的极限环。我们关注可见-可见和可见-不可见情形，其中会出现滑动环，并以统一方式处理来自切换流形两侧的环。进一步地，通过将滑动环的环数与慢散度积分的零点联系起来，我们证明了当该积分不恒为零时，紧致滑动环族的环数以二为界。与以往工作不同，我们不进行逐例研究，而是将慢散度积分的零点与适当定义的辅助分段线性系统的穿越极限环联系起来。对于可见折叠，我们给出了确保慢散度积分存在唯一简单零点的充分必要条件，这蕴含存在两个极限环。我们还表明，当滑动向量场的双曲奇点位于滑动段的边界时，环数以一为界。
英文摘要
We study limit cycles produced by Sotomayor-Teixeira regularizations of piecewise linear vector fields with generic two-fold singularities. We focus on the visible-visible and visible-invisible cases where sliding cycles occur, treating cycles from both sides of the switching manifold in a unified way. In further details, by relating the cyclicity of sliding cycles and zeros of the slow divergence integral, we prove that the cyclicity of compact families of sliding cycles is bounded by two when such integral does not vanish identically. In contrast to previous works, we do not perform a case-by-base study, but instead relate zeros of slow divergence integrals to crossing limit cycles of a suitably defined auxiliary piecewise linear system. For visible folds, we provide necessary and sufficient conditions that assure the existence of a unique simple zero of the slow divergence integral, which implies the existence of two limit cycles. We also show that, when a hyperbolic singularity of the sliding vector field lies at the boundary of the sliding segment, then the cyclicity is bounded by one.
发表机构
- Hasselt University（哈瑟尔特大学）
- Technical University of Denmark（丹麦技术大学）
- University of São Paulo (USP)（圣保罗大学）
- Universidade de Brasília（巴西利亚大学）
机构由 AI 辅助整理，请以论文原文为准。
