Persistence of Minimal Invariant Sets for Certain Set-Valued Dynamical Systems: A Boundary Mapping Approach
Persistence of Minimal Invariant Sets for Certain Set-Valued Dynamical Systems: A Boundary Mapping Approach
复制标题
某些集值动力系统的最小不变集的持久性：边界映射方法
DOI：
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发表时间：
2023
期刊：
影响因子：
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通讯作者：
D. Turaev
中科院分区：
文献类型：
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作者：
K. Kourliouros;J. Lamb;M. Rasmussen;W. H. Tey;K. Timperi;D. Turaev
我们研究了一类离散集值动力系统的具有光滑边界的最小不变集的持久性问题，这类系统自然产生于具有有界噪声的随机动力系统。特别是，我们引入了一个单值映射，所谓的边界映射，它具有的属性，这一映射的某类不变子流形是在一对一的对应与不变集相应的集值映射。我们证明了具有光滑边界的极小不变集在集值映射的小扰动下持续存在，只要相关的边界映射在边界的单位法丛处通常是双曲的。
We study the problem of persistence of minimal invariant sets with smooth boundary for a certain class of discrete-time set-valued dynamical systems, naturally arising in the context of random dynamical systems with bounded noise. In particular, we introduce a single-valued map, the so-called boundary map, which has the property that a certain class of invariant submanifolds for this map is in one-to-one correspondence with invariant sets for the corresponding set-valued map. We show that minimal invariant sets with smooth boundary persist under small perturbations of the set-valued map, provided that the associated boundary map is normally hyperbolic at the unit normal bundle of the boundary.