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一个不变流形定理
A Result of Invariant Manifolds and Its Applications

张伟年
Zhang Weinian

北京大学
(Peking University

收稿日期: 1990-06-25
出版日期: 2015-05-15

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摘要 对于微分方程的不变流形的研究,除了Coddington与Levinson,Hale,Kelley及Carr等给出了稳定(不稳定)流形、中心流形、中心稳定(不稳定)流形的结果外,这些理论还被well、Hale和Chow等推广到Banaeh空间的流或半流上。在分枝理论中,中心流形理论对处理高维动力系统的局部分枝起到了关键性作用,但对于高维系统的全
Abstract:In this paper, using the uniform contraction mapping principle we prove the existence of the Weakly Hyporbolic Invariant Manifolds (abbreviated to WHIM) for some high-dimensional dynamical systems. Unlike all known invariant manifolds, restricted in the WHIM the vector field maybe contains not only an expansive part but also a contractive part. In other word, maybe it is structurally stable at the equilibrium locally. We try to reduce a high-dimensional system to a lower-dimensional one in the WHIM to discuss conveniently its homoclinics and other singular cases, which provides a new idea to the study of global bifurcations of high-dimensional systems. In this paper we not only put forward this new idea about the WHIM but also gives a verifiable condition to guarantee its existence, C1-smooth structure and its global property. Also its exponential attractivity is discussed. As its applications an estimation of the range for centre-manifold to be represented explicitly and a condition of reducing a high-dimensional dynamical system with homocli-nics are given.
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