无穷维随机系统法向双曲的保持性及相关问题

It is said that an invariant manifold for a dynamical system is normally hyperbolic if the tangent bundle of the phase space restricted to the invariant manifold has an invariant decomposition with three subbundles and corresponding to the decomposition the tangent map has an exponential trichotomy, where the direct sum of the stable subbundle and the unstable subbundle is a normal bundle of the invariant manifold and the center subbundle is the tangent bundle of the invariant manifold. The problem of the persistence of normally hyperbolic invariant manifold includes two aspects, where one is the problem of the persistence of the invariant manifold and the other is the problem of the persistence of the normal hyperbolicity. There is an extensive literature on the persistence of normally hyperbolic invariant manifolds for autonomous systems under autonomous perturbations not only for finite dimensional cases but even for infinite dimensional ones, which is applied to study the existence of homoclinic orbits. Recently, efforts have been made to extend the result for autonomous system under random perturbations for finite dimensional cases. In this subject, we study on the problems of the persistence of normally hyperbolic invariant manifold for a system under random perturbations and the existence of random stable manifold, random unstable manifold, random stable foliation and random unstable foliation of a random normally hyperbolic invariant manifold for infinite dimensional cases, and give a sufficient condition of the existence of homoclinic orbits for a infinite dimensional random singularly perturbed system by using the new results. We do not assume the invertibility of the tangent map, which prevents us from using the approach given for the finite dimensional random case. Due to the essential feature of random dynamical systems, the problem of the measurability of the persisted invariant manifold and the measurability of the decomposition for the persisted normal hyperbolicity need to be answered. This feature prevents us from using the approach given for infinite dimensional autonomous case. Our approach is to translate the problem of measurability of the decomposition to the one of measurability of a tuple of projections and extend the roughness of exponential dichotomy, the admissibility and related results for infinite dimensional random dynamical systems to solve the problems.

法向双曲性是沿着一个不变流形在其法丛上建立的双曲性。法向双曲的保持性问题涉及在扰动下该不变流形的保持性以及法丛上双曲结构的保持性等两个方面。前人对有限维情形甚至无穷维情形的自治系统加自治扰动给出了这两种保持性结果，并用于研究同宿轨的存在性。近年来，这两种保持性在有限维情形已对自治系统加随机扰动给出了结果。本课题将针对无穷维随机系统研究这两种保持性。我们将克服无穷维情形基本解算子的不可逆性问题和随机情形不变流形和不变分解的可测性问题同时带来的困难，将集合的可测性问题转化为算子的可测性问题，通过发展无穷维随机系统指数二分的粗糙性及相关结论, 从而解决双曲分解的可测性问题。通过发展无穷维随机系统指数二分的相容性及相关结论，进而在随机法向双曲结构下研究随机稳定流形、随机不稳定流形、随机稳定叶层和随机不稳定叶层的存在性，并用于讨论无穷维随机奇异扰动系统的同宿轨问题。

按照原计划，本项目研究无穷维系统的法向双曲不变流形在扰动下的保持性及其相关问题。. 法向双曲的保持性问题涉及在扰动下该不变流形的保持性以及法丛上双曲结构的保持性等两个方面。前人对有限维情形甚至无穷维情形自治系统的一致法向双曲不变流形在自治扰动给出了这两种保持性结果。近年来，这两种保持性在有限维情形已对自治系统的一致法向双曲不变流形在随机扰动给出了结果。本课题研究了无穷维自治系统的一致法向双曲紧的不变流形在随机扰动下以及无穷维非自治系统的非一致法向双曲非紧的不变流形在扰动下的这两种保持性。针对无穷维自治系统的一致法向双曲紧的不变流形在随机扰动下的保持问题，我们通过可测伪轨跟踪和可测指数二分来克服无穷维情形基本解算子的不可逆性和随机情形不变流形和不变分解的可测性同时带来的困难，证明了该随机扰动系统具有随机法向双曲不变流形，并证明了该随机法向双曲不变流形具有随机稳定流形和随机不稳定流形。针对无穷维非自治系统的非一致法向双曲非紧的不变流形在扰动下的保持性问题，我们利用近似指数二分和伪轨方法克服了非一致性带来的困难，证明了该扰动系统具有非一致法向双曲不变流形。. 此外，我们还研究了特殊的法向双曲不变流形的相关性质，如中心流形和极限环等。我们对薄的变区域上带有色噪声驱动的随机抛物方程证明了其局部随机中心流形的C^{1,ν}收敛性；以及分别对一类H^{3}中心加扰动的平面退化系统和一类具有捕食者捕食合作的Leslie-Gower系统给出了系统的稳定极限环和不稳定极限环存在的充分条件。

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