随机动力系统中的Sacker-Sell谱与Lyapunov谱
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摘要
随机动力系统是一种斜积系统,它因为能更好的描述现实世界而引起人们越来越多的关注。本论文感兴趣的是随机动力系统中的指数二分性,Sacker-Sell谱,Lyapunov指数和随机吸引子等.指数二分性描述了系统的一种双曲性现象:状态空间可以分解成两个连续不变的子空间的直和,随着时间的正向变化,系统在其中一个子空间上表现出指数压缩行为,而在另一个上面表现出指数扩张行为。Sacker-Sell谱是基于指数二分性的一个概念,Sacker和Sell建立了Sacker-Sell谱理论.这个研究被Magalh(?)es,Sacker和Sell,Chicone和Latushkin,Chow和Leiva等推广到了无穷维动力系统中去,Cong和Siegmund还讨论了具有随机性的动力系统的Sacker-Sell谱问题.Lyapunov指数是研究动力系统渐进行为的基本工具之一,它反映了动力系统随时间演化的平均变化率。Oseledec的乘法遍历定理解决了Lyapunov指数的存在性问题,并对动力系统的动力学结构给出了更多的信息,它现在已成为动力系统理论的最基本定理之一。乘法遍历定理也被Ruelle,Ma(?)e,Thieullen,Zeng Lian和Kening Lu等学者进行了多种情形下的推广。关于Sacker-Sell谱与Lyapunov谱的关系,在有限维动力系统中,Johnson、Palmer和Sell证得了Lyapunov谱包含在Sacker-Sell谱中,而Sacker-Sell谱的边界又包含在Lyapunov谱中,并证得Oseledec谱子丛是Sacker-Sell谱子丛的加细。而后,Schreiber,Voutaz,Chicone和Latushkin等也进行了类似问题的研究.吸引子是微分方程理论和动力系统理论中一个极其重要的概念.在本论文中,我们感兴趣的是:一个紧致不变集在什么条件下可以成为一个吸引子。Ashwin在确定性系统里,利用法向Lyapunov指数讨论了这个问题.而后,他把确定性的结果推广到了随机动力系统中去,不过,他只是讨论了一个具体的随机动力系统的例子。
本论文主要研究了随机动力系统的Sacker-Sell谱理论,乘法遍历定理和随机吸引子问题.随机动力系统和确定性动力系统相比,它的底空间是一个没有任何拓扑结构的概率测度空间,这一点恰是从确定性系统到随机动力系统的一个本质困难之一,无论是对有限维的情形还是对无穷维的情形.我们克服这个困难,通过定义随机动力系统下的指数二分性,定义了随机动力系统的Sacker-Sell谱,并给出了有限维随机动力系统中的Sacker-Sell谱分解定理.在此基础上,我们比较了Sacker-Sell谱和Lyapunov谱,建立了有限维随机动力系统中两种谱的关系定理.我们也研究了无穷维的随机动力系统下的两种谱的关系。无穷维的随机动力系统和有限维的相比,其中的cocycle往往只能定义在正半时间轴上,为此我们首先对无穷维半动力系统进行了负向延拓,使得在负半时间轴上也有定义。另外,无穷维的情形还有一个难点,就是状态空间的有界的闭子集不一定是紧的。为此,我们研究了具有随机一致全连续性(紧算子是一致全连续算子的特殊情形)的随机动力系统,和具有更弱紧性(一致α-收缩性)的随机动力系统,其中用到了非紧性测度的概念.我们还给出并证明了一般的cocycle(不要求是紧算子)在可分的Banach空间的无穷维随机动力系统的乘法遍历定理,所采用的证明是基于Ma(?)e和Thieullen的方法。最后,我们讨论了非一致双曲理论中的两个问题:随机非一致指数二分性和随机一致指数二分性,紧致的随机不变集和随机吸引子.在一定条件下,我们证明了随机非一致指数二分性蕴含着随机一致指数二分性,同时,利用法向Lyapunov指数,我们还给出了一个紧致的随机不变集能成为随机吸引子的一个充分性条件,推广并改进了Ashwin的一些结果。
本论文主要用到Yongluo Cao教授在具有次可加性的随机连续函数列的结果,也用到Cao在Lyapunov指数与非一致双曲性方面的一个结果。Cao在文献中证明了具有次可加性的随机连续函数列的最大增长率能被遍历测度达到,并证明了在对Lyapunov指数施加一定条件的前提下,随机非一致双曲性实际上蕴含着随机一致双曲性。
Random dynamical system (RDS) is one kind of skew-product system. Because of more effectively describing the real world, it obtains more and more attentions. This thesis is devoted to the theories of exponential dichotomy, Sacker-Sell spectrum, Lyapunov exponents and random attractors. Exponential dichotomy describes a kind of hyperbolicity of the system. That is, the state space can be decomposed into a direct sum of two subspaces, which are continuous and invariant. And with the time evolving forward, the system is exponentially attracting in one subspace and exponentially expanding in the other subspace. Sacker-Sell spectrum is a definition in terms of exponential dichotomy. Sacker and Sell built the theory of Sacker-Sell spectrum. The theory has been developed into infinite dimensional dynamical system by many scholars, such as Magalh(a|~)es, Sacker and Sell, Chicone and Latushkin, Chow and Leiva. Also, Cong and Siegmund discussed some kind of dynamical systems in some random sense. Lyapunov exponents, one of the fundamental tools studying the asymptotic behaviors of dynamical system, reflects the average variance rate of the system with the development of time. Oseledec's Multiplicative Ergodic Theorem (MET) does not only solve the existence of Lyapunov exponents, but also show much more information of the dynamical structure. And henceforth, MET has been one of the most fundamental theorems in the theory of dynamical systems. MET has been developed by many people such as Ruelle , Ma(n|~)é, Thieullen , Zeng Lian and Kening Lu and so on. For finite dimensional dynamical system, Johnson, Palmer and Sell discussed the relations between the Sacker-Sell spectrum and the Lyapunov spectrum. They proved in particular that the Lyapunov spectrum is a subset of the Sacker-Sell spectrum, while the boudary points of the Sacker-Sell spectrum correspond to some Lyapunov exponents. They also proved that the Oseledec subbundles are the refined version of the Sacker-Sell subbundles. Attractor is one very important definition in the theory of differential equations and dynamical systems. In this thesis, we are interested in the following problem: when is a compact invariant subset a attractor? In the determined system, Ashwin, using normal Lyapunov exponents, discussed this problem. And then, he studied again the problem for one concrete example in the framework of random dynamical system.
This thesis is mainly concerned with the theory of Sacker-Sell spectrum, the mul- tiplicative ergodic theorems and random attractors for RDS. Comparing to the deterministic case, the base space of RDS is one probability space without any topology, which is one of the essential difficulties from the deterministic case to random case. We conquer the difficulty in this paper. In terms of a new exponential dichotomy for the finite dimensional RDS, we define the Sacker-Sell spectrum for RDS and show Sacker-Sell Spectrum Decomposition Theorem (SDT). Basing the SDT, we study the relations between Sacker-Sell spectrum and Lyapunov spectrum and establish the Spectrum Relationship Theorem in the framework of finite dimensional RDS. The relations between the two spectrum for infinite dimensional RDS are also obtained in this paper. In infinite dimensional RDS, different to finite case, the cocycle can be usually considered only in the positive half-time line. With this in mind, we define a backward continuation for the semi-dynamical system such that it can be extended to the whole time line. Besides this, there is another difficulty in the infinite dimensional case that a bounded and closed subset in the infinite dimensional state space is not necessarily compact. Having in mind this, we are concerned with the RDS with random uniformly completely continuous and random uniformlyα- contraction, respectively. In addition, MET for a very general infinite dimensional RDS is obtained in this paper. We take the ideas from Mane and Thieullen. In the final part, we discuss two problems in the theory of nonuniform hyperbolicity: the relation between random nonuniformly exponential dichotomy and random uniformly exponential dichotomy, the relation between a random compact and invariant subset and a random attractor. Under some conditions, we prove that random nonuniformly exponential dichotomy actually imply random uniformly exponential dichotomy. And, using the normal Lyapunov exponents, we also obtain a sufficient condition under which a random compact and invariant subset is a random forward attractor. This is a generalization of the results of Ashwin's.
Professor Yongluo Cao's two theorems, one concerning random continuous functions with sub-additivity and the other concerning nonuniform hyperbolicity, play an important role in this thesis. In the first theorem, Cao proved that the maximal growth rate of one random continuous function with sub-additivity can be achieved by some ergodic measure. In the second theorem, Cao, Luzzatto and Rios proved that a priori very weak nonuniform hyperbolicity conditions (caused by Lyapunov exponents) actually imply uniform hyperbolicity.
本论文主要研究了随机动力系统的Sacker-Sell谱理论,乘法遍历定理和随机吸引子问题.随机动力系统和确定性动力系统相比,它的底空间是一个没有任何拓扑结构的概率测度空间,这一点恰是从确定性系统到随机动力系统的一个本质困难之一,无论是对有限维的情形还是对无穷维的情形.我们克服这个困难,通过定义随机动力系统下的指数二分性,定义了随机动力系统的Sacker-Sell谱,并给出了有限维随机动力系统中的Sacker-Sell谱分解定理.在此基础上,我们比较了Sacker-Sell谱和Lyapunov谱,建立了有限维随机动力系统中两种谱的关系定理.我们也研究了无穷维的随机动力系统下的两种谱的关系。无穷维的随机动力系统和有限维的相比,其中的cocycle往往只能定义在正半时间轴上,为此我们首先对无穷维半动力系统进行了负向延拓,使得在负半时间轴上也有定义。另外,无穷维的情形还有一个难点,就是状态空间的有界的闭子集不一定是紧的。为此,我们研究了具有随机一致全连续性(紧算子是一致全连续算子的特殊情形)的随机动力系统,和具有更弱紧性(一致α-收缩性)的随机动力系统,其中用到了非紧性测度的概念.我们还给出并证明了一般的cocycle(不要求是紧算子)在可分的Banach空间的无穷维随机动力系统的乘法遍历定理,所采用的证明是基于Ma(?)e和Thieullen的方法。最后,我们讨论了非一致双曲理论中的两个问题:随机非一致指数二分性和随机一致指数二分性,紧致的随机不变集和随机吸引子.在一定条件下,我们证明了随机非一致指数二分性蕴含着随机一致指数二分性,同时,利用法向Lyapunov指数,我们还给出了一个紧致的随机不变集能成为随机吸引子的一个充分性条件,推广并改进了Ashwin的一些结果。
本论文主要用到Yongluo Cao教授在具有次可加性的随机连续函数列的结果,也用到Cao在Lyapunov指数与非一致双曲性方面的一个结果。Cao在文献中证明了具有次可加性的随机连续函数列的最大增长率能被遍历测度达到,并证明了在对Lyapunov指数施加一定条件的前提下,随机非一致双曲性实际上蕴含着随机一致双曲性。
Random dynamical system (RDS) is one kind of skew-product system. Because of more effectively describing the real world, it obtains more and more attentions. This thesis is devoted to the theories of exponential dichotomy, Sacker-Sell spectrum, Lyapunov exponents and random attractors. Exponential dichotomy describes a kind of hyperbolicity of the system. That is, the state space can be decomposed into a direct sum of two subspaces, which are continuous and invariant. And with the time evolving forward, the system is exponentially attracting in one subspace and exponentially expanding in the other subspace. Sacker-Sell spectrum is a definition in terms of exponential dichotomy. Sacker and Sell built the theory of Sacker-Sell spectrum. The theory has been developed into infinite dimensional dynamical system by many scholars, such as Magalh(a|~)es, Sacker and Sell, Chicone and Latushkin, Chow and Leiva. Also, Cong and Siegmund discussed some kind of dynamical systems in some random sense. Lyapunov exponents, one of the fundamental tools studying the asymptotic behaviors of dynamical system, reflects the average variance rate of the system with the development of time. Oseledec's Multiplicative Ergodic Theorem (MET) does not only solve the existence of Lyapunov exponents, but also show much more information of the dynamical structure. And henceforth, MET has been one of the most fundamental theorems in the theory of dynamical systems. MET has been developed by many people such as Ruelle , Ma(n|~)é, Thieullen , Zeng Lian and Kening Lu and so on. For finite dimensional dynamical system, Johnson, Palmer and Sell discussed the relations between the Sacker-Sell spectrum and the Lyapunov spectrum. They proved in particular that the Lyapunov spectrum is a subset of the Sacker-Sell spectrum, while the boudary points of the Sacker-Sell spectrum correspond to some Lyapunov exponents. They also proved that the Oseledec subbundles are the refined version of the Sacker-Sell subbundles. Attractor is one very important definition in the theory of differential equations and dynamical systems. In this thesis, we are interested in the following problem: when is a compact invariant subset a attractor? In the determined system, Ashwin, using normal Lyapunov exponents, discussed this problem. And then, he studied again the problem for one concrete example in the framework of random dynamical system.
This thesis is mainly concerned with the theory of Sacker-Sell spectrum, the mul- tiplicative ergodic theorems and random attractors for RDS. Comparing to the deterministic case, the base space of RDS is one probability space without any topology, which is one of the essential difficulties from the deterministic case to random case. We conquer the difficulty in this paper. In terms of a new exponential dichotomy for the finite dimensional RDS, we define the Sacker-Sell spectrum for RDS and show Sacker-Sell Spectrum Decomposition Theorem (SDT). Basing the SDT, we study the relations between Sacker-Sell spectrum and Lyapunov spectrum and establish the Spectrum Relationship Theorem in the framework of finite dimensional RDS. The relations between the two spectrum for infinite dimensional RDS are also obtained in this paper. In infinite dimensional RDS, different to finite case, the cocycle can be usually considered only in the positive half-time line. With this in mind, we define a backward continuation for the semi-dynamical system such that it can be extended to the whole time line. Besides this, there is another difficulty in the infinite dimensional case that a bounded and closed subset in the infinite dimensional state space is not necessarily compact. Having in mind this, we are concerned with the RDS with random uniformly completely continuous and random uniformlyα- contraction, respectively. In addition, MET for a very general infinite dimensional RDS is obtained in this paper. We take the ideas from Mane and Thieullen. In the final part, we discuss two problems in the theory of nonuniform hyperbolicity: the relation between random nonuniformly exponential dichotomy and random uniformly exponential dichotomy, the relation between a random compact and invariant subset and a random attractor. Under some conditions, we prove that random nonuniformly exponential dichotomy actually imply random uniformly exponential dichotomy. And, using the normal Lyapunov exponents, we also obtain a sufficient condition under which a random compact and invariant subset is a random forward attractor. This is a generalization of the results of Ashwin's.
Professor Yongluo Cao's two theorems, one concerning random continuous functions with sub-additivity and the other concerning nonuniform hyperbolicity, play an important role in this thesis. In the first theorem, Cao proved that the maximal growth rate of one random continuous function with sub-additivity can be achieved by some ergodic measure. In the second theorem, Cao, Luzzatto and Rios proved that a priori very weak nonuniform hyperbolicity conditions (caused by Lyapunov exponents) actually imply uniform hyperbolicity.
引文
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[2]Ashwin P.,Minimal attractors and bifurcations of random dynamical systems,Proc.R.Soc.Lond.A,455,1999,2615-2634.
[3]Ashwin P.,Buescuzx J.and Stewart I.,From attractor to chaotic saddle:a tale of transverse instability,Nonlinearity,9,1996,703-737.
[4]Barreira L.and Valls C.,Stabihty of nonautonomous differential equations,Lect.Notes in Math.1926,Springer,2008.
[5]Baxendale P.,Brownian motion in the diffeomorphism group I,Compositio Mathematica,53,1984,19-50.
[6]Berger M.,Introduction to probabihty and stochastic process.Springer-Verlag,Berlin Heidelberg New York,1993.
[7]Cao.Y.-L.,On growth rates of sub-additive functions for semi-flows:determined and random cases,J.Diff.Eq.,231,2006,1-17.
[8]Cao Y.-L.;Luzzatto S.and Rios I.,Uniform hyperbolicity for random maps with positive Lyapunov exponents,Proc.Amer.Math.Soc.,136,2008,3591-3600.
[9]Chicone C.,and Latushkin Y.,Evolution semigroups in dynamical systems and differential equations,Volume 70 of Mathematical Surveys and Monographs,American Mathematical Society,Providence,RI.,1999.
[10]Chow S.-N.,and Leiva H.,Dynamical spectrum for time dependent linear systems in Banach spaces,Jpn.J.Ind.Appl.Math.,11,1994,379-415.
[11]Chow S.-N.,and Leiva H.,Two definitions of exponential dichotomy for skewproduct semiflow in Banach spaces,Proc.Amer.Math.Soc.,124,1996,1071-1081.
[12]Coddington E.A.,Levinson N.,Theory of ordinary differential equations,McGraw-Hill,New York,1955.
[13]Cohen J.E.,Kesten H.and Newman C.M.,Oseledec's multiplicative ergodic theorem:a proof,in Random matrices and their applications,Cohen J.E.,Kesten H.and Newman C.M.Eds.,vol.50,Contemporary mathematics,Providence RI:American Mathematical Society,1986,23-30.
[14]Cong N.D.and Siegmund S.,Dichotomy spectrum of nonautonomous linear stochastic differential equations,Stoch.Dyn.,2,2002,175-201.
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