无穷维线性系统的分布混沌动力学研究

发布时间：2018-06-27 07:42

本文选题：分布混沌 + 无穷维线性系统　；参考：《华南理工大学》2016年博士论文

【摘要】：混沌作为系统复杂性的一种刻画,广泛存在于现实世界中.在有限维空间中,混沌现象与系统的非线性性密切相关,然而当相空间是无穷维时,线性系统也可以产生混沌动力学行为.无穷维线性动力学研究无穷维空间上线性算子及强连续线性算子半群(简称C0-半群)生成的半动力系统的长时间演化过程.特别地,线性算子的超循环性以及各种混沌性质的研究极大地促进了线性混沌理论的发展,为揭示混沌现象的本质提供了新的思路,并广泛应用于统计力学、量子力学、生物学、经济学、交通工程等各种实际模型中.因此,无穷维线性系统混沌行为的深入研究具有重要的理论意义与应用价值.本学位论文主要研究无穷维线性系统的分布混沌动力学行为.综合运用拓扑动力系统的一些方法与相关的算子理论,分别研究了线性算子及C0-半群的准测度、分布混沌点对、分布混沌集、分布n-混沌集的各种性质,并证明了几类线性系统的分布混沌性以及不变分布混沌线性流形的存在性.本文具体内容如下:第一章为绪论,简述了混沌动力系统的研究历史与发展现状,回顾了拓扑动力学及无穷维线性混沌理论的相关概念和结果,并简要介绍了本文的研究背景和主要结论.第二章研究无穷维Fr′echet空间上连续线性算子及C0-半群的本质分布混沌动力学.给出了C0-半群是本质分布混沌的一些充分条件,并将其应用于一个具体的例子.同时,从乘积系统的角度进一步研究了直和线性算子与直和C0-半群的准测度及分布混沌性质,并用所得结果构造了分布混沌但不是本质(或稠密)分布混沌的算子、分布混沌但不是Devaney混沌的算子.特别地,还证明了存在非超循环的分布混沌线性算子,其准测度可以小于任意给定的正数.第三章,分析了无阻尼量子调和振子模型的湮灭算符的分布混沌集性质,如不变性、多样性、代数结构、拓扑结构以及混沌集大小等,证明了该湮灭算符具有无穷多个不变的稠密分布混沌线性流形,但不存在剩余的分布混沌集.这些结论为深入探讨一般线性系统的混沌复杂性提供了新的思路.第四章考虑无穷维线性系统的分布n-混沌动力学,获得了线性算子及强连续线性算子半群的分布n-混沌集的多种性质.特别地,一个线性算子(或C0-半群)是分布混沌的等价于它是分布n-混沌的对任意整数n 2成立,该性质对拓扑动力系统未必成立.同时,通过具体的例子说明线性系统的分布n-混沌集可以是整个状态空间.最后,详细研究了一类复合算子Cφ的分布n-混沌动力学行为,证明了Cφ存在不可数的分布混沌集不是分布3-混沌的,并且即使是有限维的分布混沌线性流形也未必是分布3-混沌的.第五章,讨论一类重要的权移位算子的混沌复杂动力学行为.对于单边权移位情形,证明了该算子是分布n-混沌的且准测度为1,并通过构造得到了不变的稠密分布n-混沌线性流形.进一步,研究了双边权移位算子的分布n-混沌动力学,证明了双边权移位及其逆算子的极大分布n-混沌性,且存在稠密的线性子流形是双边权移位的不变分布n-混沌集,但不是其逆算子的分布混沌集;存在稠密的线性子流形既是双边权移位的不变分布n-混沌集,也是其逆算子的不变分布n-混沌集.特别地,构造了一个不可数集同时是双边权移位算子及其逆算子的分布混沌集,但不是这两个算子的分布3-混沌集.最后证明了权移位算子的拓扑混合性及Devaney混沌性质.
[Abstract]:As a portrayal of system complexity, chaos exists widely in the real world. In finite dimensional space, chaos is closely related to the nonlinearity of the system. However, when phase space is infinite, the linear system can also produce chaotic dynamic behavior. Infinite dimensional linear dynamics studies linear operators and strong connections on infinite dimensions. The long time evolution process of the semi dynamic system generated by the continuous linear operator semigroup (C0- semigroup). In particular, the study of the hyper cyclicity of linear operators and the studies of various chaotic properties greatly promote the development of the theory of linear chaos, and provide a new idea to reveal the essence of chaotic phenomena, and widely used in statistical mechanics and quantum mechanics. Therefore, the in-depth study of chaotic behavior of infinite dimensional linear systems has important theoretical significance and application value. This dissertation mainly studies the chaotic dynamic behavior of infinite dimensional linear systems. In theory, we study the properties of the quasi measure of linear operator and C0- semigroup, distribution chaotic point pair, distributed chaotic set, distributed n- chaotic set, and prove the existence of the chaotic distribution of several classes of linear systems and the existence of the invariant distribution chaotic linear manifold. The relevant concepts and results of topological dynamics and infinite dimensional linear chaos theory are reviewed, and the research background and main conclusions of this paper are briefly introduced. The second chapter studies the essential distribution of chaotic dynamics of continuous linear operators and C0- Semigroups on infinite dimensional Fr 'echet spaces. The essential distribution of C0- semigroups is given. Some sufficient conditions of chaos are applied to a specific example. At the same time, the quasi measure and distribution chaotic properties of direct and linear operators and direct and C0- semigroups are further studied from the angle of product system, and the operators of chaotic but not essential (or dense) distribution chaos are constructed with the results obtained, which distribute chaos but not Deva The operator of ney chaos. In particular, it is also proved that there is a non super cyclic distributed chaotic linear operator whose quasi measure can be less than any given positive number. In Chapter third, the chaotic set properties of the annihilation operator of the undamped quantum harmonic oscillator model are analyzed, such as invariance, diversity, algebraic structure, topology and chaos set size, etc. It is proved that the annihilation operator has infinitely many unchangeable dense distributed chaotic linear manifolds, but there is no remaining Distribution Chaos set. These conclusions provide a new idea for the in-depth study of the chaos complexity of general linear systems. The fourth chapter considers the distributed n- chaotic dynamics of infinite dimensional linear systems, and obtains linear operators and strong connections. A variety of properties of the distributed n- chaotic set of a linear operator semigroup. In particular, a linear operator (or C0- semigroup) is a distributed chaos equivalent to the distribution of n- chaos to arbitrary integer N 2, which may not be established for topological dynamical systems. At the same time, the distribution of n- chaotic sets of linear systems can be the whole of the distribution of the chaotic sets of linear systems. State space. Finally, the chaotic dynamic behavior of a class of complex operator C phi is studied in detail. It is proved that the existence of the non countable distributed chaotic set of C phi is not distributed 3- chaos, and even the finite dimensional distribution chaotic linear manifold is not necessarily distributed 3- chaos. In the fifth chapter, the chaotic complex of a class of important weight shift operators is discussed. In the case of unilateral weight shift, it is proved that the operator is a distributed n- chaos and the quasi measure is 1, and the unchangeable dense distributed n- chaotic linear manifold is obtained by construction. Further, the distributed n- chaotic dynamics of the bilateral weighted shift operator is studied. It is proved that the double edge weight shift and its inverse operator's maximal distribution n- chaos are proved. The dense linear submanifolds are the invariant distribution n- chaotic sets of bilateral weighted shifts, but not the distributed chaotic sets of its inverse operators. The dense linear submanifolds are both the invariant distribution n- chaotic sets of bilateral weighted shifts and the invariant distribution n- chaotic sets of its inverse operators. The weighted shift operator and its distributed chaotic set of its inverse operators are not distributed 3- chaotic sets of these two operators. Finally, the topological mixing of weighted shift operators and the properties of Devaney chaos are proved.
【学位授予单位】：华南理工大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O19

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