Statistical Properties of Deterministic Systems

verfasst von: Jiu Ding, Ph.D., Aihui Zhou, Ph.D.

Verlag: Springer Berlin Heidelberg

Buchreihe : Tsinghua University Texts

Enthalten in: Springer Professional "Wirtschaft+Technik" , Springer Professional "Technik" , Springer Professional "Wirtschaft"

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Über dieses Buch

Part of Tsinghua University Texts, "Statistical Properties of Deterministic Systems" discusses the fundamental theory and computational methods of the statistical properties of deterministic discrete dynamical systems. After introducing some basic results from ergodic theory, two problems related to the dynamical system are studied: first the existence of absolute continuous invariant measures, and then their computation. They correspond to the functional analysis and numerical analysis of the Frobenius-Perron operator associated with the dynamical system.

The book can be used as a text for graduate students in applied mathematics and in computational mathematics; it can also serve as a reference book for researchers in the physical sciences, life sciences, and engineering.

Dr. Jiu Ding is a professor at the Department of Mathematics of the University of Southern Mississippi; Dr. Aihui Zhou is a professor at the Academy of Mathematics and Systems Science of the Chinese Academy of Sciences.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Abstract

Using the famous logistic model S_r(x) = rx(1−x) as an example, we give a brief survey of discrete dynamical systems for the purpose of leading the reader on a mathematical trip from order to chaos, and then we introduce basic ideas behind the statistical study of chaos, which is the main topic of the book.

Jiu Ding, Aihui Zhou

Chapter 2. Foundations of Measure Theory

Abstract

The fundamental mathematical knowledge used in the book is reviewed in this chapter, which includes basic measure theory and Lebesgue integration theory, L¹ spaces, the classic definition of variation for functions of one variable and the modern notion of variation for functions of several variables, compactness arguments for L¹ spaces, and quasi-compact operators on a Banach space which is a compactly-imbedded dense subspace of the L¹ space.

Jiu Ding, Aihui Zhou

Chapter 3. Rudiments of Ergodic Theory

Abstract

We give a short introduction to ergodic theory and its applications to topological dynamical systems. First we study the general properties of measure preserving transformations. Then we introduce the concepts of ergodicity, mixing, and exactness that describe different levels of chaotic behavior of the deterministic dynamics. The classic Birkhoff pointwise ergodic theorem and von Neumann mean ergodic theorem are stated, and some characteristics of ergodicity, mixing and exactness in terms of function sequences convergence are also presented.

Jiu Ding, Aihui Zhou

Chapter 4. Frobenius-Perron Operators

Abstract

In this chapter we first introduce Markov operators and study their general properties. Then we define Frobenius-Perron operators, a special class of Markov operators that will be mainly studied in the book. We also present a decomposition theorem for the Frobenius-Perron operator and its spectral analysis. We employ the Frobenius-Perron operator and its dual operator, the Koopman operator, to study ergodicity, mixing, and exactness of chaotic transformations.

Jiu Ding, Aihui Zhou

Chapter 5. Invariant Measures—Existence

Abstract

The existence problem of a stationary density to a Frobenius-Perron operator will be investigated. Three general existence theorems for Markov operators are presented with the help of the concepts of compactness, quasi-compactness, and constrictiveness. A powerful spectral decomposition theorem for constrictive Markov operators will be given. Then we prove the existence results for three concrete classes of transformations. They are the class of piecewise C² and stretching interval mappings, the class of piecewise convex mappings with a weak repellor, and the class of multi-dimensional piecewise C² and expanding transformations.

Jiu Ding, Aihui Zhou

Chapter 6. Invariant Measures—Computation

Abstract

The computational problem of stationary densities of Frobenius-Perron operators will be studied. First we introduce the classic Ulam’s piecewise constant method and its direct extension to multi-dimensional transformations. Then we study the piecewise linear Markov method proposed by the authors for one-and multidimensional transformations. We present Li’s pioneering work solving Ulam’s conjecture for the Lasota-Yorke class of interval mappings, and the convergence proofs of Ulam’s method for piecewise convex mappings by Miller and for piecewise expanding transformations by Ding and Zhou.

Jiu Ding, Aihui Zhou

Chapter 7. Convergence Rate Analysis

Abstract

After showing the convergence of the two numerical methods for Frobenius-Perron operators in the previous chapter, we further investigate the convergence rate problem for them. Keller’s stochastic stability result for a class of Markov operators will be studied first, which leads to his first proof of the L¹-norm convergence rate O(ln n/n) for Ulam’s method applied to the Lasota-Yorke class of mappings. Then we introduce Murrary’s work on an explicit upper bound of the convergence rate for Ulam’s method. The convergence rate analysis for the piecewise linear Markov method under the BV -norm will be presented in the last section.

Jiu Ding, Aihui Zhou

Chapter 8. Entropy

Abstract

We introduce the concepts of Shannon’s entropy for discrete sample spaces, the Kolmogorov entropy for measurable transformations, the topological entropy for continuous transformations on compact metric spaces, and the Boltzmann entropy of density functions. We also study some relationship between the Boltzmann entropy and the iteration of a Frobenius-Perron operator.

Jiu Ding, Aihui Zhou

Chapter 9. Applications of Invariant Measures

Abstract

Some applications of absolutely continuous invariant measures are illustrated in this last chapter. First we briefly give an application to the estimation of the speed of decay of correlations. Then we present Li-Yorke’s work on the application of stationary densities of small variation to random number generation. Two modern applications in the last decade are sketched in the final two sections. One is about the transfer operator in molecular conformation dynamics and computational drug design, and the other is the direct sequence code division multiple access in the third generation wireless communications.

Jiu Ding, Aihui Zhou

Backmatter

Titel: Statistical Properties of Deterministic Systems
verfasst von: Jiu Ding, Ph.D.
Aihui Zhou, Ph.D.
Verlag: Springer Berlin Heidelberg
Electronic ISBN: 978-3-540-85367-1
Print ISBN: 978-3-540-85366-4
DOI: https://doi.org/10.1007/978-3-540-85367-1