Evolution semigroups in dynamical systems and differential equations / Carmen Chicone, Yuri Latushkin.

Format:

Book

Author/Creator:

Chicone, Carmen Charles, author.

Latushkin, Yuri, 1956- author.

Series:

Mathematical surveys and monographs ; no. 70.

Mathematical surveys and monographs, 0076-5376 ; volume 70

Language:

English

Subjects (All):

Semigroups of operators.

Evolution equations.

Differentiable dynamical systems.

Differential equations.

Physical Description:

1 online resource (375 p.)

Place of Publication:

Providence, Rhode Island : American Mathematical Society, [1999]

Language Note:

English

Summary:

The main theme of the book is the spectral theory for evolution operators and evolution semigroups, a subject tracing its origins to the classical results of J. Mather on hyperbolic dynamical systems and J. Howland on nonautonomous Cauchy problems. The authors use a wide range of methods and offer a unique presentation. The authors give a unifying approach for a study of infinite-dimensional nonautonomous problems, which is based on the consistent use of evolution semigroups. This unifying idea connects various questions in stability of semigroups, infinite-dimensional hyperbolic linear skew-product flows, translation Banach algebras, transfer operators, stability radii in control theory, Lyapunov exponents, magneto-dynamics and hydro-dynamics. Thus the book is much broader in scope than existing books on asymptotic behavior of semigroups. Included is a solid collection of examples from different areas of analysis, PDEs, and dynamical systems. This is the first monograph where the spectral theory of infinite dimensional linear skew-product flows is described together with its connection to the multiplicative ergodic theorem; the same technique is used to study evolution semigroups, kinematic dynamos, and Ruelle operators; the theory of stability radii, an important concept in control theory, is also presented. Examples are included and non-traditional applications are provided.

Contents:

""Contents""; ""Chapter 1. Introduction""; ""1.1. Motivations and main results""; ""1.2. Historical comments""; ""1.3. Organization and content""; ""Chapter 2. Semigroups on Banach Spaces and Evolution Semigroups""; ""2.1. Introduction to semigroups""; ""2.2. Evolution semigroups and hyperbolicity""; ""2.3. Bibliography and remarks""; ""Chapter 3. Evolution Families and Howland Semigroups""; ""3.1. Evolution families and dichotomy""; ""3.2. Howland semigroups on the line""; ""3.3. Howland semigroups on the half line""; ""3.4. Bibliography and remarks""

""Chapter 4. Characterizations of Dichotomy for Evolution Families""""4.1. Discrete dichotomies: an algebraic approach""; ""4.2. Green's function and evolution semigroups""; ""4.3. Dichotomy and solutions of nonhomogeneous equations""; ""4.4. Hyperbolicity and dissipativity""; ""4.5. Bibliography and remarks""; ""Chapter 5. Two Applications of Evolution Semigroups""; ""5.1. Control theory""; ""5.2. Persistence of dichotomy""; ""5.3. Bibliography and remarks""; ""Chapter 6. Linear Skew-Product Flows and Mather Evolution Semigroups""; ""6.1. Linear skew-product flows and dichotomy""

""6.2. The Mather semigroup""""6.3. Sacker-Sell spectral theory""; ""6.4. Bibliography and remarks""; ""Chapter 7. Characterizations of Dichotomy for Linear Skew-Product Flows""; ""7.1. Pointwise dichotomies""; ""7.2. The Annular Hull Theorem""; ""7.3. Dichotomy, mild solutions, and Green's function""; ""7.4. Isomorphism Theorems""; ""7.5. Dichotomy and quadratic Lyapunov function""; ""7.6. Bibliography and remarks""; ""Chapter 8. Evolution Operators and Exact Lyapunov Exponents""; ""8.1. Oseledets' Theorem and linear skew-product flows""; ""8.2. The kinematic dynamo operator""

""8.3. The Ruelle transfer operator""""8.4. Bibliography and remarks""; ""Bibliography""; ""List of Notations""; ""Index""; ""A ""; ""B ""; ""C ""; ""D ""; ""E ""; ""F ""; ""G ""; ""H ""; ""K ""; ""L ""; ""M ""; ""N ""; ""O ""; ""P ""; ""Q ""; ""R ""; ""S ""; ""T ""; ""U ""; ""V ""; ""W ""

Notes:

Description based upon print version of record.

Includes bibliographical references (pages 341-355) and index.

Description based on print version record.

ISBN:

1-4704-1297-7