**Memoirs of the American Mathematical Society**

1998;
93 pp;
Softcover

MSC: Primary 34; 35; 54;

Print ISBN: 978-0-8218-0867-2

Product Code: MEMO/136/647

List Price: $48.00

Individual Member Price: $28.80

**Electronic ISBN: 978-1-4704-0236-5
Product Code: MEMO/136/647.E**

List Price: $48.00

Individual Member Price: $28.80

# Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

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*Wenxian Shen; Yingfei Yi*

This volume is devoted to the study of almost automorphic dynamics in
differential equations. By making use of techniques from abstract topological
dynamics, it is shown that almost automorphy, a notion which was introduced by
S. Bochner in 1955, is essential and fundamental in the qualitative study of
almost periodic differential equations.

Fundamental notions from topological dynamics are introduced in the
first part of the book. Harmonic properties of almost automorphic
functions such as Fourier series and frequency module are studied. A
module containment result is provided.

In the second part, lifting dynamics of \(\omega\)-limit sets and
minimal sets of a skew-product semiflow from an almost periodic
minimal base flow are studied. Skew-product semiflows with (strongly)
order preserving or monotone natures on fibers are given particular
attention. It is proved that a linearly stable minimal set must be
almost automorphic and become almost periodic if it is also uniformly
stable. Other issues such as flow extensions and the existence of
almost periodic global attractors, etc., are also studied.

The third part of the book deals with dynamics of almost periodic
differential equations. In this part, the general theory developed in
the previous two parts is applied to study almost automorphic and
almost periodic dynamics which are lifted from certain coefficient
structures (e.g., almost automorphic or almost periodic) of
differential equations. It is shown that (harmonic or subharmonic)
almost automorphic solutions exist for a large class of almost
periodic ordinary, parabolic and delay differential
equations.

#### Table of Contents

# Table of Contents

## Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows

- Contents vii8 free
- Acknowledgment ix10
- Abstract viii9 free
- Part I. Almost Automorphy and Almost Periodicity 112 free
- Part II. Skew-product Semiflows 2334
- Part III. Applications to Differential Equations 5364
- 1. Introduction 5364
- 2. Skew-product flows or semiflows generated by differential equations 5566
- 3. Scalar ordinary differential equations and parabolic equations in 1-space dimension 5768
- 4. System of ordinary differential equations 6980
- 5. Parabolic equations in higher space dimensions 7889
- 6. Functional differential equations 8596
- 7. References 90101

#### Readership

Graduate students and research mathematicians working in dynamical systems and differential equations.