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