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Almost Automorphic and Almost Periodic Dynamics in Skew-Product Semiflows
AMS(MOS) subject classifications: 34C27, 34D05, 35B15, 35B40, 35K57, 54H20.The current series of papers, which consists of three parts, are devoted to the study of almost automorphic dynamics in differential equations. By making use of techniques from abstract topological dynamics, we show 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. 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, we study lifting dynamics of w-limit sets and minimal sets
of a skew-product semiflow from an almost periodic minimal base flow. Skewproduct
semiflows with (strongly) order preserving or monotone natures on fibers
are given a 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 series deals with dynamics of almost periodic differential
equations. In this part, we apply the general theory developed in the previous
two parts 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.Partially supported by NSF grants DMS-9207069, DMS-9402945 and DMS-9501412
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