Topological Dynamics and Differential Equations

1991 Mathematics Subject Classification. 34C27, 34C28, 54H20. First published in Contemporary Mathematics in Volume 00, 1997, published by the American Mathematical Society.By reviewing our previous works on lifting dynamics in skew-product semi-flows and also the work of Johnson on almost periodic Floquet theory, we show several significant applications of the abstract theory of topological dynamics to the qualitative study of non-autonomous differential equations. The paper also contains some detailed discussions on a conjecture of Johnson.The second author is partially supported by NSF grant DMS-9402945. The third author is partially supported by NSF grant DMS-9501412