Conley index theory and Novikov-Morse theory. (English summary) Pure Appl. Math. Q. 1 (2005), no. 4, part 3, 939–971. Summary: “We derive general Novikov-Morse-type inequalities in a Conley-type framework for flows carrying cocycles, therefore generalizing our result in [Calc. Var. Partial Differential Equations 17 (2003), no. 1, 29–73; MR1979115 (2004d:37021)] derived for integral cocycles. The condition of carrying a cocycle expresses the nontriviality of integrals of that cocycle on flow lines. Gradient-like flows are distinguished from general flows carrying a cocycle by boundedness conditions on these integrals.