Pieces of string or curves in three dimensional space may be knotted or braided. This topological tool can be used to study certain types of nonlinear differential equations. In particular, such an approach leads to forcing theorems in the spirit of the famous "period three implies chaos" for interval maps. We have identified three types of differential equations that respect the braid structure, and for these we can construct Morse-Conley-Floer type (homological) invariants. I will also illustrate how the topological arguments can be combined with a computer-assisted approach.