Almost periodic and almost automorphic dynamics for scalar convex differential equations

We study the set of bounded trajectories for the flow defined by a class of scalar convex differential equations depending on a parameter. It is found that there exists precisely one value of the parameter for which almost automorphic but not almost periodic dynamics may appear. Even for this parameter value, the occurrence of almost periodic dynamics is shown to be residual in some cases. The dependence of this parameter on the functions defining the differential equations is also studied.