This chapter generalizes the study of Chap. 6 in two main directions: on one hand, it enables the parameter k to run through the positive real numbers; on the other hand, it extends the results of Chap. 6 to a general setting covering a wide range of concrete behaviours with respect to \(y=\frac {x}{\varepsilon }\) and \(z=\frac {t}{\varepsilon ^{k}}\) , referred to as structure hypotheses, such as the almost periodicity hypothesis, and many others relevant to rapid oscillations.