  In this paper, we develop a model based on successive linearization to study
interactions between different modes in boundary layer flows. Our method
consists of two steps. First, we augment the Blasius boundary layer profile
with a disturbance field resulting from the linear Parabolized Stability
Equations (PSE) to obtain the modified base flow; and, second, we draw on
Floquet decomposition to capture the effect of mode interactions on the spatial
evolution of flow fluctuations via a sequence of linear progressions. The
resulting Parabolized Floquet Equations (PFE) can be conveniently advanced
downstream to examine the interaction between different modes in slowly varying
shear flows. We apply our framework to two canonical settings of transition in
boundary layers; the H-type transition scenario that is initiated by
exponential instabilities, and streamwise elongated laminar streaks that are
triggered by the lift-up mechanism. We demonstrate that the PFE capture the
growth of various harmonics and provide excellent agreement with the results
obtained in direct numerical simulations and in experiments.
