  Hierarchical quadruple systems arise naturally in stellar binaries and
triples that harbour planets. Examples are hot Jupiters (HJs) in stellar triple
systems, and planetary companions to HJs in stellar binaries. The secular
dynamical evolution of these systems is generally complex, with secular chaotic
motion possible in certain parameter regimes. The latter can lead to extremely
high eccentricities and, therefore, strong interactions such as efficient tidal
evolution. These interactions are believed to play an important role in the
formation of HJs through high-eccentricity migration. Nevertheless, a deeper
understanding of the secular dynamics of these systems is still lacking. Here
we study in detail the secular dynamics of a special case of hierarchical
quadruple systems in either the `2+2' or `3+1' configurations. We show how the
equations of motion can be cast in a form representing a perturbed hierarchical
three-body system, in which the outer orbital angular momentum vector is
precessing steadily around a fixed axis. In this case, we show that
eccentricity excitation can be significantly enhanced when the precession
period is comparable to the Lidov-Kozai oscillation time-scale of the inner
orbit. This arises from an induced large mutual inclination between the inner
and outer orbits driven by the precession of the outer orbit, even if the
initial mutual inclination is small. We present a simplified semi-analytic
model that describes the latter phenomenon.
