  We study the chaotic and secular evolution of hierarchical quadruple systems
in the $3+1$ configuration, focusing on the evolution of mutual inclination of
the inner binaries as the system undergoes coupled Lidov-Kozai (LK)
oscillations. We include short-range forces (SRF; such as those due to tidal
and rotational distortions) that control the eccentricity excitation of the
inner binary. The evolution of mutual inclination is described, a priori, by
two dimensionless parameters, $\pazocal{R}_0$, the ratio between the inner and
outer LK time-scales and $\epsilon_{SRF}$, the ratio between the SRF precession
and the inner LK precession rates. We find that the chaotic zones for the
mutual inclination depend mainly on $\pazocal{R}_0$, while $\epsilon_{SRF}$
controls mainly the range of eccentricity excitation. The mutual inclination
evolves chaotically for $1\lesssim \pazocal{R}_0\lesssim 10$, leading to large
misalignments. For $0.4 \lesssim \pazocal{R}_0 \lesssim 0.8$, the system could
be weakly excited and produce bimodal distribution of mutual inclination
angles. Our results can be applied to exomoons-planets in stellar binaries and
Warm/Hot Jupiters in stellar triples. Such systems could develop large mutual
inclination angles if the inner binary is tight enough, and also high
eccentricities, depending of the strength of the short-range forces. Future
detections of tilted Warm/Hot Jupiters and exomoons could put our mechanism
under observational tests.
