  Permanently deformed objects in binary systems can experience complex
rotation evolution, arising from the extensively studied effect of spin-orbit
coupling as well as more nuanced dynamics arising from spin-spin interactions.
The ability of an object to sustain an aspheroidal shape largely determines
whether or not it will exhibit non-trivial rotational behavior. In this work,
we adopt a simplified model of a gravitationally interacting primary and
satellite pair, where each body's quadrupole moment is approximated by two
diametrically opposed point masses. After calculating the net gravitational
torque on the satellite from the primary, and the associated equations of
motion, we employ a Hamiltonian formalism which allows for a perturbative
treatment of the spin-orbit and retrograde and prograde spin-spin coupling
states. By analyzing the resonances individually and collectively, we determine
the criteria for resonance overlap and the onset of chaos, as a function of
orbital and geometric properties of the binary. We extend the 2D planar
geometry to calculate the obliquity evolution, and find that satellites in
spin-spin resonances undergo precession when inclined out of the plane, but do
not tumble. We apply our resonance overlap criteria to the contact binary
system (216) Kleopatra, and find that its satellites, Cleoselene and
Alexhelios, may plausibly be exhibiting chaotic rotational dynamics from the
overlap of the spin-orbit and retrograde spin-spin resonances. While this model
is by construction generalizable to any binary system, it will be particularly
useful to study small bodies in the solar system, whose irregular shapes make
them ideal candidates for exotic rotational states.
