  We present 3D hydrodynamic models of the interaction between the outflows of
evolved, pulsating, Asymptotic Giant Branch (AGB) stars and nearby ($< 3$
stellar radii) sub-stellar companions ($M_{\mathrm{comp}} \lesssim 40$ M$_J$).
Our models show that due to resonances between the orbital period of the
companion and the pulsation period of the AGB star, multiple spiral structures
can form; the shocks driven by the pulsations are enhanced periodically in
different regions as they encounter the denser material created by the
sub-stellar companion's wake. We discuss the properties of these spiral
structures and the effect of the companion parameters on them. We also
demonstrate that the gravitational potential of the nearby companion enhances
the mass loss from the AGB star. For more massive ($M_{\mathrm{comp}} > 40$
M$_J$) and more distant companions ($> 4$ stellar radii), a single spiral arm
forms. We discuss the possibility of observing these structures with the new
generations of high-resolution, high-sensitivity instruments, and using them to
`find' sub-stellar companions around bright, evolved stars. Our results also
highlight possible structures that could form in our solar system when the Sun
turns into an AGB star.
