  The NASA Double Asteroid Redirection Test (DART) mission is a planetary
defense-driven test of a kinetic impactor on Dimorphos, the satellite of the
binary asteroid 65803 Didymos. DART will intercept Dimorphos at a relative
speed of ${\sim}6.5 \text{ km s}^{-1}$, perturbing Dimorphos's orbital velocity
and changing the binary orbital period. We present three independent methods
(one analytic and two numerical) to investigate the post-impact attitude
stability of Dimorphos as a function of its axial ratios, $a/b$ and $b/c$ ($a
\ge b \ge c$), and the momentum transfer efficiency $\beta$. The first method
uses a novel analytic approach in which we assume a circular orbit and a
point-mass primary that identifies four fundamental frequencies of motion
corresponding to the secondary's mean motion, libration, precession, and
nutation frequencies. At resonance locations among these four frequencies, we
find that attitude instabilities are possible. Using two independent numerical
codes, we recover many of the resonances predicted by the analytic model and
indeed show attitude instability. With one code, we use fast Lyapunov
indicators to show that the secondary's attitude can evolve chaotically near
the resonance locations. Then, using a high-fidelity numerical model, we find
that Dimorphos enters a chaotic tumbling state near the resonance locations and
is especially prone to unstable rotation about its long axis, which can be
confirmed by ESA's Hera mission arriving at Didymos in late 2026. We also show
that a fully coupled treatment of the spin and orbital evolution of both bodies
is crucial to accurately model the long-term evolution of the secondary's spin
state and libration amplitude. Finally, we discuss the implications of a
post-impact tumbling or rolling state, including the possibility of terminating
BYORP evolution if Dimorphos is no longer in synchronous rotation.
