  Many exoplanetary systems containing hot Jupiters are observed to have highly
misaligned orbital axes relative to the stellar spin axes. Kozai-Lidov
oscillations of orbital eccentricity/inclination induced by a binary companion,
in conjunction with tidal dissipation, is a major channel for the production of
hot Jupiters. We demonstrate that gravitational interaction between the planet
and its oblate host star can lead to chaotic evolution of the stellar spin axis
during Kozai cycles. As parameters such as the planet mass and stellar rotation
period vary, periodic islands can appear in an ocean of chaos, in a manner
reminiscent of other dynamical systems. In the presence of tidal dissipation,
the complex spin evolution can leave an imprint on the final spin-orbit
misalignment angles.
