  The Johannsen-Psaltis spacetime is a perturbation of the Kerr spacetime
designed to avoid pathologies like naked singularities and closed timelike
curves. This spacetime depends not only on the mass and the spin of the compact
object, but also on extra parameters, making the spacetime deviate from Kerr;
in this work we consider only the lowest order physically meaningful extra
parameter. We use numerical examples to show that geodesic motion in this
spacetime can exhibit chaotic behavior. We study the corresponding phase space
by using Poincar\'{e} sections and rotation numbers to show chaotic behavior,
and we use Lyapunov exponents to directly estimate the sensitivity to initial
conditions for chaotic orbits.
