  The knuckleball is perhaps the most enigmatic pitch in baseball. Relying on
the presence of raised seams on the surface of the ball to create asymmetric
flow, a knuckleball's trajectory has proven very challenging to predict
compared to other baseball pitches, such as fastballs or curveballs. Previous
experimental tracking of large numbers of knuckleballs has shown that they can
move in essentially any direction relative to what would be expected from a
drag-only trajectory. This has led to speculation that knuckleballs exhibit
chaotic motion. Here we develop a relatively simple model of a knuckleball that
includes quadratic drag and lift from asymmetric flow which is taken from
experimental measurements of slowly rotating baseballs. Our models can indeed
exhibit dynamical chaos as long In contrast, models that omit torques on the
ball in flight do not show chaotic behavior. Uncertainties in the phase space
position of the knuckleball are shown to grow by factors as large as $10^6$
over the flight of the ball from the pitcher to home plate. We quantify the
impact of our model parameters on the chaos realized in our models,
specifically showing that maximum Lyapunov exponent is roughly proportional to
the square root of the effective lever arm of the torque, and also roughly
proportional to the initial velocity of the pitch. We demonstrate the existence
of bifurcations that can produce changes in the location of the ball when it
reaches the plate of as much as 1.2 m for specific initial conditions similar
to those used by professional knuckleball pitchers. As we introduce additional
complexity in the form of more faithful representations of the empirical
asymmetry force measurements, we find that a larger fraction of the possible
initial conditions result in dynamical chaos.
