Dynamical Chaos in a Simple Model of a Knuckleball

TL;DR

This study models a knuckleball's flight incorporating asymmetric flow and quadratic drag, demonstrating that such a simple model can exhibit chaotic motion, explaining the unpredictable trajectories observed in real pitches.

Contribution

The paper introduces a simple dynamical model of a knuckleball that includes asymmetric flow effects, showing that chaos arises from these factors, unlike models without torques.

Findings

Models with asymmetric flow exhibit chaos, unlike torque-free models.

Uncertainty in the ball's position can grow by factors up to 10^6 during flight.

Bifurcations can cause significant deviations in the ball's landing position.

Abstract

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.