Elliptical Pursuit and Evasion -Extended Version-

Abstract

Many studies on one-on-one pursuit-evasion problems have shown that formulas about the pursuer's trajectory can be solved by supposing three conditions. First, the evader follows specific figures. Second, the pursuer's velocity vector always points toward the evader's position. Third, the ratio of their respective speed remains constant. However, previous studies often assumed that the evader moves at a steady speed. This study aims to investigate how changes in the evader's speed affect the pursuer's trajectory. We hypothesized that the pursuer's trajectory would remain unchanged. First, the pursuer's trajectories were obtained from three scenarios where the evader orbits an ellipse with different speeds and angular velocities. These trajectories coincided. Second, changes in the evader's speed correspond to changes in the evader's trajectory parameters. Replacing the evader's parameter is proven to be replacing the pursuer's parameter. It is shown that replacing the evader's parameter is equivalent to replacing the pursuer's parameter. Consequently, the shape of the pursuer's trajectory is unaffected by the evader's speed; only the speed ratio matters in the game. This version includes additional sections on the dynamical system that were not present in the original version. If the evader's speed is always one, a dynamical system can be derived from the three conditions of pursuit and evasion. When the evader orbits a circle, this dynamical system is autonomous and has an asymptotically stable equilibrium point. However, when the evader orbits an ellipse, the dynamical system becomes non-autonomous, and the solution trajectory converges to a closed curve. Additionally, we present a second-order nonlinear differential equation describing the angular difference between the velocity vectors of both players.