Dynamical Systems in Elliptical Pursuit and Evasion

Abstract

This paper investigates the elliptical case of one-on-one pursuit and evasion problems. Using the simultaneous differential equations derived by Barton and Eliezer, we derive a dynamical system based on the assumption that the shape of the pursuer's trajectory is unaffected by the evader's speed. The dynamical system involves the angular difference between the velocity vectors of the players and their separation distance. We examine whether the pursuer lies inside or outside the evader's elliptical path, and how the angle between the directions of motion of the two players evolves. Moreover, using a complex variable that includes information about the logarithmic distance and the angular difference, we transform the system into a single ordinary differential equation along which the distance between any two solutions never increases. We obtain four main results. First, when the evader is slower than the pursuer, the pursuer captures the evader in finite time, and we derive an explicit upper bound for the capture time. Second, when the evader is faster than the pursuer, a pursuer starting inside the evader's elliptical path is never captured, whereas capture does occur from just outside it on a nearly head-on course. Third, in the same case, the system has a unique periodic solution, to which every solution that avoids capture converges. Fourth, this periodic solution is hyperbolic and exponentially stable, and the product of its Floquet multipliers is given explicitly.

arXiv commentv3: major revision. Theorem 3 of v2 (global convergence for n<1) is corrected: the periodic solution attracts every solution that avoids capture, but capture can occur for pursuers starting just outside the ellipse. The proofs are rewritten and extended. Sections 4 and 5 of v2 (simulation results) were published separately as S. Yoshihara, JSIAM Letters 18 (2026) 109-112, and are removed here