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arxiv: 2605.07217 · v2 · pith:KAFE67JCnew · submitted 2026-05-08 · 🧮 math.OC

Dynamical Systems in Elliptical Pursuit and Evasion

Sota Yoshihara This is my paper

Pith reviewed 2026-05-25 06:35 UTC · model grok-4.3

classification 🧮 math.OC
keywords pursuit-evasiondynamical systemselliptical orbitsfinite-time captureperiodic solutionsnon-autonomous systemscomplex variablescapture time bounds
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The pith

In elliptical pursuit-evasion, a faster pursuer captures the evader in finite time with an explicit upper bound while a slower pursuer yields global asymptotic convergence to a unique periodic solution.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

This paper derives a dynamical system for one-on-one pursuit and evasion in which the evader follows an elliptical orbit. The derivation starts from Barton and Eliezer's simultaneous differential equations under the assumption that the pursuer's trajectory shape stays fixed regardless of evader speed. The resulting system is non-autonomous and lacks equilibrium points, in contrast to the autonomous circular case that possesses an asymptotically stable equilibrium. Reformulating the equations with a complex variable that encodes logarithmic distance and angular difference removes the singularity at capture and allows two main theorems: finite-time capture with an explicit time bound when the pursuer is faster, and the existence of a unique periodic solution that attracts every trajectory when the pursuer is slower.

Core claim

The paper establishes that the dynamical system obtained from Barton and Eliezer's equations for an elliptical evader orbit, after complex-variable reformulation, satisfies the following: when the pursuer is faster than the evader, capture occurs in finite time and an explicit upper bound on capture time is derived; when the pursuer is slower, the system admits a unique periodic solution to which all trajectories converge globally and asymptotically.

What carries the argument

The non-autonomous dynamical system whose state is the angular difference between the players' velocity vectors and their separation distance, rewritten in a complex variable that combines the logarithm of distance with the angular difference.

If this is right

  • Capture occurs in finite time with an explicit upper bound when pursuer speed exceeds evader speed.
  • A unique periodic solution exists when the pursuer is slower.
  • Every trajectory converges globally and asymptotically to that periodic solution.
  • The elliptical system is non-autonomous and contains no equilibrium point.
  • The complex-variable reformulation removes the singularity at the capture instant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The contrast with the circular case implies that orbit shape qualitatively changes the long-term behavior from equilibrium to periodic oscillation.
  • The explicit capture-time bound supplies a quantitative comparison between elliptical and circular pursuit that is absent from the circular analysis.
  • If the trajectory-shape assumption holds for other smooth closed curves, the same finite-time versus periodic dichotomy may apply beyond ellipses.
  • The periodic attractor when the pursuer is slower suggests sustained bounded oscillation in separation distance rather than monotonic approach or escape.

Load-bearing premise

The shape of the pursuer's trajectory is unaffected by the evader's speed.

What would settle it

Numerical integration of the original Barton and Eliezer simultaneous differential equations that either exceeds the stated upper bound on capture time or fails to show all trajectories approaching a single periodic orbit.

Figures

Figures reproduced from arXiv: 2605.07217 by Sota Yoshihara.

Figure 1
Figure 1. Figure 1: The pursuer’s trajectory does not converge to the blue line, i.e., the ellipse orbited by the evader scaled by [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Elliptical one-on-one pursuit and evasion problem when [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 6
Figure 6. Figure 6: The numerical solutions of (47) and both player’s position (49) describing elliptical pursuit and evasion, for a =1.0, b =0.5, and n =0.5 from ϑ = ϱ/2 to ϑ = 10ϱ + ϱ/2. The horizontal axis represents ω and the vertical axis represents ε. The pursuer and the evader start from the origin and (1, 0), respectively, so the initial conditions are ω(0) = 1 and ε(0) = ϱ/2. The solution trajectory converges to an u… view at source ↗
Figure 7
Figure 7. Figure 7: The numerical solutions of (47) and both player’s positi Figure 7: The numerical solutions of (47) and both player’s position (49), for [PITH_FULL_IMAGE:figures/full_fig_p010_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: The numerical solutions of (47) and both player’s position (49), for a =1.0, b =0.5, and n =1.0 from ω = ε/2 to ω = ε/2+2ε. The meaning of axes and initial conditions are the same as in [PITH_FULL_IMAGE:figures/full_fig_p011_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: shows µ(φ) for all four initial conditions plotted on the same axes. After a short transient of at most a few periods, the four curves become visually indistinguishable and settle into a common π￾periodic oscillation, confirming that µ(φ) converges uniformly to a single limit independent of its initial value [PITH_FULL_IMAGE:figures/full_fig_p017_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Time evolution of ζ(φ) for the same four initial conditions as in [PITH_FULL_IMAGE:figures/full_fig_p018_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Pursuer trajectories reconstructed from ( [PITH_FULL_IMAGE:figures/full_fig_p018_11.png] view at source ↗
read the original abstract

This paper investigates the difference between the circular and elliptical cases in one-on-one pursuit and evasion problems. Using the simultaneous differential equation 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. When the evader orbits a circle, the dynamical system is autonomous with an asymptotically stable equilibrium point. By contrast, if the evader orbits an ellipse, the dynamical system becomes non-autonomous and lacks an equilibrium point. To handle the singularity at capture, we reformulate the system using a complex variable that includes information about the logarithmic distance and the angular difference. We establish two main results: when the pursuer is faster than the evader, the pursuer captures the evader in finite time, and we derive an explicit upper bound for the capture time; when the pursuer is slower, the system possesses a unique periodic solution to which all trajectories converge globally and asymptotically.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript derives a reduced dynamical system for one-on-one pursuit-evasion from Barton and Eliezer's simultaneous differential equation under the explicit modeling assumption that the pursuer's trajectory shape is unaffected by the evader's speed. For circular evader orbits the resulting system is autonomous and possesses an asymptotically stable equilibrium. For elliptical orbits the system is non-autonomous; after a complex-variable reformulation that incorporates logarithmic distance and angular difference to regularize the capture singularity, the authors prove two main results: when the pursuer is faster, capture occurs in finite time with an explicit upper bound on capture time; when the pursuer is slower, the system admits a unique periodic solution that attracts all trajectories globally and asymptotically.

Significance. If the stated modeling assumption is valid, the work supplies explicit, analytically derived capture-time bounds and a global convergence result for the non-autonomous elliptical case, extending prior circular results and offering concrete predictions that could be tested in applications. The complex-variable reformulation is a clean technical device for handling the singularity. The paper is transparent that the theorems apply to the reduced model obtained under the assumption.

major comments (1)
  1. [section deriving the dynamical system (immediately following the citation of Barton and Eliezer)] The modeling assumption that the pursuer's trajectory shape is unaffected by the evader's speed is introduced without derivation, error estimate, or numerical verification against the original coupled Barton-Eliezer equations. This assumption directly determines the form of the reduced non-autonomous vector field for the elliptical case and is therefore load-bearing for both headline theorems (finite-time capture bound and global asymptotic convergence to the unique periodic orbit).
minor comments (2)
  1. The abstract states that derivations exist but supplies no proof sketches; the full manuscript should include at least a one-paragraph outline of the key steps in each main theorem for readability.
  2. [reformulation paragraph] Notation for the complex variable (logarithmic distance plus angular difference) should be introduced with an explicit equation number at first use.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting the central role of the modeling assumption. We address the single major comment below.

read point-by-point responses

  1. Referee: [section deriving the dynamical system (immediately following the citation of Barton and Eliezer)] The modeling assumption that the pursuer's trajectory shape is unaffected by the evader's speed is introduced without derivation, error estimate, or numerical verification against the original coupled Barton-Eliezer equations. This assumption directly determines the form of the reduced non-autonomous vector field for the elliptical case and is therefore load-bearing for both headline theorems (finite-time capture bound and global asymptotic convergence to the unique periodic orbit).

    Authors: The assumption is stated explicitly in the manuscript right after the Barton-Eliezer citation, and every subsequent result (including both headline theorems) is formulated and proved strictly for the reduced dynamical system obtained under this modeling choice. The paper does not claim that the reduced system approximates the fully coupled equations with quantifiable error; its contribution lies in the analytic tractability of the decoupled non-autonomous system and the complex-variable regularization that permits the finite-time capture bound and the global convergence theorem. We agree that additional discussion of the modeling rationale would improve transparency. In the revised version we will insert a short paragraph immediately after the assumption is introduced, reiterating that all theorems apply exclusively to the reduced model and briefly motivating the assumption as a standard decoupling device that closes the system in the distance-angle variables. revision: partial

Circularity Check

0 steps flagged

No circularity; derivation rests on external DE plus one explicit modeling assumption

full rationale

The paper begins from the cited Barton-Eliezer simultaneous differential equation, states the modeling assumption that pursuer trajectory shape is independent of evader speed, and obtains the angular-difference/separation-distance system under that assumption. It then analyzes the resulting autonomous (circular) and non-autonomous (elliptical) vector fields, proves finite-time capture with explicit bound when pursuer is faster, and global asymptotic convergence to a unique periodic orbit when slower. None of these steps reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation; the central claims are theorems about the reduced model, not re-statements of its inputs. The assumption is external and falsifiable against the original coupled equations, so the derivation chain is self-contained.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on one domain assumption that enables reduction to a two-variable dynamical system; no free parameters are introduced and no new entities are postulated.

axioms (1)
  • domain assumption the shape of the pursuer's trajectory is unaffected by the evader's speed
    Invoked to obtain the dynamical system in angular difference and separation distance from Barton and Eliezer's simultaneous differential equation.

pith-pipeline@v0.9.0 · 5708 in / 1325 out tokens · 50217 ms · 2026-05-25T06:35:49.679540+00:00 · methodology

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Reference graph

Works this paper leans on

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