[Paper Review] A small variation of the Taylor Method and periodic solutions of the 3-body problem

Óscar Perdomo|arXiv (Cornell University)|Jul 4, 2015

Spacecraft Dynamics and Control4 references3 citations

TL;DR

This paper introduces a modified Taylor method that tracks global error and round-off errors without requiring the exact solution, enabling rigorous numerical proofs in dynamical systems. It applies this method to rigorously prove the existence of a new periodic solution in the three-body problem with masses 100, 100, and 200, period T ≈ 96.241, confirmed via validated numerical integration and interval arithmetic.

ABSTRACT

In this paper we define a small variation of the Taylor method and a formula for the global error of this new numerical method that allows us to keep track of the round-off error and does not require previous knowledge of the exact solution. As an application we provide a rigorous proof of the construction/existence of a periodic solution of the three body problem. Some images of this periodic motion can be seen at https://www.youtube.com/watch?v=fSmQyeKcj5k

Motivation & Objective

• To develop a numerically stable and rigorous method for solving ODEs in dynamical systems, particularly when exact solutions are unknown.
• To address the challenge of tracking round-off errors in long numerical integrations, especially in chaotic systems like the three-body problem.
• To provide a validated numerical proof of existence for a new periodic solution in the three-body problem, building on numerically observed but unproven solutions.
• To enable the use of computer algebra systems for high-precision numerical integration without exponential growth in computational cost due to symbolic rational arithmetic.
• To demonstrate that numerically observed periodic solutions can be rigorously proven periodic using a new error-bound formula.

Proposed method

• A small variation of the Taylor method is proposed, where each step computes a high-precision approximation using mathematical operations and then bounds the result within a rational interval of width H.
• The method uses interval arithmetic to bound the result of each iteration within a rational interval of user-defined width H, ensuring error control without symbolic exact arithmetic.
• A key theoretical contribution is Theorem 2.3, which provides a global error bound in terms of step size h and rounding tolerance H, without requiring prior knowledge of the solution's bounds.
• The method leverages computer algebra systems' ability to compute tight bounds on expressions involving rational numbers and square roots, enabling high-precision interval arithmetic.
• The approach combines Taylor series integration with validated numerical methods, using interval enclosures to track both truncation and round-off errors.
• The method is applied to the three-body problem by integrating the equations of motion with controlled error bounds, verifying that the state at time T returns within a small interval of the initial state.

Experimental results

Research questions

• RQ1 Can a modified Taylor method be designed to rigorously track global and round-off errors without requiring the exact solution?
• RQ2 Can this method be used to validate that a numerically observed periodic solution in the three-body problem is indeed mathematically periodic?
• RQ3 What is the minimal computational cost to achieve rigorous numerical validation of periodicity in chaotic systems?
• RQ4 How can interval arithmetic be effectively combined with Taylor series integration to maintain precision without exponential growth in expression size?
• RQ5 Is it possible to prove the existence of a periodic solution in the three-body problem using only numerical integration with error bounds?

Key findings

• A new periodic solution of the three-body problem is rigorously proven to exist, with masses 100, 100, and 200, and period T ≈ 96.241.
• The solution is numerically observed to return to a state within 10^-14 of the initial condition after one period, and this is validated as exact using the new method.
• The method successfully bounds the global error using only h (step size) and H (rounding tolerance), without requiring a priori knowledge of solution bounds.
• The approach avoids the exponential growth in expression size seen in traditional symbolic rational arithmetic, making long integrations feasible.
• The method enables the use of standard computer algebra systems for validated integration by bounding intermediate results within rational intervals of controlled width.
• The proof confirms that the numerically observed solution from Figure 1.1 is indeed periodic, marking a significant step in validating numerical solutions in celestial mechanics.