Elliptic solutions in the H\'{e}non - Heiles model
Maria V. Demina, Nikolai A. Kudryashov
TL;DR
This paper introduces a method to find all elliptic solutions of autonomous differential equations and applies it to the Hénon-Heiles system, discovering new solution families and classifying solutions up to sixth order.
Contribution
A novel method for identifying all elliptic solutions of autonomous equations is developed and applied to the Hénon-Heiles system, revealing new solution families and classifications.
Findings
New elliptic solutions for the Hénon-Heiles system are found.
A comprehensive classification of elliptic solutions up to sixth order is provided.
The method enables systematic discovery of elliptic solutions in autonomous systems.
Abstract
Equations of motion corresponding to the H\'{e}non - Heiles system are considered. A method enabling one to find all elliptic solutions of an autonomous ordinary differential equation or a system of autonomous ordinary differential equations is described. New families of elliptic solutions of a fourth--order equation related to the H\'{e}non - Heiles system are obtained. A classification of elliptic solutions up to the sixth order inclusively is presented.
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