Elliptic solutions in the H{e}non - Heiles model

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.

💡 Research Summary

The paper tackles the classic Hénon–Heiles Hamiltonian system, a two‑degree‑of‑freedom nonlinear model that exhibits both regular and chaotic dynamics. Starting from the Lagrangian, the authors reduce the coupled second‑order equations to a single autonomous fourth‑order ordinary differential equation (ODE) for one coordinate, incorporating the conserved energy as a parameter. This reduction is a standard step, but the resulting ODE is highly nonlinear and does not pass the Painlevé test, which means that conventional analytic methods yield only a few special solutions.

To overcome this limitation, the authors develop a systematic procedure for constructing all elliptic (Weierstrass‑function) solutions of an autonomous ODE or a system of ODEs. The core idea is to assume that a solution can be expressed as a polynomial in the Weierstrass ℘‑function and its derivative ℘′, i.e.

\