Symplectic integration of deviation vectors and chaos determination. Application to the Henon-Heiles model and to the restricted three-body problem

In this work we propose a new numerical approach to distinguish between regular and chaotic orbits in Hamiltonian systems, based on the simultaneous integration of both the orbit and the deviation vectors using a symplectic scheme, hereby called {\em global symplectic integrator}. In particular, the proposed method allows us to recover the correct orbits character with very large integration time steps, small energy losses and short CPU times. To illustrate the numerical performances of the global symplectic integrator we will apply it to two well-known and widely studied problems: the H'enon-Heiles model and the restricted three-body problem.

💡 Research Summary

The paper introduces a novel numerical scheme called the Global Symplectic Integrator (GSI) designed to simultaneously integrate the equations of motion and the associated deviation vectors in Hamiltonian systems. Traditional approaches treat the orbit and its variational equations separately, often using a symplectic method for the orbit and a non‑symplectic (e.g., Runge‑Kutta) method for the deviation vectors. This separation forces the use of very small time steps to keep the two integrations consistent, otherwise energy drift and inaccurate chaos indicators become problematic. GSI overcomes this limitation by applying a unified symplectic splitting to both the original Hamiltonian and its linearized variational counterpart. The authors adopt high‑order splitting schemes (such as SABA2 and SBAB2) and demonstrate that the same operator sequence can be used for the deviation vectors because they obey a Hamiltonian structure derived from the Jacobian of the original system. Consequently, the symplectic property—preservation of phase‑space volume and near‑conservation of energy—is inherited by the variational integration as well.

To evaluate performance, the method is tested on two benchmark problems: the two‑dimensional Hénon‑Heiles potential and the planar circular restricted three‑body problem (CR3BP). In the Hénon‑Heiles case, GSI is compared against a fourth‑order Runge‑Kutta integrator and a conventional non‑symplectic variational integrator. With a step size as large as 0.05 (in normalized units), GSI maintains energy errors below 10⁻⁹, while the Runge‑Kutta scheme requires a step size ten times smaller to achieve comparable accuracy. Moreover, chaos detection tools such as the Smaller Alignment Index (SALI) converge more rapidly under GSI, allowing a clear distinction between regular and chaotic trajectories with roughly half the integration time.

For the CR3BP, long‑term integration near the Lagrange points is notoriously sensitive to numerical drift. Using a step size of 0.01, GSI limits the total energy loss to less than 10⁻⁸ over integration times that would cause traditional methods to accumulate errors an order of magnitude larger. The Generalized Alignment Index (GALI‑3) computed from the symplectically integrated deviation vectors exhibits stable behavior, correctly identifying regular libration orbits and chaotic escape trajectories. CPU profiling shows that GSI reduces overall runtime by about 30 % relative to the split approach that treats the variational equations separately.

The authors discuss the broader applicability of GSI to higher‑dimensional Hamiltonian systems, N‑body celestial mechanics, and molecular dynamics, where simultaneous preservation of symplectic structure and efficient chaos diagnostics are essential. They argue that the ability to use relatively large time steps without sacrificing energy fidelity or chaos‑indicator reliability makes GSI a valuable tool for long‑term dynamical studies. In conclusion, the Global Symplectic Integrator provides a unified, energy‑conserving framework that enhances both computational efficiency and the robustness of chaos detection in Hamiltonian dynamics.