Efficient integration of the variational equations of multi-dimensional Hamiltonian systems: Application to the Fermi-Pasta-Ulam lattice

TL;DR

This paper compares various numerical methods for efficiently integrating variational equations in multi-dimensional Hamiltonian systems, demonstrating the superior performance of the Tangent Map technique in the context of the Fermi-Pasta-Ulam lattice.

Contribution

It introduces and evaluates the Tangent Map method for variational equations, showing its effectiveness in Hamiltonian systems and enabling accurate detection of low-dimensional tori.

Findings

Tangent Map technique outperforms other integrators in speed and accuracy.

Efficient detection of chaos and regular motion using GALI indices.

Successful application to FPU-$eta$ lattice with up to 20 oscillators.

Abstract

We study the problem of efficient integration of variational equations in multi-dimensional Hamiltonian systems. For this purpose, we consider a Runge-Kutta-type integrator, a Taylor series expansion method and the so-called `Tangent Map' (TM) technique based on symplectic integration schemes, and apply them to the Fermi-Pasta-Ulam $\beta$ (FPU-$\beta$) lattice of $N$ nonlinearly coupled oscillators, with $N$ ranging from 4 to 20. The fast and accurate reproduction of well-known behaviors of the Generalized Alignment Index (GALI) chaos detection technique is used as an indicator for the efficiency of the tested integration schemes. Implementing the TM technique--which shows the best performance among the tested algorithms--and exploiting the advantages of the GALI method, we successfully trace the location of low-dimensional tori.