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

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.

14 pages, 6 figures

References in corpus (3)

• Detecting chaos, determining the dimensions of tori and predicting slow diffusion in Fermi--Pasta--Ulam lattices by the Generalized Alignment Index method
• Regular and chaotic orbits in barred galaxies - I. Applying the SALI/GALI method to explore their distribution in several models
• Probing the local dynamics of periodic orbits by the generalized alignment index (GALI) method

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