Efficient Integration of the variational equations of Multidimensional Hamiltonian Systems: Application to the Fermi-PASTA-Ulam Lattice

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 β (FPU-β) 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.