Efficient integration of the variational equations of multi-dimensional 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 $\beta$ (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.Comment: 14 pages, 6 figure

© IC-SCCE NUMERICAL SOLUTION OF THE BOUSSINESQ EQUATION USING SPECTRAL METHODS AND STABILITY OF SOLITARY WAVE PROPAGATION

Abstract. We study numerically the propagation and stability properties of solitary waves (solitons) of the Boussinesq equation in one space dimension, using a combination of finite differences in time and spectral methods in space. Our schemes follow very accurately these solutions, which are given by simple closed formulas and are known to be stable under small perturbations, for small enough velocities. Studying the interaction of two such solitons, we determine in the velocity parameter plane a sharp curve beyond which they become unstable. This is achieved by applying a precise criterion, which predicts when the observed amplitude growth of the waves is caused by a dynamical instability rather than the accumulation of numerical errors.

© IC-SCCE TRACING PERIODIC ORBITS IN 3D GALACTIC POTENTIALS BY THE PARTICLE SWARM OPTIMIZATION METHOD

Abstract. The class of Swarm Intelligence algorithms consists of stochastic optimization methods that exploit a population of interacting individuals to probe the search space simultaneously. The ability to work on nondifferentiable and discontinuous functions using only function value information renders these algorithms a useful tool, especially in cases where classical optimization methods fail. In this contribution, we apply Particle Swarm Optimization for locating periodic orbits in a 3D Ferrers bar model. An appropriate scheme that transforms the problem of finding periodic orbits to the corresponding problem of detecting the global minimizers of a function defined on the Poincaré Surface of Section of the Hamiltonian system is employed. We succeeded in tracing systematically several periodic orbits of the system, a large fraction of which is reported for the first time. In particular, we found families of 2D and 3D periodic orbits, associated with inner resonance higher than the 8:1 resonance, and not belonging to the x1-tree. We were also able to locate a plethora of pperiodic orbits with p>1