Particle swarm optimization: an efficient method for tracing periodic orbits in three-dimensional galactic potentials

Abstract

We propose particle swarm optimization (PSO) as an alternative method for locating periodic orbits in a three-dimensional (3D) model of barred galaxies. We develop an appropriate scheme that transforms the problem of finding periodic orbits into the problem of detecting global minimizers of a function, which is defined on the Poincare surface section of the Hamiltonian system. By combining the PSO method with deflection techniques, we succeeded in tracing systematically several periodic orbits of the system. The method succeeded in tracing the initial conditions of periodic orbits in cases where Newton iterative techniques had difficulties. In particular, we found families of two- and three-dimensional periodic orbits associated with the inner 8:1 to 12:1 resonances, between the radial 4:1 and corotation resonances of our 3D Ferrers bar model. The main advantages of the proposed algorithm are its simplicity, its ability to work using function values solely, and its ability to locate many periodic orbits per run at a given Jacobian constant. Ke yw ords: methods: numerical - galaxies: kinematics and dynamics - galaxies: structure.