Stable and unstable periodic orbits in the one-dimensional lattice ϕ^{4} theory.

Abstract

Periodic orbits for the classical ϕ^{4} theory on the one-dimensional lattice are systematically constructed by extending the normal modes of the harmonic theory, for periodic, free and fixed boundary conditions. Through the process, we investigate which normal modes of the linear theory can or cannot be extended to the full nonlinear theory and why. We then analyze the stability of these orbits, clarifying the link between the stability, parametric resonance, and Lyapunov spectra for these orbits. The construction of the periodic orbits and the stability analysis is applicable to theories governed by Hamiltonians with quadratic intersite potentials and a general on-site potential. We also apply the analysis to theories with on-site potentials that have qualitatively different behavior from the ϕ^{4} theory, with some concrete examples.