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Periodic orbits for the classical $\phi^4$ theory on the one dimensionallattice are systematically constructed by extending the normal modes of theharmonic theory, for periodic, fixed and free boundary conditions. Through theprocess, we clarify, in general, which normal modes of the linear theory can orcan not be extended to the full non-linear theory. We then analyze thestability of these orbits, clarifying the link between the stability,parametric resonance and the Lyapunov spectra for these orbits.The constructionof the periodic orbits and the stability analysis is applicable to theoriesgoverned by Hamiltonians with quadratic inter-site potentials and a generalon-site potential. We also apply the analysis to theories with on-sitepotentials that have qualitatively different behavior from the $\phi^4$ theory,with some concrete examples.

Stable and unstable periodic orbits in the one dimensional lattice $\phi^4$ theory