  Periodic orbits for the classical $\phi^4$ theory on the one dimensional
lattice are systematically constructed by extending the normal modes of the
harmonic theory, for periodic, fixed and free boundary conditions. Through the
process, we clarify, in general, which normal modes of the linear theory can or
can not be extended to the full non-linear theory. We then analyze the
stability of these orbits, clarifying the link between the stability,
parametric resonance and the Lyapunov spectra for these orbits.The construction
of the periodic orbits and the stability analysis is applicable to theories
governed by Hamiltonians with quadratic inter-site 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 $\phi^4$ theory,
with some concrete examples.
