  We revisit the equilibrium one-dimensional $\phi^4$ model from the dynamical
systems point of view. We find an infinite number of periodic orbits which are
computationally stable. At the same time some of the orbits are found to
exhibit positive Lyapunov exponents! The periodic orbits confine every particle
in a periodic chain to trace out either the same or a mirror-image trajectory
in its two-dimensional phase space. These "computationally stable" sets of
pairs of single-particle orbits are either symmetric or antisymmetric to the
very last computational bit. In such a periodic chain the odd-numbered and
even-numbered particles' coordinates and momenta are either identical or differ
only in sign. "Positive Lyapunov exponents" can and do result if an
infinitesimal perturbation breaking a perfect two-dimensional antisymmetry is
introduced so that the motion expands into a four-dimensional phase space. In
that extended space a positive exponent results.
  We formulate a standard initial condition for the investigation of the
microcanonical chaotic number dependence of the model. We speculate on the
uniqueness of the model's chaotic sea and on the connection of such collections
of deterministic and time-reversible states to the Second Law of
Thermodynamics.
