  We study which and how a periodic orbit in phase space links to both the
largest Lyapunov exponent and the expectation values of macroscopic variables
in a Hamiltonian system with many degrees of freedom. The model which we use in
this paper is the discrete nonlinear Schr\"odinger equation. Using a method
based on the modulational estimate of a periodic orbit, we predict the largest
Lyapunov exponent and the expectation value of a macroscopic variable. We show
that (i) the predicted largest Lyapunov exponent generally depends on the
periodic orbit which we employ, and (ii) the predicted expectation value of the
macroscopic variable does not depend on the periodic orbit at least in a high
energy regime. In addition, the physical meanings of these dependencies are
considered.
