  Given an autonomous system of ordinary differential equations (ODE), we
consider developing practical models for the deterministic, slow/coarse
behavior of the ODE system. Two types of coarse variables are considered. The
first type consists of running finite time averages of phase functions.
Approaches to construct the coarse evolution equation for this type are
discussed and implemented on a 'Forced' Lorenz system and a singularly
perturbed system whose fast flow does not necessarily converge to an
equilibrium. We explore two strategies. In one, we compute (locally) invariant
manifolds of the fast dynamics, parameterized by the slow variables. In the
other, the choice of our coarse variables automatically guarantees them to be
'slow' in a precise sense. This allows their evolution to be phrased in terms
of averaging utilizing limit measures (probability distributions) of the fast
flow. Coarse evolution equations are constructed based on these approaches and
tested against coarse response of the 'microscopic' models. The second type of
coarse variables are defined as (non-trivial) scalar state functions that are
required by design to evolve autonomously, to the extent possible, with the
goal of being candidate state functions for unambiguously initializable coarse
dynamics. The question motivates a mathematical restatement in terms of a
first-order PDE. A computational approximation is developed and tested on the
Lorenz system and the Hald Hamiltonian system.
