  We present a novel approach for studying the global dynamics of a
vibro-impact pair, that is, a ball moving in a harmonically forced capsule.
Motivated by a specific context of vibro-impact energy harvesting, we develop
the method with broader non-smooth systems in mind. The seeming complications
of the impacts of the ball with the capsule are exploited as useful non-smooth
features in selecting appropriate return maps. This choice yields a
computationally efficient framework for constructing return maps on short-time
realizations from the state space of possible initial conditions rather than
via long-time simulations often used to generate more traditional maps. The
different dynamics in sub-regions in the state space yield a small collection
of reduced polynomial approximations. Combined into a piecewise composite map,
these capture transient and attracting behaviors and reproduce bifurcation
sequences of the full system. Further ``separable'' reductions of the composite
map provide insight into both transient and global dynamics. This composite map
is valuable for cobweb analysis, which opens the door to computer-assisted
global analysis and is realized via conservative auxiliary maps based on the
extreme bounds of the maps in each subregion. We study the global dynamics of
energetically favorable states and illustrate the potential of this approach in
broader classes of dynamics.
