  Three similar convergence notions are considered. Two of them are the long
established notions of convergent dynamics and incremental stability. The other
is the more recent notion of contraction analysis. All three convergence
notions require that all solutions of a system converge to each other. In this
paper, we investigate the differences between these convergence properties for
discrete-time, time-varying nonlinear systems by comparing the properties in
pairs and using examples. We also demonstrate a time-varying smooth Lyapunov
function characterization for each of these convergence notions, and, with
appropriate assumptions, we provide several sufficient conditions to establish
relationships between these properties in terms of Lyapunov functions.
