  In this paper, we provide a novel contraction-theoretic approach to analyze
two-time scale systems. In our proposed framework, systems enjoy several
robustness properties, which can lead to a more complete characterization of
their behaviors. Key assumptions are the contractivity of the fast sub-system
and of the reduced model, combined with an explicit upper bound on the
time-scale parameter. For two-time scale systems subject to disturbances, we
show that the distance between solutions of the nominal system and solutions of
its reduced model is uniformly upper bounded by a function of contraction
rates, Lipschitz constants, the time-scale parameter, and the time variability
of the disturbances. We also show local contractivity of the two-time scale
system and give sufficient conditions for global contractivity. We then
consider two special cases: for autonomous nonlinear systems we obtain sharper
bounds than our general results and for linear time-invariant systems we
present novel bounds based upon log norms and induced norms. Finally, we apply
our theory to two application areas -- online feedback optimization and
Stackelberg games -- and obtain new individual tracking error bounds showing
that solutions converge to their (time-varying) optimizer and computing overall
contraction rates.
