  This paper is concerned with the study of the stability of dynamical systems
evolving on time scales. We first {formalize the notion of matrix measures on
time scales, prove some of their key properties and make use of this notion to
study both linear and nonlinear dynamical systems on time scales.}
Specifically, we start with considering linear time-varying systems and, for
these, we prove a time scale analogous of an upper bound due to Coppel. We make
use of this upper bound to give stability and input-to-state stability
conditions for linear time-varying systems. {Then, we consider nonlinear
time-varying dynamical systems on time scales and} establish a sufficient
condition for the convergence of the solutions. Finally, after linking our
results to the existence of a Lyapunov function, we make use of our approach to
study certain epidemic dynamics and complex networks. For the former, we give a
sufficient condition on the parameters of a SIQR model on time scales ensuring
that its solutions converge to the disease-free solution. For the latter, we
first give a sufficient condition for pinning controllability of complex time
scale networks and then use this condition to study certain collective opinion
dynamics. The theoretical results are complemented with simulations.
