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

Matrix measures, stability and contraction theory for dynamical systems on time scales