2 research outputs found
On the matrix measure as a tool to study the stability of linear and nonlinear dynamical systems on time scales
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.Comment: 28 pages, 3 figure