  This paper aims to improve existing results about using averaging method for
analysis of dynamic systems on time scales. We obtain a more accurate estimate
for proximity between solutions of original and averaged systems regarding
$\Delta$-periodic and $\Delta$-quasiperiodic systems, which are introduced for
the first time. To illustrate the application of the averaging theorem for such
kind of system we considered an example and conducted numerical modelling.
Obtained results extend an application area for previously developed
numerically-asymptotic method of solution for optimal control problems on time
scales.
