Asymptotically сlose slutions to an ODE system

We consider a system of ordinary differential equations of a very general form. Let we have found a power-logarithmic expansion of its solution with the power asymptotics. Here we show how by methods of Power Geometry one can compute all power and exponential additions to the expansion, corresponding to such solutions to the system that are close to the found one. This allows finding some exponentially small effects by means of Power Geometry. We give examples of the calculations. The main attention is given to explanations of the computational algorithms.