The asymptotics of solutions of systems of differential equations with a small parameter for large times

For a single-frequency system of differential equations with small perturbation, the possibility of constructing and justifying of the asymptotic expansion of the solution for large times is studied. Under additional restrictions, it is proved that the asymptotic solution constructed by the known method of separation of variables approximates the true solution on the time interval $[0,\varepsilon^{-k}]$ with accuracy up to any fixed power of the small parameter. Another construction and justification of the asymptotics is carried out with the help of the two-scale expansion on the phase plane. Calculations are performed for the classical Van der Pol equation; they show that the usual two-scale method of expansion is inapplicable to time of order $t^{-2}$ and that the method considered here provides a good approximation for this time.