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We use the averaging method and Levinson's fundamental theorem to study phenomenon of parametric resonance in some new equations from the class of adiabatic oscillators.

We use the method of averaging and the discrete analogue of Levinson's theorem to construct the asymptotics for solutions of the difference Schrödinger equation. Moreover, we present the general form for the averaging change of variable.

A system of linear differential equations with oscillatory decreasing coefficients is considered. The coefficients have the form t-aa(t), a > O, where a(t) is a trigonometric polynomial with an arbitrary set of frequencies. The asymptotic behavior of the solutions of this system as t ---, co is studied. We construct an invertible (for sufficiently large t) change of variables that takes the original system to a system not containing oscillatory coefficients in its principal part. The study of the asymptotic behavior of the solutions of the transformed system is a simpler problem. As an example, the following equation is considered:where ,~ and a, 0 < a _< 1, are real numbers.KEY WORDS: asymptotic integration of linear differential systems, linear differential equation with oscillatory decreasing coefficients, invertible change of variables.The stability and asymptotic behavior of the solutions of the linear system of differential equationswhere A is a constant square matrix and B(t) is a square matrix small in a certain sense as t ---* cr was studied by many authors (see [1][2][3][4][5][6], as well as the monographs [7-11]). Shtokalo [12,13] studied the stability of the solutions of the following system of differential equations:

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