Geodesic
The asymptotic behaviour of the solution to a system of differential equations with a small parameter and singular initial point
Sbornik. Mathematics, Tome 201 (2010) no. 1, pp. 79-101
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The initial value problem for a system of nonlinear ordinary differential equations with a small parameter multiplying the highest derivative is investigated. In a neighbourhood of the initial point the asymptotic behaviour of the solution has quite a complicated structure. A uniform asymptotic approximation to the solution up to an arbitrary power of the small parameter is constructed and substantiated. Bibliography: 3 titles.
Keywords: asymptotic expansion, small parameter, initial value problem, matching method. 
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A. M. Il'in; Yu. A. Leonychev; O. Yu. Khachay. The asymptotic behaviour of the solution to a system of differential equations with a small parameter and singular initial point. Sbornik. Mathematics, Tome 201 (2010) no. 1, pp. 79-101. http://geodesic.mathdoc.fr/item/SM_2010_201_1_a3/

[1] A. B. Vasileva, V. F. Butuzov, Asimptoticheskie metody v teorii singulyarnykh vozmuschenii, Vysshaya shk., M., 1990 | MR | Zbl

[2] A. M. Il'in, Matching of asymptotic expansions of solutions of boundary value problems, Transl. Math. Monogr., 102, Amer. Math. Soc., Providence, RI, 1992 | MR | Zbl | Zbl

[3] M. V. Fedoryuk, Asimptotika: integraly i ryady, Nauka, M., 1987 | MR | Zbl