Geodesic
Normal form of differential equations with a~small parameter
Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 407-414

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In this paper we study a system of ordinary differential equations with a small parameter in the neighborhood of a fixed solution. We find a normal form for such a system. Then for the case of a small parameter and a single resonance we show that the formal integral manifold, found by V. I. Arnol'd (see Referativnyi Zhurnal Matematika, 8B678), is not always analytic. We discuss the conditions under which it is analytic. 
@article{MZM_1974_16_3_a7,
     author = {A. D. Bruno},
     title = {Normal form of differential equations with a~small parameter},
     journal = {Matemati\v{c}eskie zametki},
     pages = {407--414},
     publisher = {mathdoc},
     volume = {16},
     number = {3},
     year = {1974},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a7/}
}
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A. D. Bruno. Normal form of differential equations with a~small parameter. Matematičeskie zametki, Tome 16 (1974) no. 3, pp. 407-414. http://geodesic.mathdoc.fr/item/MZM_1974_16_3_a7/