Abstract
In this paper, in a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.
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V. S. Samovol, “Normal form of autonomous systems with one zero eigenvalue,” Mat. Zametki 75(5), 711–720 (2004) [Math. Notes 75 (5–6), 660–668 (2004)].
V. S. Samovol, “On new resonances and normal form of autonomous systems with one zero eigenvalue,” Mat. Zametki (in press).
Ph. Hartman, Ordinary Differential Equations (Wiley, New York, 1964; Mir, Moscow, 1970).
V. S. Samovol, “Equivalence of systems of differential equations in the neighborhood of a singular point,” in Trudy Moskov.Mat. Obshch. (Izd. Moskov. Univ., Moscow, 1982), Vol. 44, pp. 213–234 [in Russian].
A. N. Kuznetsov, “Differentiable solutions to degenerate systems of ordinary equations,” Funktsional. Anal. i Prilozhen. 6(2), 41–51 (1972) [Functional Anal. Appl. 6, 119–127 (1972)].
G. R. Belitskii, “Smooth equivalence of germs of vector fields with a single zero eigenvalue or a pair of purely imaginary eigenvalues,” Funktsional. Anal. i Prilozhen. 20(4), 1–8 (1986) [Functional Anal. Appl. 20, 253–259 (1986)].
W. Wazow, Asymptotic Expansions for Ordinary Differential Equations (Wiley, New York-London-Sydney, 1965; Mir,Moscow, 1968).
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Original Russian Text © V. S. Samovol, 2010, published in Matematicheskie Zametki, 2010, Vol. 88, No. 2, pp. 913–925.
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Samovol, V.S. Finitely smooth local equivalence of autonomous systems with one zero root. Math Notes 88, 251–261 (2010). https://doi.org/10.1134/S0001434610070230
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DOI: https://doi.org/10.1134/S0001434610070230