2017
Том 69
№ 6

All Issues

Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold

Lykova O. B.

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Abstract

For a system of $n + m$ equations $$\frac{dx}{dt} = X(y)x + \varepsilon X*(t, x, y),$$ $$\frac{dy}{dt} = \varepsilon Y(t, x, y),$$ where $x, X*, y, Y$ are respectively $n$ and $m$ vectors, $X — n \times n$ is the matrix, $\varepsilon$ is a small parameter, the author proves the theorem of the existence and properties of a two-dimensional local integral manifold in the neighbourhood of family of periodic solutions $$x = 0,\; y = y^0(\psi, a)$$ oi the lollowing auxiliary system $$\frac{dx}{dt} = X(y)x,$$ $$\frac{dy}{dt} = \varepsilon Y_0(x, y),$$ where $$Y_0(x, y) = \lim_{T\rightarrow 0}\int_0^T Y(t, x,y)dt.$$

Citation Example: Lykova O. B. Investigation of the solutions of a system of $n + m$ nonlineai differential equations in the vicinity of an integral manifold // Ukr. Mat. Zh. - 1964. - 16, № 1. - pp. 13-30.

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