2017
Том 69
№ 7

All Issues

Behaviour of particularly perturbed autonomous nonlinear differential systems in the differential systems in the neighbourhood of a family of cylinders

Zadiraka K. V.

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Abstract

The author considers the system of differential equations $$\frac{dx}{dt} = f(x,z), \quad \varepsilon\frac{dz}{dt} = F(x, z),\quad (1)$$ and the corresponding degenerated system $$\frac{d\bar{x}}{dt} = f(\bar{x},\varphi(\bar{x})),\quad \bar{z} = \varphi(\bar{x}),\quad (2)$$ where $z = \varphi(x)$ is an isolated solution of the system $F{x,z) = 0$.

It is proved that if system (2) has a family of solutions $$\bar{x} = \bar{x}^0(\theta, c), \quad \bar{z} = \bar{z}^0(\theta, c),$$ periodic in $\theta = \omega (c)t + \varphi_0$ with a period of $2\pi$, family (1) has a stable family of periodic solutions $$x = x^0(\theta, c, \varepsilon), z = z^0(\theta, c, \varepsilon)$$ with the same period; furthermore $|x^0 - \bar{x}^0| \rightarrow 0, \quad |z^0 - \bar{z}^0| \rightarrow 0$ together with $\varepsilon$.

Citation Example: Zadiraka K. V. Behaviour of particularly perturbed autonomous nonlinear differential systems in the differential systems in the neighbourhood of a family of cylinders // Ukr. Mat. Zh. - 1962. - 14, № 3. - pp. 235-249.

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