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2013
Impact Factor

Alexandr Kovalev

74, R.Luxemburg, 83114, Donetsk, Ukraine
Institute of Applied Mathematics and Mechanics of the National Academy of Sciences of Ukraine

Publications:

Kovalev A. M.
Invariant and integral manifolds of dynamical systems and the problem of integration of the Euler–Poisson equations
2004, vol. 9, no. 1, pp.  59-72
Abstract
The problem on inclusion of an invariant manifold of a dynamical system into a set of integral manifolds is considered. It is shown that this inclusion is always possible unless the invariant manifold is a singular one (consisting of singular points of the system) of codimension one. It enables us to study invariant manifolds using the equation for integrals instead of the Levi-Civita equations containing uncertain factors. The defining role of the singular manifolds in shaping a phase portrait of a dynamical system is established and the following from this integrals properties are obtained. The results are applied to the analysis of solutions of the Euler–Poisson equations. A new treatment of the fourth integrals in Euler's, Lagrange's, Kovalevskaya's cases is proposed. It is proved that under the Hess conditions there is a fourth integral of which special cases are Euler's and Lagrange's integrals as well as the Hess and the Dokshevich solutions.
Citation: Kovalev A. M.,  Invariant and integral manifolds of dynamical systems and the problem of integration of the Euler–Poisson equations, Regular and Chaotic Dynamics, 2004, vol. 9, no. 1, pp. 59-72
DOI:10.1070/RD2004v009n01ABEH000265

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