**MSC2010** 34A34, 34C25, 34C45

### To the problem of existence of integral manifolds systems of differential equations not solved with respect to derivatives

#### M. I. Kuptsov^{1}, M. T. Terekhin^{2}, V. V. Tenyaev^{3}

Annotation | The issue of finding a local non-zero integral manifold of a nonlinear (n + m) -dimensional system of ordinary differential equations not solved with respect to derivatives. It is assumed that the researching system has n-dimensional trivial integral manifold with all parameter points and corresponding linear subsystem has the m-parametric family of periodic solutions. In particular it means that the linear system does not have the property of exponential dichotomy. It is assumed that the matrix of linear approximation of the system with zero parameter points is a function of the independent variable. The problem of existence of integral manifolds is reduced to the issue of operator equations solution in the space of bounded Lipschitz-continuous periodic vector functions. Linearization is used to prove the existence of integral manifolds of the original system, where the method converts the matrix. The method of transforming the matrix has been used including the case of absence of a linear in the parameter of the operator equations members. The sufficient conditions of existence in the neighborhood of the equilibrium state of the system n-dimensional nonzero periodic integral manifolds were received. |
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Keywords | the method of transforming matrix, integral manifold, ordinary differential equations system, operator equation, dimensional reduction of phase space |

^{1}**Michail I. Kuptsov**, Head of the Department of Mathematics and Information Technology, The Academy of Law Management of the Federal Penal Service of Russia, (1 Sennaya Str., Ryazan 390000, Ryazan Region, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-6814-6423, kuptsov_michail@mail.ru.^{2}**Michail T. Terekhin** Professor of the Department of mathematics and methods of teaching of mathematical disciplines, Ryazan State University S.A. Esenin (46 Svobody Str., Ryazan 390000, Ryazan Region, Russia), Doctor of Physical and Mathematical Sciences, ORCID: http://orcid.org/0000-0002-5657-0155, m.terehin@rsu.edu.ru.^{3}**Victor V. Tenyaev**, Deputy Head of the Department of Mathematics and Information Technology, The Academy of Law Management of the Federal Penal Service of Russia (1 Sennaya Str., Ryazan 390000, Ryazan Region, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-3359-7152, tenyaevvv@yandex.ru.

**Citation**: M. I. Kuptsov, M. T. Terekhin, V. V. Tenyaev, "[To the problem of existence of integral manifolds systems of differential equations not solved with respect to derivatives]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:2 (2017) 76–84 (In Russian)

**DOI** 10.15507/2079-6900.19.201701.076-084