DOI 10.15507/2079-6900.26.202403.280-293

*Original article*

ISSN 2079-6900 (Print)

ISSN 2587-7496 (Online)

*MSC2020 34D20*

#### On the partial instability of the zero solution of nonlinear systems to the first approximation

**P. A. Shamanaev**

*National Research Mordovia State University (Saransk, Russian Federation)*

**Abstract.** Sufficient conditions for instability with respect to a part of the variables of the zero solution of a nonlinear system in the linear approximation are obtained. The results are presented when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. The results are given for the cases when the right-hand of the system under study is presented both in the most general form and in the form of a vector polynomial. As a first approximation, a linear system of ordinary differential equations with a constant matrix is taken, whose eigenvalues may have zero real parts. Moreover, algebraic and geometric multiplicities of these eigenvalues may not coincide. The approach is based on establishing some correspondence between the solutions of the system under study and its linear approximation. If such correspondence exists, solutions of such systems starting in a sufficiently small neighborhood of zero have some identical component-wise asymptotic properties. In particular, this article focuses on solution instability with respect to some variables, which is one of such properties. Conditions are given for the case when the instability properties of the zero solution of one system are preserved upon transition to another system. The paper gives an example of instability with respect to a part of variables of the zero solution of a nonlinear system, whose linear approximation matrix contains one positive, one negative and one zero eigenvalue, and algebraic and geometric multiplicities of the zero eigenvalue do not coincide.

**Key Words:** ordinary differential equations, partial instability, uniform local componen-twise asymptotic equivalence, first approximation

**For citation:** *P. A. Shamanaev.* On the partial instability of the zero solution of nonlinear systems to the first approximation. *Zhurnal Srednevolzhskogo matematicheskogo obshchestva*. 26:3(2024), 280–293. DOI: https://doi.org/10.15507/2079-6900.26.202403.280-293

*
Submitted: 21.05.2024;
Revised: 10.07.2024;
Accepted: 28.08.2024 *

*Information about the author:*

**Pavel A. Shamanaev**, Ph. D. in Phys. and Math., Associate Professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68 Bolshevistskaya Str., Saransk 430005, Republic of Mordovia, Russia), ORCID: http://orcid.org/0000-0002-0135-317X, korspa@yandex.ru

*The author have read and approved the final manuscript.*

*Conflict of interest: The author declare no conflict of interest.*