The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations
P. A. Shamanaev1, O. S. Yazovtseva2
|The article states the sufficient polystability conditions for part of variables for nonlinear systems of ordinary differential equations with a sufficiently smooth right-hand side. The obtained theorem proof is based on the establishment of a local componentwise Brauer asymptotic equivalence. An operator in the Banach space that connects the solutions of the nonlinear system and its linear approximation is constructed. This operator satisfies the conditions of the Schauder principle, therefore, it has at least one fixed point. Further, using the estimates of the non-zero elements of the fundamental matrix, conditions that ensure the transition of the properties of polystability are obtained, if the trivial solution of the linear approximation system to solutions of a nonlinear system that is locally componentwise asymptotically equivalent to its linear approximation. There are given examples, that illustrate the application of proven sufficient conditions to the study of polystability of zero solutions of nonlinear systems of ordinary differential equations, including in the critical case, and also in the presence of positive eigenvalues.
|nonlinear ordinary differential equations, local componentwise Brauer asymptotic equivalence, the Shauder principle for a fixed point, stability with respect to a part of variables
1Pavel A. Shamanaev, Associate Professor, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-0135-317X, email@example.com
2Olga S. Yazovtseva, Postgraduate student, Department of Applied Mathematics, Differential Equations and Theoretical Mechanics, National Research Mordovia State University (68/1 Bolshevistskaya St., Saransk 430005, Russia), ORCID: http://orcid.org/0000-0001-8075-4491, firstname.lastname@example.org
Citation: P. A. Shamanaev, O. S. Yazovtseva, "[The sufficient conditions for polystability of solutions of~nonlinear systems of ordinary differential equations]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,20:3 (2018) 304–317 (In Russian)