**MSC2010** 34C20

### The sufficient conditions of local asymptotic equivalence of nonlinear systems of ordinary differential equations and its application for investigation of stability respect to part of variables

#### P. A. Shamanaev^{1}, O. S. Yazovtseva^{2}

Annotation | The article states sufficient conditions of local component-wise asymptotic equivalence for nonlinear systems of ordinary differential equations with perturbations in form of vector polynoms. The proof method is based on constructing of operator in Banach space, which connects solutions of nonlinear system and its linear approach, and using the Shauder principle for fixed point. The existing of constructed operator is proved by using component-wise estimates of elements of fundamental matrix of linear approach. The operator allows to construct mapping, which establishes relation between initial points of nonlinear system and initial points of its linear approach. Sufficient conditions for the stability (asymptotic stability) of zero solutions of locally componentwise asymptotically equivalent systems according to Brauer are presented. The problem of stability with respect to a part of variables of equilibrium of system of nonlinear equations, which corresponds to a kinetic model of certain stages of compact scheme of propane pyrolisys reaction, is considered as application. The assigned task is reduced to investigation of trivial equilibrium of nonlinear system, which coincides with explored system. Then, there is shown, that nonlinear system is local component-wise equivalent by Brauer to its linear approach. Considering, that the trivial solution of linear approach is asymptotic stable with respect to the first two variables and has asymptotic equilibrium with respect to the other variables, the conclusion, that each of equilibriums of explored system has the same properties, is drawn. |
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Keywords | nonlinear ordinary differential equations, local component-wise Brauer asymptotic equivalence, the Shauder principle for a fixed point, stability with respect to a part of variables, chemical kinetics. |

^{1}**Pavel A. Shamanaev**, 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), Ph.D. (Physics and Mathematics), ORCID: http://orcid.org/0000-0002-0135-317X, korspa@yandex.ru^{2}**Olga S. Yazovtseva**, Postgraduate student, 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/0000-0001-8075-4491, kurinaos@gmail.com

**Citation**: P. A. Shamanaev, O. S. Yazovtseva, "[The sufficient conditions of local asymptotic equivalence of nonlinear systems of ordinary differential equations and its application for investigation of stability respect to part of variables]", Zhurnal Srednevolzhskogo matematicheskogo obshchestva,19:1 (2017) 102–115 (In Russian)

**DOI** 10.15507/2079-6900.19.2017.01.102-115