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

The article states sufficient conditions of local component-wise asymptotic equivalence for nonlinear systems of ordinary differential equations with perturbations in form of vector polynomials. The proof method is based on constructing of operator in Banach space, which connects solutions of nonlinear system and of its linear approximation, and on using the Shauder principle for fixed point. The existance of constructed operator is proved by using component-wise estimates for elements of fundamental matrix of linear approximation. The operator allows to construct mapping which establishes relation between initial points of nonlinear system and initial points of its linear approximation. Sufficient conditions for the stability (asymptotic stability) of zero solutions of locally component-wise asymptotically equivalent systems according to Brauer are presented. As an application of the theory built the nonlinear equations’system is considered which corresponds to the kinetic model of certain stages of compact scheme of propane pyrolysis reaction. The stability of equilibrium state of this system is investigated. The assigned task reduces to investigation of trivial equilibrium of nonlinear system coinciding with explored system. Then it is shown that nonlinear system is locally component-wise equivalent according to Brauer to its linear approximation. Taking in mind that trivial solution of linear approach is asymptotically stable with respect to the first two variables and has asymptotic equilibrium with respect to the other variables the conclusion is drawn that allthe equilibria of explored system have the same properties.