Stability and Asymptotic Equivalence of Perturbations of Nonlinear Systems of Differential Equations
A nonlinear variation of constants method was introduced by Alekseev  and applications of this formula to questions of stability and asymptotic equivalence of differential systems was demonstrated by Brauer [2,3,4]. In  a different approach to the nonlinear variation of constants method is given. This new approach involves determining the solution of the perturbed system by variation of the starting vector in the unperturbed system. Conceptually this is the method used in obtaining the classical variation of constants formula for perturbations of linear systems. In  the method yields two different formulas, one of which is equivalent to the Alekseev formula under the hypothesis which guarantees the Alekseev representation. Also, in  some applications to stability and asymptotic equilibrium are given. The approach introduced in  was shown to be applicable for the study of integral and integro-differential systems in  and for the study of difference equations in . In this paper some further applications of the nonlinear variation of constants result of  are obtained for differential equations. The result on asymptotic equivalence is related to that given by Brauer  and is shown to complement those results.