An estimation of decay for solutions of nonlinear systems with equal frequencies

Abstract

This paper continues a series of studies on the stability and asymptotic behavior of solutions to systems of nonlinear differential equations whose matrix of linear approximation has purely imaginary eigenvalues and eigenvalues with negative real parts. In this paper, we consider the case of several purely imaginary eigenvalues related via second-order resonances. For such a system, sufficient conditions for the asymptotic stability regardless of forms higher than the third order are obtained. The main result provides a power estimate for the norm of solutions in the case of a diagonalizable matrix of linear approximation. The results obtained are illustrated by an example of a 7-DOF pendulum system.