On the influence of resonances on the asymptotic behavior of trajectories of nonlinear systems in critical cases

This paper is devoted to the study of the asymptotic behavior of an essentially nonlinear system with resonant frequencies. Namely, it is assumed that the matrix of linear approximation of the system has several subsets of multiple purely imaginary eigenvalues. For such systems, the paper presents sufficient conditions for the asymptotic stability of the equilibrium regardless of forms higher than the third order. The main result is a power estimate for the norm of solutions of a system. A method for computing the coefficient of such an estimate is also proposed with use of the center manifold reduction and the normal form theory. It is shown that the order of the decay estimate varies for cases of a diagonalizable matrix of linear approximation and for a matrix containing a 2×2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$2 \times 2$$\end{document} Jordan block. As an example, the decay estimate and the Lyapunov function are constructed explicitly for a spring-pendulum system with partial dissipation.