Asymptotic behavior of solutions of a nonlinear system in the critical case of qq pairs of purely imaginary eigenvalues

This paper is devoted to the stability problem of a nonlinear system in the critical case of qq pairs of purely imaginary eigenvalues. The center manifold theory and the normal form method are exploited in this study. The main result of the paper is a power estimate for the solutions in the case when stability is ensured by third order forms. A Lyapunov function is constructed explicitly as a part of the proof of this estimate. The result obtained is illustrated by an example of a double pendulum with partial dissipation.

► An explicit estimate of the solutions is proposed for a class of nonlinear systems.
► A Lyapunov function is constructed in the critical case of q pairs of imaginary roots.
► The decay rate is analyzed for a double pendulum with partial dissipation.