Unbounded solutions to systems of differential equations at resonance

TL;DR

This paper proves the existence of unbounded solutions in a resonant system of coupled differential equations using Lyapunov functions, under certain conditions on the coupling terms.

Contribution

It introduces a Lyapunov function approach for discrete dynamical systems to establish unbounded solutions at resonance for coupled ODEs.

Findings

Existence of unbounded solutions at resonance.

Lyapunov function method applied to coupled systems.

Conditions on coupling terms for unbounded solutions.

Abstract

We deal with a weakly coupled system of ODEs of the type $$ x_j'' + n_j^2 \,x_j + h_j(x_1,\ldots,x_d) = p_j(t), \qquad j=1,\ldots,d, $$ with $h_j$ locally Lipschitz continuous and bounded, $p_j$ continuous and $2\pi$-periodic, $n_j \in \mathbb{N}$ (so that the system is at resonance). By means of a Lyapunov function approach for discrete dynamical systems, we prove the existence of unbounded solutions, when either global or asymptotic conditions on the coupling terms $h_1,\ldots,h_d$ are assumed.