Nonlinear perturbations of Hill's systems and the pendulum with variable length

The stability of the trivial solution to the nonlinear Hill equation has been extensively studied in the literature. In this paper, we provide an approach mainly based on the application of KAM theory to relate the stability of the nonlinear equation to the stability of the linearised equation. In addition, we extend the stability result to the case where a quasi-periodic perturbation is added to the periodic forcing of Hill's equation. We focus on the pendulum with variable length, both because of its physical interest and for the sake of concreteness, but the analysis may be extended to any nonlinear Hill system. Therefore, rather than looking for optimal estimates, which strongly depend on the considered system, we emphasise the general strategy for studying the stability of both the linearised equations and the full nonlinear equations.