Nonintegrability of forced nonlinear oscillators

In recent papers by the authors (Motonaga and Yagasaki, Arch. Ration. Mech. Anal. 247:44 (2023), J. Nonlinear Sci. 32:43 (2022)), two different techniques which allow us to prove real-analytic or complex-meromorphic nonintegrability of forced nonlinear oscillators having form time-periodic perturbations single-degree-of-freedom Hamiltonian systems were provided. Here concept in Bogoyavlenskij sense is adopted first integrals commutative vector fields are also required depend real-analytically complex-meromorphically on small parameter. this paper we review theories continue demonstrate their usefulness. particular, consider periodically damped pendulum, provides an especially important differential equation not only dynamical mechanics but other such as mechanical electrical engineering robotics, its above meaning.