Nonintegrability of forced nonlinear oscillators

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