Nonintegrability of Forced Nonlinear Oscillators

TL;DR

This paper reviews and extends techniques to prove the nonintegrability of forced nonlinear oscillators, specifically demonstrating the nonintegrability of the periodically forced damped pendulum using real-analytic and complex-meromorphic methods.

Contribution

It introduces a unified approach to establish nonintegrability of forced oscillators, applying it to the forced damped pendulum and emphasizing the role of small parameters.

Findings

Proved nonintegrability of the forced damped pendulum.

Validated the effectiveness of the techniques for real-analytic and complex-meromorphic nonintegrability.

Extended previous methods to include damping and forcing in oscillator analysis.

Abstract

In recent papers by the authors (S.~Motonaga and K.~Yagasaki, Obstructions to integrability of nearly integrable dynamical systems near regular level sets, submitted for publication, and K.~Yagasaki, Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, submitted for publication), 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 and prove its nonintegrability in the above meaning.