Nonintegrability of nearly integrable dynamical systems near resonant periodic orbits

TL;DR

This paper extends a technique to prove nonintegrability of nearly integrable dynamical systems near resonant periodic orbits, demonstrating its application to various perturbed Hamiltonian systems and their stability analysis.

Contribution

It applies and illustrates a technique for establishing nonintegrability in perturbed Hamiltonian systems, linking it with the subharmonic Melnikov method.

Findings

The technique detects nonintegrability near resonant orbits.

Application to the periodically forced Duffing oscillator.

Analysis of stability and periodic orbits in example systems.

Abstract

In a recent paper by the author (K. Yagasaki, Nonintegrability of the restricted three-body problem, submitted for publication), a technique was developed for determining whether nearly integrable systems are not meromorphically Bogoyavlenskij-integrable such that the first integrals and commutative vector fields also depend meromorphically on the small parameter. Here we continue to demonstrate the technique for some classes of dynamical systems. In particular, we consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and discuss a relationship of the technique with the subharmonic Melnikov method, which enables us to detect the existence of periodic orbits and their stability. We illustrate the theory for the periodically forced Duffing oscillator and two more additional examples: second-order coupled oscillators and a two-dimensional system of pendulum-type subjected to a constant torque.