Nonintegrability of time-periodic perturbations of single-degree-of-freedom Hamiltonian systems near homo- and heteroclinic orbits

TL;DR

This paper investigates the nonintegrability of time-periodic perturbations in single-degree-of-freedom Hamiltonian systems near special orbits, using advanced Morales-Ramis theory and Melnikov functions, with applications to Duffing oscillators.

Contribution

It extends nonintegrability results to heteroclinic orbits and strengthens conclusions for homoclinic orbits in perturbed Hamiltonian systems.

Findings

Non-constant Melnikov functions imply nonintegrability near special orbits.

Results apply to systems with finite Fourier series perturbations.

Illustrations include forced Duffing oscillators and 2D systems.

Abstract

We consider time-periodic perturbations of single-degree-of-freedom Hamiltonian systems and study their real-meromorphic nonintegrability in the Bogoyavlenskij sense using a generalized version due to Ayoul and Zung of the Morales-Ramis theory. The perturbation terms are assumed to have finite Fourier series in time, and the perturbed systems are rewritten as higher-dimensional autonomous systems having the small parameter as a state variable.We show that if the Melnikov functions are not constant, then the autonomous systems are not real-meromorphically integrable near homo- and heteroclinic orbits. Our result is not just an extension of previous results for homocliic orbits to heteroclinic orbits and provides a stronger conclusion than them for the case of homoclinic orbits. We illustrate the theory for two periodically forced Duffing oscillators and a periodically forced two-dimensional system.