Periodic orbits in analytically perturbed Poisson systems

TL;DR

This paper demonstrates that analytically perturbed Poisson systems can be transformed into a standard form, enabling the use of averaging theory to analyze periodic orbits and bifurcations in various oscillator models.

Contribution

It shows that perturbed Poisson systems are analytically orbitally conjugate to planar harmonic oscillators, facilitating the study of periodic orbits through averaging theory.

Findings

Perturbed systems are conjugate to harmonic oscillators on symplectic leaves.

Averaging theory up to second order helps analyze periodic orbits.

Examples include harmonic, Duffing oscillators, and zero-Hopf singularities.

Abstract

Analytical perturbations of a family of finite-dimensional Poisson systems are considered. It is shown that the family is analytically orbitally conjugate in $U \subset \mathbb{R}^n$ to a planar harmonic oscillator defined on the symplectic leaves. As a consequence, the perturbed vector field can be transformed in the domain $U$ to the Lagrange standard form. On the latter, use can be made of averaging theory up to second order to study the existence, number and bifurcation phenomena of periodic orbits. Examples are given ranging from harmonic oscillators with a potential and Duffing oscillators, to a kind of zero-Hopf singularity analytic normal form.