Periodic orbits in analytically perturbed Poisson systems

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.

References in corpus (6)

• Generalization of solutions of the Jacobi PDEs associated to time reparametrizations of Poisson systems
• Perturbed Euler top and bifurcation of limit cycles on invariant Casimir surfaces
• Generalized results on the role of new-time transformations in finite-dimensional Poisson systems
• Generalization of the separation of variables in the Jacobi identities for finite-dimensional Poisson systems
• New global solutions of the Jacobi partial differential equations
• Poisson systems as the natural framework for additional first integrals via Darboux invariant hypersurfaces

Cited by in corpus (3)

• Inverse Jacobi multiplier as a link between conservative systems and Poisson structures
• Congruence method for global Darboux reduction of finite-dimensional Poisson systems
• Perturbed rank 2 Poisson systems and periodic orbits on Casimir invariant manifolds