Studies of perturbed three vortex dynamics


Journal Article

It is well known that the dynamics of three point vortices moving in an ideal fluid in the plane can be expressed in Hamiltonian form, where the resulting equations of motion are completely integrable in the sense of Liouville and Arnold. The focus of this investigation is on the persistence of regular behavior (especially periodic motion) associated with completely integrable systems for certain (admissible) kinds of Hamiltonian perturbations of the three vortex system in a plane. After a brief survey of the dynamics of the integrable planar three vortex system, it is shown that the admissible class of perturbed systems is broad enough to include three vortices in a half plane, three coaxial slender vortex rings in three space, and "restricted" four vortex dynamics in a plane. Included are two basic categories of results for admissible perturbations: (i) general theorems for the persistence of invariant tori and periodic orbits using Kolmogorov-Arnold-Moser- and Poincaƕ-Birkhoff-type arguments and (ii) more specific and quantitative conclusions of a classical perturbation theory nature guaranteeing the existence of periodic orbits of the perturbed system close to cycles of the unperturbed system, which occur in abundance near centers. In addition, several numerical simulations are provided to illustrate the validity of the theorems as well as indicating their limitations as manifested by transitions to chaotic dynamics. © 2007 American Institute of Physics.

Full Text

Cited Authors

  • Blackmore, D; Ting, L; Knio, O

Published Date

  • June 25, 2007

Published In

Volume / Issue

  • 48 / 6

International Standard Serial Number (ISSN)

  • 0022-2488

Digital Object Identifier (DOI)

  • 10.1063/1.2428272

Citation Source

  • Scopus