Chaos and resonances in a rotating flow disturbed by а periodic motion of a point vortex

A. I. Gudimenko

Russian journal of nonlinear dynamics, Tome 3 (2007) no. 1, pp. 33-48.

Résumé

A Hamiltonian dynamical system describing a rotating incompressible two-dimensional flow, disturbed by an oscillating point vortex, is studied numerically and analytically. It is shown numerically that under perturbation the region of strongly mixed trajectories of the system forms. As the amplitude of perturbation increases, the region grows in size due to the destruction and absorption of the nearest resonances. The order and multiplicity of the resonances are determined mainly by the relation $\omega/\Omega$, where $\omega$ is the perturbation frequency and $\Omega$ is the rotation frequency of the flow. The patterns of the resonances differ essentially whether this quantity is integer or fractional. The results of the numerical experiment are justified analytically. In the domain that is sufficiently far from the vortex, the Hamiltonian is represented in the angle–action variables. Based on the representation, the arrangement of the resonances on the phase plane is analyzed. In particular, a classification of the resonances, which is adequate to the numerical patterns, is proposed. The widths of the resonances are calculated. It is shown that, at large distances from the vortex, global chaotization of trajectories of the system is impossible.

Détail

Mots-clés : point vortex
Keywords: chaotic dynamics, nonlinear resonances.