A generalized Poincar&eacute;-Birkhoff theorem with applications to coaxial vortex ring motion

Denis Blackmore; Jyoti Champanerkar; Chengwen Wang

doi:10.3934/dcdsb.2005.5.15

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2005, Volume 5, Issue 1: 15-33. Doi: 10.3934/dcdsb.2005.5.15

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A generalized Poincaré-Birkhoff theorem with applications to coaxial vortex ring motion

1.
Department of Mathematical Sciences and Center for Applied Mathematics and Statistics, New Jersey Institute of Technology, Newark, NJ 07102-1984
2.
Department of Mathematics and Computer Science, Rutgers University-Newark, Newark, NJ 07102

Received: August 31, 2003

Revised: January 31, 2004

Published: January 2005

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Abstract

A new generalization of the Poincaré-Birkhoff fixed point theorem applying to small perturbations of finite-dimensional, completely integrable Hamiltonian systems is formulated and proved. The motivation for this theorem is an extension of some recent results of Blackmore and Knio on the dynamics of three coaxial vortex rings in an ideal fluid. In particular, it is proved using KAM theory and this new fixed point theorem that if $n>3$ coaxial rings all having vortex strengths of the same sign are initially in certain positions sufficiently close to one another in a three-dimensional ideal fluid environment, their motion with respect to the center of vorticity exhibits invariant $(n-1)$-dimensional tori comprised of quasiperiodic orbits together with interspersed periodic trajectories.

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Mathematics Subject Classification: 37J10, 37J40, 37J45, 37N10, 76B47.

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