KAM theory analysis of the dynamics of three coaxial vortex rings

Denis Blackmore*, Omar Knio

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

15 Scopus citations


The equations of motion of three coaxial vortex rings in Euclidean 3-space are formulated as a Hamiltonian system. It is shown that the Hamiltonian function for this system can be written as the sum of a completely integrable part H0 (related to the motion of three point vortices in the plane) and a non-integrable perturbation H1. Then it is proved that when the vortex strengths all have the same sign and the ratio of the mean distances among the rings is very small in comparison to the mean radius of the rings, H1/H01. Moreover, it is shown that H1/H0 is very small for all time for certain initial positions of the rings under the same assumptions. It is proved that the decomposition of the Hamiltonian and the estimates carry over to a reduced form of the system in coordinates moving with the center of vorticity and having one less degree of freedom. Then KAM theory is applied to prove the existence of invariant two-dimensional tori containing quasiperiodic motions. The existence of periodic solutions is also demonstrated. Several examples are solved numerically to show transitions from quasiperiodic and periodic to chaotic regimes in accordance with the theoretical results.

Original languageEnglish (US)
Pages (from-to)321-348
Number of pages28
JournalPhysica D: Nonlinear Phenomena
Issue number3-4
StatePublished - Jun 15 2000
Externally publishedYes


  • Hamiltonian system
  • KAM theory
  • Point vortices
  • Vortex rings

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics
  • Condensed Matter Physics
  • Applied Mathematics


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