Motion of Unstable Polygonal Ring of Vortex Points on Sphere With Pole Vortices

Sakajo, Takashi

doi:10.1007/1-4020-4181-0_34

Cited by 1 publication

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“…Generally speaking, the projection conditions (20) and (21) just restrict the system to the eigenspaces, and fail to reduce the N -ring system to the two-dimensional invariant dynamical system. That is to say, the projected iso-surface of the Hamiltonian disagree with the actual orbit of the N -ring evolution.…”

Section: Hamiltonian Projection Methodsmentioning

confidence: 99%

“…Hence, the unstable manifold corresponding to λ + M has the twodimensional tangent space, and thus the motion of the perturbed N -ring could be complicated for the odd case. Actually, numerical computation of the 3-ring at the equator [20] pointed out the existence of a heteroclinic structure in the highdimensional phase space, which results in a non-trivial recurrent evolution. In the present paper, we confirm the numerical conjecture by a proposing projection method.…”

Section: Introductionmentioning

confidence: 99%

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High-dimensional heteroclinic and homoclinic connections in odd point-vortex ring on a sphere

Sakajo

2005

Nonlinearity

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We consider the motion of the N -vortex points that are equally spaced along a line of latitude on sphere with fixed pole vortices, called "N -ring". We are especially interested in the case when the number of the vortex points is odd. Since the eigenvalues that determine the stability of the odd Nring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional. Hence, it is generally difficult to describe the global structure of the manifolds. In this article, based on the linear stability analysis, we propose a projection method to observe the structure of the iso-surface of the Hamiltonian, in which the orbit of the vortex points evolves. Applying the projection method to the motion of the 3-ring and 5-ring, we characterize the complex evolution of the unstable odd N -ring from the topological structure of the iso-surface of the Hamiltonian.

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Section: Hamiltonian Projection Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%