Lyapunov orbits in the n-vortex problem

Carvalho, Adecarlos C.; Cabral, Hildeberto E.

doi:10.1134/s156035471403006x

Cited by 8 publications

(7 citation statements)

References 27 publications

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“…Bifurcation from the equilateral triangle: Recall that the minimum of H CP n-2 is achieved when the 4 vortices form the centred square (4-polygon). Thus the Moser-Weinstein theorem should show the existence of relative periodic solutions of short period, bifurcating from the square [6,7,5];…”

Section: Comparison With Other Methodsmentioning

confidence: 99%

“…For example methods based on the Lyapunov centre theorem around relative equilibria, or based on superposition principles (i.e. substituting a number of bodies in small relative equilibrium configuration for another body) have been applied successfully to find non-equilibrium periodic solutions in the n-vortex problem from Euler equation [5,7,3,4]. Even some particular qualitative properties (for instance some discrete symmetry of these orbits) could be deduced for these solutions.…”

Section: Absolute and Relative Periodic Orbitsmentioning

confidence: 99%

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Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Wang

2018

Preprint

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We study the existence of (relative) simple choreographies for a class of Hamiltonian systems describing the interaction of particles in the plane motivated mainly by the nvortex type problem. In particular, by constructing choreographic pseudo-holomorphic spheres, we prove that there exist infinitely many non-trivial relative choreographies for the identical n-vortex problem arising from both the Euler equation and the Gross-Pitaevskii equation.

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Section: Comparison With Other Methodsmentioning

confidence: 99%

Section: Absolute and Relative Periodic Orbitsmentioning

confidence: 99%

Choreographic Holomorphic Spheres with Application to Hamiltonian Systems of $N$-Vortex Type

Wang

2018

Preprint

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“…Due to rotational invariance there is always a zero frequency, ν 0 = 0. Since the frequencies have 1 : 1 resonances (ν k = ν n−k ), straightforward application of the Lyapunov center theorem is not possible, except for some cases with k = n/2 for n = 2, 4, 6, see [12]. For example, for n = 6 we have…”

Section: The N-polygon Of Vortices In the Planementioning

confidence: 99%

Choreographies in the n-vortex Problem

2018

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We consider the equations of motion of n vortices of equal circulation in the plane, in a disk and on a sphere. The vortices form a polygonal equilibrium in a rotating frame of reference. We use numerical continuation in a boundary value setting to determine the Lyapunov families of periodic orbits that arise from the polygonal relative equilibrium. When the frequency of a Lyapunov orbit and the frequency of the rotating frame have a rational relationship then the orbit is also periodic in the inertial frame. A dense set of Lyapunov orbits, with frequencies satisfying a diophantine equation, corresponds to choreographies of the n vortices. We include numerical results for all cases, for various values of n, and we provide key details on the computational approach.

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“…We will call such a relative equilibrium a ring of vortices. The planar case of a ring of vortices with and without a central vortex as well as the case of a ring of vortices on the sphere without a vortex at north pole were treated in the paper [10].…”

Section: Introductionmentioning

confidence: 99%