Periodic Solutions of N-Vortex Type Hamiltonian Systems near the Domain Boundary

Dai, Qianhui; Gebhard, Björn; Bartsch, Thomas

doi:10.1137/16m1107085

Cited by 7 publications

(3 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The concept of choreography in vortex theory was used for the first time in [65]. Modern studies of choreography as applied to 2D vortex dynamics are presented in [65][66][67][68][69], and for a two-layer rotating fluid in [56,57,61].…”

Section: Examples Of Trajectories: Absolute and Relative Choreographiesmentioning

confidence: 99%

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

Koshel,

Sokolovskiy,

Dritschel

et al. 2024

Fluids

View full text Add to dashboard Cite

This paper seeks and examines N-symmetric vortical solutions of the two-layer geostrophic model for the special case when the vortices (or eddies) have vanishing summed strength (circulation anomaly). This study is an extension [Sokolovskiy et al. Phys. Fluids 2020, 32, 09660], where the general formulation for arbitrary N was given, but the analysis was only carried out for N=2. Here, families of stationary solutions are obtained and their properties, including asymptotic ones, are investigated in detail. From the point of view of geophysical applications, the results may help interpret the propagation of thermal anomalies in the oceans.

show abstract

Section: Examples Of Trajectories: Absolute and Relative Choreographiesmentioning

confidence: 99%

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

Koshel,

Sokolovskiy,

Dritschel

et al. 2024

Fluids

View full text Add to dashboard Cite

show abstract

“…Surprisingly, choreographies have been constructed recently for n-vortices in general bounded domains, using blow-up techniques. They are located close to a stagnation point of a vortex in the domain [4] and close to the boundary of the domain [15].…”

Section: Introductionmentioning

confidence: 99%

Choreographies in the n-vortex Problem

2018

View full text Add to dashboard Cite

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.

show abstract

“…Recently, choreographies have also been constructed for n vortices in general bounded domains with blow-up techniques. They are located close to stagnation points of one vortex in the domain [3], and close to its boundary [13].…”

mentioning

confidence: 98%

Relative periodic solutions of the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-vortex problem on the sphere

García-Azpeitia¹

2019

Journal of Geometric Mechanics

View full text Add to dashboard Cite

This paper gives an analysis of the movement of n vortices on the sphere. When the vortices have equal circulation, there is a polygonal solution that rotates uniformly around its center. The main result concerns the global existence of relative periodic solutions that emerge from this polygonal relative equilibrium. In addition, it is proved that the families of relative periodic solutions contain dense sets of choreographies.

show abstract

Periodic Solutions of N-Vortex Type Hamiltonian Systems near the Domain Boundary

Cited by 7 publications

References 24 publications

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

N-Symmetric Interaction of N Hetons, II: Analysis of the Case of Arbitrary N

Choreographies in the n-vortex Problem

Relative periodic solutions of the <inline-formula><tex-math id="M1">\begin{document}$ n $\end{document}</tex-math></inline-formula>-vortex problem on the sphere

Contact Info

Product

Resources

About