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Stability of Periodic Solutions of the N-vortex Problem in General Domains

Abstract

We investigate stability properties of a type of periodic solutions of the N-vortex problem on general domains Ω ⊂ ℝ2. The solutions in question bifurcate from rigidly rotating configurations of the whole-plane vortex system and a critical point a0 ∈ Ω of the Robin function associated to the Dirichlet Laplacian of Ω. Under a linear stability condition on the initial rotating configuration, which can be verified for examples consisting of up to 4 vortices, we show that the linear stability of the induced solutions is solely determined by the type of the critical point a0. If a0 is a saddle, they are unstable. If a0 is a nondegenerate maximum or minimum, they are stable in a certain linear sense. Since nondegenerate minima exist generically, our results apply to most domains Ω. The influence of the general domain Ω can be seen as a perturbation breaking the symmetries of the N-vortex system on ℝ2. Symplectic reduction is not applicable and our analysis on linearized stability relies on the notion of approximate eigenvectors. Beyond linear stability, Herman’s last geometric theorem allows us to prove the existence of isoenergetically orbitally stable solutions in the case of N = 2 vortices.

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Acknowledgments

B. G. is very grateful for the hospitality of R. O. and the group of Differential Equations during his research stay in Granada.

Funding

B. G. has been supported by DAAD grant 57314604. R. O. has been supported by MTM2017-82348-C2-1-P (Spain).

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Correspondence to Björn Gebhard or Rafael Ortega.

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Gebhard, B., Ortega, R. Stability of Periodic Solutions of the N-vortex Problem in General Domains. Regul. Chaot. Dyn. 24, 649–670 (2019). https://doi.org/10.1134/S1560354719060054

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Keywords

  • vortex dynamics
  • periodic solutions
  • stability
  • Floquet multipliers
  • bifurcation
  • Poincaré section

MSC2010 numbers

  • 37J25
  • 37J45
  • 37N10
  • 76B47