Bifurcation of relative equilibria generated by a circular vortex path in a circular domain

David Rojas; Pedro J. Torres

doi:10.3934/dcdsb.2019265

Article Contents

2020, Volume 25, Issue 2: 749-760. Doi: 10.3934/dcdsb.2019265

This issue Previous Article On the approximation of fixed points for non-self mappings on metric spaces Next Article On an optimal control problem of time-fractional advection-diffusion equation

Bifurcation of relative equilibria generated by a circular vortex path in a circular domain

David Rojas^1, , and
Pedro J. Torres^2,

1.
Departament d'Informàtica, Matemàtica Aplicada i Estadística, Universitat de Girona, 17003 Girona, Spain
2.
Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

^* Corresponding author: David Rojas (david.rojas@udg.edu)
^* Corresponding author: David Rojas (david.rojas@udg.edu)

Received: November 30, 2018

Revised: February 28, 2019

Early access: November 2019

Published: February 2020

All the authors are partially supported by the MINECO/FEDER grant MTM2017-82348-C2-1-P. The first author is also partially supported by the MINECO/FEDER grant MTM2017-86795-C3-1-P

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Abstract

We study the passive particle transport generated by a circular vortex path in a 2D ideal flow confined in a circular domain. Taking the strength and angular velocity of the vortex path as main parameters, the bifurcation scheme of relative equilibria is identified. For a perturbed path, an infinite number of orbits around the centers are persistent, giving rise to periodic solutions with zero winding number.

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Mathematics Subject Classification: Primary: 25C34, 37N10, 76B47.

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Figure 1. Phase portrait of system (3) depending on the parameters according to Theorem 2.1

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Figure 2. Bifurcation diagram of the phase-portrait of system (3) on $ D_R $. The bold curve corresponds to bifurcation parameters $ \mathcal B $, whereas the remaining ones correspond to regular parameters. In Theorem 2.1 the phase portrait at each region is given

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