Figure 1.
Phase portrait of system (3) depending on the parameters according to Theorem 2.1
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February
2020, 25(2): 749-760.
doi: 10.3934/dcdsb.2019265
Bifurcation of relative equilibria generated by a circular vortex path in a circular domain
| 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 |
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.
Keywords:
Vortex,
passive transport,
stirring protocol,
stream function,
periodic orbit,
winding number.
Mathematics Subject Classification:
Primary: 25C34, 37N10, 76B47.
Citation:
David Rojas, Pedro J. Torres. Bifurcation of relative equilibria generated by a circular vortex path in a circular domain. Discrete and Continuous Dynamical Systems - B,
2020, 25
(2)
: 749-760.
doi: 10.3934/dcdsb.2019265
![]()
References:
| [1] |
H. Aref,
Stirring by chaotic advection, J. Fluid Mech., 143 (1984), 1-21.
doi: 10.1017/S0022112084001233. |
| [2] |
H. Aref, J. Roenby, M. A. Stremler and L. Tophøj,
Nonlinear excursions of particles in ideal 2D flows, Phys. D, 240 (2011), 199-207.
doi: 10.1016/j.physd.2010.08.007. |
| [3] |
A. Boscaggin and P. J. Torres,
Periodic motions of fluid particles induced by a prescribed vortex path in a circular domain, Phys. D, 261 (2013), 81-84.
doi: 10.1016/j.physd.2013.07.004. |
| [4] |
T. Carletti and A. Margheri,
Measuring the mixing efficiency in a simple model of stirring: Some analytical results and a quantitative study via frequency map analysis, J. Phys. A, 39 (2006), 299-312.
doi: 10.1088/0305-4470/39/2/002. |
| [5] |
A. Fonda, M. Sabatini and F. Zanolin,
Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff Theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.
|
| [6] |
P. Franzese and L. Zannetti,
Advection by a point vortex in closed domains, European J. Mech. B Fluids, 12 (1993), 43-67.
|
| [7] |
R. Ortega, V. Ortega and P. J. Torres,
Point-vortex stability under the influence of an external periodic flow, Nonlinearity, 31 (2018), 1849-1867.
doi: 10.1088/1361-6544/aaa5e2. |
| [8] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.
doi: 10.1017/CBO9780511624063.
|
| [9] |
P. J. Torres, Mathematical Models With Singularities, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015.
doi: 10.2991/978-94-6239-106-2.
|
| [10] |
S. Wiggins and J. M. Ottino,
Foundations of chaotic mixing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362 (2004), 937-970.
doi: 10.1098/rsta.2003.1356. |
show all references
References:
| [1] |
H. Aref,
Stirring by chaotic advection, J. Fluid Mech., 143 (1984), 1-21.
doi: 10.1017/S0022112084001233. |
| [2] |
H. Aref, J. Roenby, M. A. Stremler and L. Tophøj,
Nonlinear excursions of particles in ideal 2D flows, Phys. D, 240 (2011), 199-207.
doi: 10.1016/j.physd.2010.08.007. |
| [3] |
A. Boscaggin and P. J. Torres,
Periodic motions of fluid particles induced by a prescribed vortex path in a circular domain, Phys. D, 261 (2013), 81-84.
doi: 10.1016/j.physd.2013.07.004. |
| [4] |
T. Carletti and A. Margheri,
Measuring the mixing efficiency in a simple model of stirring: Some analytical results and a quantitative study via frequency map analysis, J. Phys. A, 39 (2006), 299-312.
doi: 10.1088/0305-4470/39/2/002. |
| [5] |
A. Fonda, M. Sabatini and F. Zanolin,
Periodic solutions of perturbed Hamiltonian systems in the plane by the use of the Poincaré-Birkhoff Theorem, Topol. Methods Nonlinear Anal., 40 (2012), 29-52.
|
| [6] |
P. Franzese and L. Zannetti,
Advection by a point vortex in closed domains, European J. Mech. B Fluids, 12 (1993), 43-67.
|
| [7] |
R. Ortega, V. Ortega and P. J. Torres,
Point-vortex stability under the influence of an external periodic flow, Nonlinearity, 31 (2018), 1849-1867.
doi: 10.1088/1361-6544/aaa5e2. |
| [8] |
P. G. Saffman, Vortex Dynamics, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press, New York, 1992.
doi: 10.1017/CBO9780511624063.
|
| [9] |
P. J. Torres, Mathematical Models With Singularities, Atlantis Briefs in Differential Equations, 1, Atlantis Press, Paris, 2015.
doi: 10.2991/978-94-6239-106-2.
|
| [10] |
S. Wiggins and J. M. Ottino,
Foundations of chaotic mixing, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 362 (2004), 937-970.
doi: 10.1098/rsta.2003.1356. |
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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