Abstract:
This paper is concerned with a system of equations that describes the motion of two point vortices in a flow possessing constant uniform vorticity and perturbed by an acoustic wave. The system is shown to have both regular and chaotic regimes of motion. In addition, simple and chaotic attractors are found in the system. Attention is given to bifurcations of fixed points of a Poincaré map which lead to the appearance of these regimes. It is shown that, in the case where the total vortex strength changes, the “reversible pitch-fork” bifurcation is a typical scenario of emergence of asymptotically stable fixed and periodic points. As a result of this bifurcation, a saddle point, a stable and an unstable point of the same period emerge from an elliptic point of some period. By constructing and analyzing charts of dynamical regimes and bifurcation diagrams we show that a cascade of period-doubling bifurcations is a typical scenario of transition to chaos in the system under consideration.
Keywords:
point vortices, shear flow, perturbation by an acoustic wave, bifurcations, reversible pitch-fork, period doubling.
The work of E.V.Vetchanin (Sections 4, 5 and 6) was supported by RSF grant № 15-12-20035. The work of I.S.Mamaev (Introduction and Sections 1, 2 and 3) was carried out within the framework of the state assignment of the Ministry of Education and Science of Russia (1.2405.2017/4.6) and supported by RFBR grant № 15-38-20879 mol_a_ved.
Citation:
Evgeny V. Vetchanin, Ivan S. Mamaev, “Dynamics of Two Point Vortices in an External Compressible Shear Flow”, Regul. Chaotic Dyn., 22:8 (2017), 893–908
\Bibitem{VetMam17}
\by Evgeny V. Vetchanin, Ivan S. Mamaev
\paper Dynamics of Two Point Vortices in an External Compressible Shear Flow
\jour Regul. Chaotic Dyn.
\yr 2017
\vol 22
\issue 8
\pages 893--908
\mathnet{http://mi.mathnet.ru/rcd298}
\crossref{https://doi.org/10.1134/S1560354717080019}
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Linking options:
https://www.mathnet.ru/eng/rcd298
https://www.mathnet.ru/eng/rcd/v22/i8/p893
This publication is cited in the following 10 articles:
Elizaveta M. Artemova, Evgeny V. Vetchanin, “The Motion of an Unbalanced Circular Disk
in the Field of a Point Source”, Regul. Chaotic Dyn., 27:1 (2022), 24–42
E. M. Artemova, E. V. Vetchanin, “Control of the motion of a circular cylinder in an ideal fluid using a source”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 30:4 (2020), 604–617
S. P. Kuznetsov, V. P. Kruglov, A. V. Borisov, “Chaplygin sleigh in the quadratic potential field”, EPL, 132:2 (2020), 20008
V. Chigarev, A. Kazakov, A. Pikovsky, “Kantorovich-Rubinstein-Wasserstein distance between overlapping attractor and repeller”, Chaos, 30:7 (2020)
A. Kazakov, “Merger of a Henon-like attractor with a Henon-like repeller in a model of vortex dynamics”, Chaos, 30:1 (2020), 011105
Vyacheslav P. Kruglov, Sergey P. Kuznetsov, “Topaj – Pikovsky Involution in the Hamiltonian Lattice of Locally Coupled Oscillators”, Regul. Chaotic Dyn., 24:6 (2019), 725–738
Alexey V. Borisov, Ivan S. Mamaev, Eugeny V. Vetchanin, “Dynamics of a Smooth Profile in a Medium with Friction in the Presence of Parametric Excitation”, Regul. Chaotic Dyn., 23:4 (2018), 480–502
Alexey V. Borisov, Ivan S. Mamaev, Ivan A. Bizyaev, “Three Vortices in Spaces of Constant Curvature: Reduction, Poisson Geometry, and Stability”, Regul. Chaotic Dyn., 23:5 (2018), 613–636
Alexey V. Borisov, Ivan S. Mamaev, Evgeny V. Vetchanin, “Self-propulsion of a Smooth Body in a Viscous Fluid Under Periodic Oscillations of a Rotor and Circulation”, Regul. Chaotic Dyn., 23:7-8 (2018), 850–874
Ivan S. Mamaev, Evgeny V. Vetchanin, “The Self-propulsion of a Foil with a Sharp Edge in a Viscous Fluid Under the Action of a Periodically Oscillating Rotor”, Regul. Chaotic Dyn., 23:7-8 (2018), 875–886
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