Strange repeller in the dynamics of an elliptical foil with an attached vortex in an ideal fluid

Kilin A.A., Artemova E.M., Gavrilova A.M.
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This paper addresses the problem of the plane-parallel motion of an elliptic foil with an attached point vortex of constant strength in an ideal fluid. It is assumed that the position of the vortex relative to the foil remains unchanged during motion. The flow of the fluid outside the body is assumed to be potential (except for the singularity corresponding to a point vortex), and the flow around the body is noncirculatory. Special attention is given to the general position case in which the point vortex does not lie on the continuations of the semiaxes of the ellipse. The problem under consideration is described by a system of six first-order differential equations. After reduction by the motion group of the plane E(2) it reduces to a system of three differential equations. An analysis of this reduced system is made. It is shown that this system admits one to five fixed points which correspond to motions of the ellipse in various circles. By numerically investigating the phase flow of the reduced system near fixed points, it is shown that, in the general case, the system admits no invariant measure with a smooth positive definite density. Parameter values are found for which one of the fixed points of the reduced system is an unstable node-focus. It is shown that, as the variation of the parameters is continued, an unstable limit cycle can arise from an unstable fixed point via an Andronov – Hopf bifurcation. An analysis is made of bifurcations of this limit cycle for the case where the position of the point vortex relative to the ellipse changes. By constructing a parametric bifurcation diagram, it is shown that, as the system’s parameters are varied, the limit cycle undergoes a cascade of period-doubling bifurcations, giving rise to a chaotic repeller (a reversed-time attractor). To carry out a numerical analysis of the problem, the method of constructing a twodimensional Poincaré map is used. The search for and analysis of simple and strange repellers were performed backward in time.

Keywords: ideal fluid, elliptic foil, point vortex, chaos, strange attractor
Citation in English: Kilin A.A., Artemova E.M., Gavrilova A.M. Strange repeller in the dynamics of an elliptical foil with an attached vortex in an ideal fluid // Computer Research and Modeling, 2025, vol. 17, no. 6, pp. 1051-1067
Citation in English: Kilin A.A., Artemova E.M., Gavrilova A.M. Strange repeller in the dynamics of an elliptical foil with an attached vortex in an ideal fluid // Computer Research and Modeling, 2025, vol. 17, no. 6, pp. 1051-1067
DOI: 10.20537/2076-7633-2025-17-6-1051-1067
URL: http://crm-en.ics.org.ru/journal/article/3666/
Creative Commons License This work is licensed under a Creative Commons Attribution-NoDerivs 3.0 Unported License.

Copyright © 2025 Kilin A.A., Artemova E.M., Gavrilova A.M.

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