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Russia Moscow
Year
2019
Volume
29
Issue
1
Pages
3-18
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Section Mathematics
Title Bifurcation study of transition to chaos in the oscillatory system of motion of a plate in a liquid
Author(-s) Gurina T.A.a
Affiliations Moscow Aviation Institutea
Abstract We consider the model of chaotic motion of a plate in a viscous fluid, described by an oscillatory system of three ordinary differential equations with a quadratic nonlinearity. In the course of the bifurcation study of singular points of the system, maps of the types of singular points are constructed and a surface equation is found in the space of dissipation and circulation parameters on which the Andronov-Hopf bifurcation of the limit cycle creation takes place. With a further change in the parameters near the Andronov-Hopf surface, cascades of the period doubling doubling of the Feigenbaum cycle and the Sharkovsky subharmonic cascades, ending with the creation of a cycle of period three, are found. Expressions are obtained for saddle numbers of the saddle-node and two saddle-foci and their plots are plotted in the parameter space. It is shown that homoclinic cascades of bifurcations are realized in the system with the destruction of homoclinic trajectories of saddle-foci. The existence of homoclinic trajectories of saddle-foci is proved by a numerical-analytical method. The graphs of the largest Lyapunov exponent and the bifurcation diagrams show that when the dissipation coefficients change, the system switches to chaos in several stages.
Keywords motion of a body in a liquid, singular point, limit cycle, homoclinic trajectory, cascade of bifurcations, attractor, chaos, largest Lyapunov exponent
UDC 517.938, 531.36, 534.1
MSC 34C15, 34C23, 34C25
DOI 10.20537/vm190101
Received 17 October 2018
Language Russian
Citation Gurina T.A. Bifurcation study of transition to chaos in the oscillatory system of motion of a plate in a liquid, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2019, vol. 29, issue 1, pp. 3-18.
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