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 Contemporary Mathematics and Its Applications, 2016, Volume 100, Pages 76–133 (Mi cma408)

Pendulum systems with dynamical symmetry

B. Ya. Lokshin, V. A. Samsonov, M. V. Shamolin

Lomonosov Moscow State University, Institute of Mechanics

Abstract: In this paper, we consider a class of oscillatory mechanical systems described by nonlinear second-order differential equations that contain parameters with variable dissipation and possess the property of dynamic symmetry. We study properties of mathematical models of such systems depending on parameters. Special attention is paid to bifurcations and reconstructions of phase portraits and the appearance and disappearance of cycles that envelope the phase cylinder or lie entirely on its covering. We present a complete parametric analysis of system with position-viscous friction and four parameters, where the force action linearly depends on the speed. The space of parameters is split into domains in which the topological behavior of the system is preserved. For some classes of pendulum systems whose right-hand sides depend on smooth functions, a qualitative analysis is performed, i.e., the space of systems considered is split into domains of different behavior of trajectories on the phase cylinder of quasi-velocities. We also perform a systematic analysis of the problem on an aerodynamic pendulum and detect periodic modes unknown earlier; these modes correspond to cycles in the mathematical model.

Full text: PDF file (6529 kB)

English version:
Journal of Mathematical Sciences, 2017, 227:4, 461–519

UDC: 517+531.01

Citation: B. Ya. Lokshin, V. A. Samsonov, M. V. Shamolin, “Pendulum systems with dynamical symmetry”, Contemporary Mathematics and Its Applications, 100 (2016), 76–133; Journal of Mathematical Sciences, 227:4 (2017), 461–519

Citation in format AMSBIB
\Bibitem{LokSamSha16} \by B.~Ya.~Lokshin, V.~A.~Samsonov, M.~V.~Shamolin \paper Pendulum systems with dynamical symmetry \jour Contemporary Mathematics and Its Applications \yr 2016 \vol 100 \pages 76--133 \mathnet{http://mi.mathnet.ru/cma408} \transl \jour Journal of Mathematical Sciences \yr 2017 \vol 227 \issue 4 \pages 461--519 \crossref{https://doi.org/10.1007/s10958-017-3597-8}