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 Nelin. Dinam., 2012, Volume 8, Number 2, Pages 231–247 (Mi nd319)

Analysis of discontinuous bifurcations in nonsmooth dynamical systems

A. P. Ivanov

Moscow Institute of Physics and Technology, Inststitutskii per. 9, Dolgoprudnyi, 141700, Russia

Abstract: Dynamical systems with discontinuous right-hand sides are considered. It is well known that the trajectories of such systems are nonsmooth and the fundamental solution matrix is discontinuous. This implies the presence of the so-called discontinuous bifurcations, resulting in a discontinuous change in the multipliers. A method of stepwise smoothing is proposed allowing the reduction of discontinuous bifurcations to a sequence of typical bifurcations: saddle-node, period doubling and Hopf bifurcations. The results obtained are applied to the analysis of the well-known system with friction a block on the moving belt, which serves as a popular model for the description of selfexcited frictional oscillations of a brake shoe. Numerical techniques used in previous investigations of this model did not allow general conclusions to be drawn as to the presence of self-excited oscillations. The new method makes it possible to carry out a complete qualitative investigation of possible types of discontinuous bifurcations in this system and to point out the regions of parameters which correspond to stable periodic regimes.

Keywords: non-smooth dynamical systems, discontinuous bifurcations, oscillator with dry friction

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UDC: 531.4
MSC: 37G15, 37G25
Accepted:07.05.2012

Citation: A. P. Ivanov, “Analysis of discontinuous bifurcations in nonsmooth dynamical systems”, Nelin. Dinam., 8:2 (2012), 231–247

Citation in format AMSBIB
\Bibitem{Iva12} \by A.~P.~Ivanov \paper Analysis of discontinuous bifurcations in nonsmooth dynamical systems \jour Nelin. Dinam. \yr 2012 \vol 8 \issue 2 \pages 231--247 \mathnet{http://mi.mathnet.ru/nd319} 

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• http://mi.mathnet.ru/eng/nd/v8/i2/p231

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This publication is cited in the following articles:
1. A. P. Ivanov, “O variatsionnoi formulirovke dinamiki sistem s treniem”, Nelineinaya dinam., 9:3 (2013), 478–498
2. E. M. Mukhamsdiev, I. D. Nurov, M. Sh. Khalilova, “Limiting cycles of piece-linear second order differential equations”, Ufa Math. J., 6:1 (2014), 80–89
3. M. K. Arabov, E. M. Mukhamadiev, I. D. Nurov, Kh. I. Sobirov, “Existence tests for limiting cycles of second order differential equations”, Ufa Math. J., 9:4 (2017), 3–11
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