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

This article is cited in 3 scientific papers (total in 3 papers)

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

Full text: PDF file (420 kB)
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UDC: 531.4
MSC: 37G15, 37G25
Received: 14.03.2012
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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    Citing articles on Google Scholar: Russian citations, English citations
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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  mathnet
    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  mathnet  crossref  elib
    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  mathnet  crossref  isi  elib
  • Нелинейная динамика
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