One-dimensional chaos in a system with dry friction: analytical approach


We introduce a new analytical method, which allows to find chaotic regimes in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered. The corresponding mathematical model is being studied. We show that the considered dynamical system is a skew product over a piecewise smooth mapping of a segment (the so-called base map). For this base map we demonstrate existence of a domain of parameters where a chaotic dynamics can be observed. We prove existence of an infinite set of periodic points of arbitrarily big period. Moreover, a reduction of the considered map to a compact subset of the segment is semi-conjugated to a shift on the set of one-sided infinite boolean sequences. We find conditions, sufficient for existence of a superstable periodic point of the base map. The obtained result partially solves a general problem: theoretical confirmation of chaotic and periodic regimes numerically and experimentally observed for models of percussion drilling.

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This work has been partially supported by Russian Foundation for Basic Researches, Grants 12-01-00275-a and 14-01-00202, by Saint-Petersburg State University under Thematic Plans and, by the UK Royal Society (joint research project of University of Aberdeen and Saint-Petersburg State University), by Centre for Research and by FEDER funds through COMPETE Operational Programme Factors of Competitiveness (Programa Operacional Factores de Competitividade) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications and the Portuguese Foundation for Science and Technology (FCT \(-\) Fundação para a Ciência e a Tecnologia), within projects PTDC/MAT/113470/2009 FCT and PEst-C/MAT/UI4106/2011 with COMPETE number FCOMP-01-0124-FEDER-022690. Authors are grateful to Prof. Ron Chen, Prof. Marian Wiercigroch, Prof. Viktor Avrutin, Dr. James Ing and anonymous Referees for their precious remarks.

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Correspondence to Sergey Kryzhevich.

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Begun, N., Kryzhevich, S. One-dimensional chaos in a system with dry friction: analytical approach. Meccanica 50, 1935–1948 (2015).

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  • Chaos
  • Mappings of segments
  • Dry friction
  • Filippov systems
  • Turbulence