TY - JOUR

T1 - One-dimensional chaos in a system with dry friction: analytical approach

AU - Begun, N.

AU - Kryzhevich, S.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Chaos 􏱮 Mappings of segments 􏱮 Dry friction 􏱮 Filippov systems 􏱮 Turbulence

U2 - 10.1007/s11012-014-0071-2

DO - 10.1007/s11012-014-0071-2

M3 - Article

VL - 50

SP - 1935

EP - 1948

JO - Meccanica

JF - Meccanica

SN - 0025-6455

IS - 8

ER -