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