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

TL;DR

This paper presents an analytical method to identify chaotic dynamics in a non-smooth mechanical system with dry friction, confirming the existence of chaos and periodic regimes through rigorous mathematical proofs.

Contribution

The authors develop a new analytical approach to demonstrate chaos in a non-smooth system, providing rigorous proofs of chaotic and periodic regimes.

Findings

Existence of robust chaotic dynamics in the system

Presence of an infinite set of periodic points with arbitrarily large periods

Conditions for superstable periodic points

Abstract

We introduce a new analytical method, which allows to find out chaotic dynamics in non-smooth dynamical systems. A simple mechanical system consisting of a mass and a dry friction element is considered as an example. The corresponding mathematical model is represented. We show that the considered dynamical system is a skew product over a piecewise smooth mapping of a segment (a base map). For this base map we demonstrate there is a domain of parameters where a robust chaotic dynamics can be observed. Namely, we prove existence of an infinite set of periodic points of arbitrarily big period. Moreover, a reduction of the considered map is semi-conjugated to a shift on the set of one-sided infinite boolean sequences. Also, we find conditions, sufficient for existence of a superstable periodic point of this 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.