On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction
type: Item
instance of: scholarly article
title: On Lyapunov exponents for non-smooth dynamical systems with an application to a pendulum with dry friction
zbMATH Open document ID: 0954.34045
DOI: 10.1023/A:1009046702601
published in: Journal of Dynamics and Differential Equations
publication date: 2001-01-23T00:00:00Z
review text: This very large article contains an impressive number of very technical concepts and results attempting to construct a general theory of Lyapunov exponents for nonsmooth (measurable) semiflows and to apply it to the example of the pendulum with dry friction that is modelled by discontinuous differential equations of the form: \[ x''+x+sgn(x')=\gamma .\sin (\eta .t), \;\gamma , \eta \geq 0. \] The ``measurable semiflow'' introduced by the author and described by some ten ``axioms'' seems to be, essentially, a finite ``concatenation'' of smooth flows, with smooth ``switching'' times and surfaces that are related in such a way that a reasonable ``canonical cocycle'', \(T(.,.)\), that is replacing the derivative, \(T(t,y)=\partial_y\phi (t,y)\), in the case of a smooth semiflow, and therefore the Lyapunov exponents: \[ \lambda (y_0,y)=\lim_{t\to \infty} {{1}\over {t}}\ln|T(t,y_0)y|, \;y_0,y\in R^d, \] could be defined and the so-called ``Oseledets' multiplicative ergodic theorem'' could be applied. A large part of the paper (some 35pp) is dedicated to the study of the continuous but nonsmooth flow of the pendulum with dry friction, verifying the ten axioms of the general concept of measurable semiflow, proving the existence and stability of periodic solutions and the fact that its Lyapunov exponents are nonpositive and concluding that the system does not show ``chaotic behaviour''.
Mathematics Subject Classification ID: 34D08
Mathematics Subject Classification ID: 34C23
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34A36
Mathematics Subject Classification ID: 70E15
zbMATH DE Number: 1453228
zbMATH Keywords: discontinuous differential system
zbMATH Keywords: measurable semiflow
zbMATH Keywords: canonical cocycle
zbMATH Keywords: Lyapunov exponent
zbMATH Keywords: Oseledets' multiplicative ergodic theorem
zbMATH Keywords: pendulum with dry friction
author: Markus Kunze
reviewed by: Ştefan Mirică
MaRDI profile type: Publication
full work available at URL: a:1009046702601
OpenAlex ID: W200182404
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description: scientific article; zbMATH DE number 1453228