Non-smooth dynamical systems
type: Item
instance of: scholarly article
title: Non-smooth dynamical systems
zbMATH Open document ID: 0965.34026
published in: Lecture Notes in Mathematics
publication date: 2000-10-17T00:00:00Z
review text: The author deals with nonsmooth phenomena in dynamical systems theory. The canonical example under consideration is a pendulum-type equation with dry (sometimes also called Coulomb's) friction, expressed by the term \(\text{sgn }x'\), namely \[ x''+ \text{sgn }x'+ x= p(t). \] The influence of the forcing term \(p\) on the structure of the related phase plane is investigated for \(p\) being bounded, periodic, almost-periodic, etc. In fact, the above model is only a simplification (the first approximation) of the pendulum-type equation \[ x''+ \text{sgn }x'+ \sin x= p(t), \] whose study is much more difficult. Nevertheless, it is quite sufficient for demonstrating the complexity of the related nonsmooth phenomena for which the standard methods fail. The methods rely mostly on some basic theorems from set-valued analysis, when using the approximation techniques, or so. The impresssion of various topics and approaches can be made from the names of nine chapters in the book: 1. Introduction; 2. Some general theory of differential inclusions; 3. Bounded, unbounded, periodic, and almost-periodic solutions; 4. Lyapunov exponents of nonsmooth dynamical systems; 5. On the application of Conley index theory to nonsmooth dynamical systems; 6. On the application of KAM theory to nonsmooth dynamical systems; 7. Planar nonsmooth dynamical systems; 8. Melnikov's mehod for nonsmooth dynamical systems; 9. Further topics and notes. The material in the book is well arranged and self-consistent. Therefore, it can be used as a suitable introduction to the study of nonsmooth dynamics.
reviewed by: Jan Andres
Mathematics Subject Classification ID: 34C15
Mathematics Subject Classification ID: 34-02
Mathematics Subject Classification ID: 37B30
Mathematics Subject Classification ID: 70K40
Mathematics Subject Classification ID: 37D25
Mathematics Subject Classification ID: 37J40
Mathematics Subject Classification ID: 70H05
zbMATH DE Number: 1517962
zbMATH Keywords: dry friction problems
zbMATH Keywords: bounded periodic and almost-periodic solutions
zbMATH Keywords: Conley index
zbMATH Keywords: KAM theory
zbMATH Keywords: Melnikov's method
author: Markus Kunze
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1517962