Piecewise-smooth Dynamical Systems (eBook)

Theory and Applications

Mario Bernardo, Chris Budd, Alan Richard Champneys, Piotr Kowalczyk (Autoren)

eBook Download: PDF

2008 | 1. Auflage
XXII, 482 Seiten
Springer London (Verlag)
978-1-84628-708-4 (ISBN)

Infos

This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction expounds the ubiquity of such models via numerous. The results are presented in an informal style, and illustrated with many examples. The book is aimed at a wide audience of applied mathematicians, engineers and scientists at the beginning postgraduate level. Almost no mathematical background is assumed other than basic calculus and algebra.

Inhalt

Preface 6
Acknowledgments 11
Contents 14
Glossary 20
1 Introduction 23
1.1 Why piecewise smooth? 23
1.2 Impact oscillators 25
1.3 Other examples of piecewise-smooth systems 50
1.4 Non-smooth one-dimensional maps 61
2 Qualitative theory of non-smooth dynamical systems 69
2.1 Smooth dynamical systems 69
2.2 Piecewise-smooth dynamical systems 93
2.3 Other formalisms for non-smooth systems 105
2.4 Stability and bifurcation of non-smooth systems 115
2.5 Discontinuity mappings 125
2.6 Numerical methods 136
3 Border-collision in piecewise-linear continuous maps 143
3.1 Locally piecewise-linear continuous maps 143
3.2 Bifurcation of the simplest orbits 150
3.3 Equivalence of border-collision classi.cation methods 159
3.4 One-dimensional piecewise-linear maps 165
3.5 Two-dimensional piecewise-linear normal form maps 176
3.6 Maps that are noninvertible on one side 181
3.7 Effects of nonlinear perturbations 191
4 Bifurcations in general piecewise-smooth maps 193
4.1 Types of piecewise-smooth maps 193
4.2 Piecewise-smooth discontinuous maps 196
4.3 Square-root maps 210
4.4 Higher-order piecewise-smooth maps 232
5 Boundary equilibrium bifurcations in flows 241
5.1 Piecewise-smooth continuous flows 241
5.2 Filippov flows 255
5.3 Equilibria of impacting hybrid systems 267
6 Limit cycle bifurcations in impacting systems 275
6.1 The impacting class of hybrid systems 275
6.2 Discontinuity mappings near grazing 287
6.3 Grazing bifurcations of periodic orbits 301
6.4 Chattering and the geometry of the grazing manifold 317
6.5 Multiple collision bifurcation 324
7 Limit cycle bifurcations in piecewise-smooth flows 329
7.1 De.nitions and examples 329
7.2 Grazing with a smooth boundary 340
7.3 Boundary-intersection crossing bifurcations 362
8 Sliding bifurcations in Filippov systems 377
8.1 Four possible cases 377
8.2 Motivating example: a relay feedback system 386
8.3 Derivation of the discontinuity mappings 395
8.4 Mapping for a whole period: normal form maps 405
8.5 Unfolding the grazing-sliding bifurcation 418
8.6 Other cases 425
9 Further applications and extensions 431
9.1 Experimental impact oscillators: noise and parameter sensitivity 431
9.2 Rattling gear teeth: the similarity of impacting and piecewise- smooth systems 444
9.3 A hydraulic damper: non-smooth invariant tori 456
9.4 Two-parameter sliding bifurcations in friction oscillators 470
References 481
Index 497

Produkt-Details

• Erscheint lt. Verlag: 1.1.2008
• Reihe/Serie: Applied Mathematical Sciences
• Zusatzinfo: XXII, 482 p.
• Verlagsort: London
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Analysis
• Themenwelt: Technik ► Bauwesen
• Themenwelt: Technik ► Elektrotechnik / Energietechnik
• Schlagworte: Bifurcation • Calculus • differential equation • Dynamical Systems • Electronics • Piecewise-Smooth • Standard
• ISBN-10: 1-84628-708-1 / 1846287081
• ISBN-13: 978-1-84628-708-4 / 9781846287084