: Mario Bernardo, Chris Budd, Alan Richard Champneys, Piotr Kowalczyk
: Piecewise-smooth Dynamical Systems Theory and Applications
: Springer-Verlag
: 9781846287084
: 1
: CHF 149.20
:
: Analysis
: English
: 482
: Wasserzeichen/DRM
: PC/MAC/eReader/Tablet
: PDF

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.
Preface6
Acknowledgments11
Contents14
Glossary20
1 Introduction23
1.1 Why piecewise smooth?23
1.2 Impact oscillators25
1.3 Other examples of piecewise-smooth systems50
1.4 Non-smooth one-dimensional maps61
2 Qualitative theory of non-smooth dynamical systems69
2.1 Smooth dynamical systems69
2.2 Piecewise-smooth dynamical systems93
2.3 Other formalisms for non-smooth systems105
2.4 Stability and bifurcation of non-smooth systems115
2.5 Discontinuity mappings125
2.6 Numerical methods136
3 Border-collision in piecewise-linear continuous maps143
3.1 Locally piecewise-linear continuous maps143
3.2 Bifurcation of the simplest orbits150
3.3 Equivalence of border-collision classi.cation methods159
3.4 One-dimensional piecewise-linear maps165
3.5 Two-dimensional piecewise-linear normal form maps176
3.6 Maps that are noninvertible on one side181
3.7 Effects of nonlinear perturbations191
4 Bifurcations in general piecewise-smooth maps193
4.1 Types of piecewise-smooth maps193
4.2 Piecewise-smooth discontinuous maps196
4.3 Square-root maps210
4.4 Higher-order piecewise-smooth maps232
5 Boundary equilibrium bifurcations in flows241
5.1 Piecewise-smooth continuous flows241
5.2 Filippov flows255
5.3 Equilibria of impacting hybrid systems267
6 Limit cycle bifurcations in impacting systems275
6.1 The impacting class of hybrid systems275
6.2 Discontinuity mappings near grazing287
6.3 Grazing bifurcations of periodic orbits301
6.4 Chattering and the geometry of the grazing manifold317
6.5 Multiple collision bifurcation324
7 Limit cycle bifurcations in piecewise-smooth flows329
7.1 De.nitions and examples329
7.2 Grazing with a smooth boundary340
7.3 Boundary-intersection crossing bifurcations362
8 Sliding bifurcations in Filippov systems377
8.1 Four possible cases377
8.2 Motivating example: a relay feedback system386
8.3 Derivation of the discontinuity mappings395
8.4 Mapping for a whole period: normal form maps405
8.5 Unfolding the grazing-sliding bifurcation418
8.6 Other cases425
9 Further applications and extensions431
9.1 Experimental impact oscillators: noise and parameter sensitivity431
9.2 Rattling gear teeth: the similarity of impacting and piecewise- smooth systems444
9.3 A hydraulic damper: non-smooth invariant tori456
9.4 Two-parameter sliding bifurcations in friction oscillators470
References481
Index497