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Piecewise-smooth dynamical systems : theory and applications

Author: M Di Bernardo
Publisher: London : Springer, ©2008.
Series: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 163.
Edition/Format:   Print book : EnglishView all editions and formats
Summary:
"The primary purpose of this book is to present a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. An informal introduction asserts the ubiquity of such models with examples drawn from mechanics, electronics, control theory and physiology. The main thrust is to classify complex behavior via bifurcation theory in a systematic yet applicable way. The key concept is that of a  Read more...
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Material Type: Internet resource
Document Type: Book, Internet Resource
All Authors / Contributors: M Di Bernardo
ISBN: 9781846280399 1846280397
OCLC Number: 144515657
Description: xxi, 481 pages : illustrations ; 25 cm.
Contents: 1.1 Why piecewise smooth? 1 --
1.2 Impact oscillators 3 --
1.2.1 Case study I: A one-degree-of-freedom impact oscillator 6 --
1.2.2 Periodic motion 13 --
1.2.3 What do we actually see? 18 --
1.2.4 Case study II: A bilinear oscillator 26 --
1.3 Other examples of piecewise-smooth systems 28 --
1.3.1 Case study III: Relay control systems 28 --
1.3.2 Case study IV: A dry-friction oscillator 32 --
1.3.3 Case study V: A DC-DC converter 34 --
1.4 Non-smooth one-dimensional maps 39 --
1.4.1 Case study VI: A simple model of irregular heartbeats 39 --
1.4.2 Case study VII: A square-root map 42 --
1.4.3 Case study VIII: A continuous piecewise-linear map 44 --
2 Qualitative theory of non-smooth dynamical systems 47 --
2.1 Smooth dynamical systems 47 --
2.1.1 Ordinary differential equations (flows) 49 --
2.1.2 Iterated maps 53 --
2.1.3 Asymptotic stability 58 --
2.1.4 Structural stability 59 --
2.1.5 Periodic orbits and Poincare maps 63 --
2.1.6 Bifurcations of smooth systems 67 --
2.2 Piecewise-smooth dynamical systems 71 --
2.2.1 Piecewise-smooth maps 71 --
2.2.2 Piecewise-smooth ODEs 73 --
2.2.3 Filippov systems 75 --
2.2.4 Hybrid dynamical systems 78 --
2.3 Other formalisms for non-smooth systems 83 --
2.3.1 Complementarity systems 83 --
2.3.2 Differential inclusions 88 --
2.3.3 Control systems 91 --
2.4 Stability and bifurcation of non-smooth systems 93 --
2.4.1 Asymptotic stability 94 --
2.4.2 Structural stability and bifurcation 96 --
2.4.3 Types of discontinuity-induced bifurcations 100 --
2.5 Discontinuity mappings 103 --
2.5.1 Transversal intersections; a motivating calculation 105 --
2.5.2 Transversal intersections; the general case 107 --
2.5.3 Non-transversal (grazing) intersections 111 --
2.6 Numerical methods 114 --
2.6.1 Direct numerical simulation 115 --
2.6.2 Path-following 118 --
3 Border-collision in piecewise-linear continuous maps 121 --
3.1 Locally piecewise-linear continuous maps 121 --
3.1.2 Possible dynamical scenarios 125 --
3.1.3 Border-collision normal form map 127 --
3.2 Bifurcation of the simplest orbits 128 --
3.2.1 A general classification theorem 128 --
3.2.2 Notation for bifurcation classification 131 --
3.3 Equivalence of border-collision classification methods 137 --
3.3.1 Observer canonical form 137 --
3.3.2 Proof of Theorem 3.1 140 --
3.4 One-dimensional piecewise-linear maps 143 --
3.4.1 Periodic orbits of the map 145 --
3.4.2 Bifurcations between higher modes 147 --
3.4.3 Robust chaos 149 --
3.5 Two-dimensional piecewise-linear normal form maps 154 --
3.5.1 Border-collision scenarios 155 --
3.5.2 Complex bifurcation sequences 157 --
3.6 Maps that are noninvertible on one side 159 --
3.6.1 Robust chaos 159 --
3.6.2 Numerical examples 164 --
3.7 Effects of nonlinear perturbations 169 --
4 Bifurcations in general piecewise-smooth maps 171 --
4.1 Types of piecewise-smooth maps 171 --
4.2 Piecewise-smooth discontinuous maps 174 --
4.2.1 The general case 174 --
4.2.2 One-dimensional discontinuous maps 176 --
4.2.3 Periodic behavior: l = -1, v[subscript 1]> 0, v[subscript 2] <1 180 --
4.2.4 Chaotic behavior: l = -1, v[subscript 1]> 0, 1 <v[subscript 2] <2 185 --
4.3 Square-root maps 188 --
4.3.1 The one-dimensional square-root map 188 --
4.3.2 Quasi one-dimensional behavior 193 --
4.3.3 Periodic orbits bifurcating from the border-collision 199 --
4.3.4 Two-dimensional square-root maps 205 --
4.4 Higher-order piecewise-smooth maps 210 --
4.4.1 Case I: [gamma] = 2 211 --
4.4.2 Case II: [gamma] = 3/2 213 --
4.4.3 Period-adding scenarios 214 --
4.4.4 Location of the saddle-node bifurcations 217 --
5 Boundary equilibrium bifurcations in flows 219 --
5.1 Piecewise-smooth continuous flows 219 --
5.1.1 Classification of simplest BEB scenarios 221 --
5.1.2 Existence of other attractors 225 --
5.1.3 Planar piecewise-smooth continuous systems 226 --
5.1.4 Higher-dimensional systems 229 --
5.1.5 Global phenomena for persistent boundary equilibria 232 --
5.2 Filippov flows 233 --
5.2.1 Classification of the possible cases 235 --
5.2.2 Planar Filippov systems 237 --
5.2.3 Some global and non-generic phenomena 242 --
5.3 Equilibria of impacting hybrid systems 245 --
5.3.1 Classification of the simplest BEB scenarios 246 --
5.3.2 The existence of other invariant sets 249 --
6 Limit cycle bifurcations in impacting systems 253 --
6.1 The impacting class of hybrid systems 253 --
6.1.2 Poincare maps related to hybrid systems 261 --
6.2 Discontinuity mappings near grazing 265 --
6.2.1 The geometry near a grazing point 266 --
6.2.2 Approximate calculation of the discontinuity mappings 271 --
6.2.3 Calculating the PDM 271 --
6.2.4 Approximate calculation of the ZDM 273 --
6.2.5 Derivation of the ZDM and PDM using Lie derivatives 274 --
6.3 Grazing bifurcations of periodic orbits 279 --
6.3.1 Constructing compound Poincare maps 280 --
6.3.2 Unfolding the dynamics of the map 284 --
6.4 Chattering and the geometry of the grazing manifold 295 --
6.4.1 Geometry of the stroboscopic map 295 --
6.4.2 Global behavior of the grazing manifold G 296 --
6.4.3 Chattering and the set G[superscript ([infinity])] 299 --
6.5 Multiple collision bifurcation 302 --
7 Limit cycle bifurcations in piecewise-smooth flows 307 --
7.2 Grazing with a smooth boundary 318 --
7.2.1 Geometry near a grazing point 319 --
7.2.2 Discontinuity mappings at grazing 321 --
7.2.3 Grazing bifurcations of periodic orbits 325 --
7.2.5 Detailed derivation of the discontinuity mappings 334 --
7.3 Boundary-intersection crossing bifurcations 340 --
7.3.1 The discontinuity mapping in the general case 341 --
7.3.2 Derivation of the discontinuity mapping in the corner-collision case 346 --
8 Sliding bifurcations in Filippov systems 355 --
8.1 Four possible cases 355 --
8.1.1 The geometry of sliding bifurcations 356 --
8.1.2 Normal form maps for sliding bifurcations 359 --
8.2 Motivating example: a relay feedback system 364 --
8.2.1 An adding-sliding route to chaos 366 --
8.2.2 An adding-sliding bifurcation cascade 368 --
8.2.3 A grazing-sliding cascade 370 --
8.3 Derivation of the discontinuity mappings 373 --
8.3.1 Crossing-sliding bifurcation 375 --
8.3.2 Grazing-sliding bifurcation 377 --
8.3.3 Switching-sliding bifurcation 381 --
8.3.4 Adding-sliding bifurcation 382 --
8.4 Mapping for a whole period: normal form maps 383 --
8.4.1 Crossing-sliding bifurcation 384 --
8.4.2 Grazing-sliding bifurcation 390 --
8.4.3 Switching-sliding bifurcation 393 --
8.4.4 Adding-sliding bifurcation 395 --
8.5 Unfolding the grazing-sliding bifurcation 396 --
8.5.1 Non-sliding period-one orbits 396 --
8.5.2 Sliding orbit of period-one 397 --
8.5.3 Conditions for persistence or a non-smooth fold 399 --
8.5.4 A dry-friction example 399 --
8.6 Other cases 403 --
8.6.1 Grazing-sliding with a repelling sliding region --
catastrophe 403 --
8.6.2 Higher-order sliding 404 --
9 Further applications and extensions 409 --
9.1 Experimental impact oscillators: noise and parameter sensitivity 409 --
9.1.1 Noise 410 --
9.1.2 An impacting pendulum: experimental grazing bifurcations 412 --
9.1.3 Parameter uncertainty 419 --
9.2 Rattling gear teeth: the similarity of impacting and piecewise-smooth systems 422 --
9.2.1 Equations of motion 423 --
9.2.2 An illustrative case 425 --
9.2.3 Using an impacting contact model 426 --
9.2.4 Using a piecewise-linear contact model 431 --
9.3 A hydraulic damper: non-smooth invariant tori 434 --
9.3.1 The model 436 --
9.3.2 Grazing bifurcations 438 --
9.3.3 A grazing bifurcation analysis for invariant tori 441 --
9.4 Two-parameter sliding bifurcations in friction oscillators 448 --
9.4.1 A degenerate crossing-sliding bifurcation 449 --
9.4.2 Fold bifurcations of grazing-sliding limit cycles 453 --
9.4.3 Two simultaneous grazings 455.
Series Title: Applied mathematical sciences (Springer-Verlag New York Inc.), v. 163.
Responsibility: M. di Bernardo [and others].
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Abstract:

This book presents a coherent framework for understanding the dynamics of piecewise-smooth and hybrid systems. The results are presented in an informal style, and illustrated with many examples. The  Read more...

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