1 Introduction .................................................. 1
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 Poincaré 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
3.1.1 Definitions ....................................... 124
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, v1 > 0, v2 < 1 ......... 180
4.2.4 Chaotic behavior: l = -1, v1 > 0, 1 < v2 < 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: γ = 2 ..................................... 211
4.4.2 Case II: γ = 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
6.1.1 Examples .......................................... 255
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
6.3.3 Examples .......................................... 285
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 ........... 296
6.4.3 Chattering and the set G(∞) ........................ 299
6.5 Multiple collision bifurcation .......................... 302
7 Limit cycle bifurcations in piecewise-smooth flows .......... 307
7.1 Definitions and examples ................................ 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
7.2.4 Examples .......................................... 327
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
7.3.3 Examples .......................................... 347
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
References .................................................... 459
Index ......................................................... 475
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