Piecewise-smooth dynamical systems: theory and applications
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| Format: | Book |
|---|---|
| Language: | English |
| Published: |
London
Springer
2008
|
| Series: | Applied mathematical sciences
163 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XXI, 481 S. Ill., graph. Darst. |
| ISBN: | 9781846280399 |
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| 245 | 1 | 0 | |a Piecewise-smooth dynamical systems |b theory and applications |c M. di Bernardo ... |
| 264 | 1 | |a London |b Springer |c 2008 | |
| 300 | |a XXI, 481 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Applied mathematical sciences |v 163 | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 0 | 7 | |a Dynamisches System |0 (DE-588)4013396-5 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
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| 689 | 0 | |5 DE-604 | |
| 700 | 1 | |a Di Bernardo, Mario |d 1970- |e Sonstige |0 (DE-588)136110754 |4 oth | |
| 830 | 0 | |a Applied mathematical sciences |v 163 |w (DE-604)BV000005274 |9 163 | |
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Record in the Search Index
| _version_ | 1834641487421243392 |
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| adam_txt |
Contents
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
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
XIV Contents
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
.
Ill
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: I
= —1,
v\
> 0,
v2
< 1.180
4.2.4
Chaotic behavior: I
= —1,
v\
> 0, 1 <
v2
< 2.185
4.3
Square-root maps
.188
Contents
XV
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:
7 = 2.211
4.4.2
Case II:
7 = 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
Poincaré
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
Poincaré
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
G
.296
6.4.3
Chattering and the set G(oo)
.299
6.5
Multiple collision bifurcation
.302
XVI Contents
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
Contents XVII
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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| id | DE-604.BV023110707 |
| illustrated | Illustrated |
| index_date | 2024-07-02T19:48:13Z |
| indexdate | 2025-06-11T14:01:06Z |
| institution | BVB |
| isbn | 9781846280399 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016313303 |
| oclc_num | 255415987 |
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| physical | XXI, 481 S. Ill., graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
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| publisher | Springer |
| record_format | marc |
| series | Applied mathematical sciences |
| series2 | Applied mathematical sciences |
| spelling | Piecewise-smooth dynamical systems theory and applications M. di Bernardo ... London Springer 2008 XXI, 481 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Applied mathematical sciences 163 Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Dynamisches System (DE-588)4013396-5 s Verzweigung Mathematik (DE-588)4078889-1 s Chaotisches System (DE-588)4316104-2 s DE-604 Di Bernardo, Mario 1970- Sonstige (DE-588)136110754 oth Applied mathematical sciences 163 (DE-604)BV000005274 163 Digitalisierung UB Regensburg https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016313303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Piecewise-smooth dynamical systems theory and applications Applied mathematical sciences Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4013396-5 (DE-588)4078889-1 (DE-588)4316104-2 |
| title | Piecewise-smooth dynamical systems theory and applications |
| title_auth | Piecewise-smooth dynamical systems theory and applications |
| title_exact_search | Piecewise-smooth dynamical systems theory and applications |
| title_exact_search_txtP | Piecewise-smooth dynamical systems theory and applications |
| title_full | Piecewise-smooth dynamical systems theory and applications M. di Bernardo ... |
| title_fullStr | Piecewise-smooth dynamical systems theory and applications M. di Bernardo ... |
| title_full_unstemmed | Piecewise-smooth dynamical systems theory and applications M. di Bernardo ... |
| title_short | Piecewise-smooth dynamical systems |
| title_sort | piecewise smooth dynamical systems theory and applications |
| title_sub | theory and applications |
| topic | Bifurcation theory Differentiable dynamical systems Dynamisches System (DE-588)4013396-5 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Bifurcation theory Differentiable dynamical systems Dynamisches System Verzweigung Mathematik Chaotisches System |
| url | https://bvbr.bib-bvb.de:443/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016313303&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000005274 |
| work_keys_str_mv | AT dibernardomario piecewisesmoothdynamicalsystemstheoryandapplications |