Nonlinear stability and bifurcation theory
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Wien u.a.
Springer
1991
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MARC
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| 100 | 1 | |a Troger, Hans |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Nonlinear stability and bifurcation theory |c Hans Troger ; Alois Steindl |
| 264 | 1 | |a Wien u.a. |b Springer |c 1991 | |
| 300 | |a XI, 407 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 500 | |a Literaturverz. S. 389 - 401 | ||
| 650 | 0 | 7 | |a Stabilität |0 (DE-588)4056693-6 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Steindl, Alois |e Verfasser |4 aut | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0510/ohne Sign. 0740/M.3.3705 0001/92 A 81 |
|---|---|
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| DE-BY-TUM_media_number | TEMP2887502 040000921046 TEMP2887501 |
| _version_ | 1816711586091368449 |
| adam_text | Contents
1 Introduction 1
2 Representation of systems 23
2.1 Dynamical systems 24
2.1.1 Time continuous system 24
2.1.2 Time discrete system 26
2.2 Statical systems 31
2.3 Definitions of stability 36
2.3.1 Stability in the sense of Ljapunov 37
2.3.2 Structural stability (robustness, coarseness) .... 40
3 Reduction process, bifurcation equations 46
3.1 Finite dimensional dynamical systems 46
3.1.1 Steady states 46
3.1.2 Periodic motions 60
3.2 Infinite dimensional statical and dynamical systems .... 62
3.2.1 Statical systems 63
3.2.2 Dynamical systems 66
4 Application of the reduction process 69
4.1 Equilibria of finite dimensional systems 69
4.1.1 Double pendulum with axially elastic rods and fol¬
lower force loading 69
4.1.2 Double pendulum with elastic end support and fol¬
lower force loading 74
4.1.3 Double pendulum under aerodynamic excitation . 79
4.1.4 Loss of stability of the straight line motion of a
tractor semitrailer 82
4.1.5 Loss of stability of the straight line motion of a
railway vehicle 85
4.1.6 Summary of Section 4.1 87
4.2 Periodic solutions of finite dimensional systems 88
4.2.1 Mechanical model and equations of motion .... 89
viii
Contents ix
4.2.2 Calculation of the power series expansion of the
Poincare mapping 91
4.2.3 Stability boundary in parameter space 93
4.2.4 Center manifold reduction 95
4.3 Finite and infinite dimensional statical systems 97
4.3.1 Buckling of a rod: discrete model 97
4.3.2 Buckling of a rod: continuous model 101
4.3.3 Buckling of a circular ring 105
4.3.4 Buckling at a double eigenvalue: rectangular plate 110
4.3.5 The pattern formation problem: buckling of com¬
plete spherical shells 124
5 Bifurcations under symmetries 143
5.1 Introduction 143
5.2 Finite dimensional dynamical systems 151
5.2.1 Two zero roots 155
5.2.2 Two purely imaginary pairs 157
5.3 Infinite dimensional statical systems 159
5.4 Infinite dimensional dynamical systems 167
6 Discussion of the bifurcation equations 179
6.1 Transformation to normal form 180
6.1.1 Time continuous dynamical systems 180
6.1.2 Time discrete dynamical systems 192
6.1.3 Statical systems 195
6.2 Codimension 198
6.2.1 Static bifurcation 200
6.2.2 Dynamic bifurcation 206
6.3 Determinacy 208
6.4 Unfolding 212
6.5 Classification 214
6.5.1 Dynamic bifurcation 214
6.5.2 Static bifurcation: elementary catastrophe theory . 219
6.5.3 The unfolding theory of Golubitsky and Schaeffer . 220
6.5.4 Restricted generic bifurcation 226
6.6 Bifurcation diagrams 227
6.6.1 Statical systems 227
6.6.2 Time continuous dynamical systems 246
6.6.3 Time discrete dynamical systems 269
6.6.4 Symmetric dynamical systems 275
6.6.5 Symmetric statical systems 283
x Contents
Appendix 287
A Linear spaces and linear operators 287
A.I Linear spaces 287
A.2 Linear operators 292
B Transformation of matrices to Jordan form 297
C Adjoint and self adjoint linear differential operators 301
C.I Calculation of the adjoint operator 301
C.2 Self adjoint differential operators 303
D Projection operators 306
D.I General considerations 306
D.2 Projection for non self adjoint operators 308
D.3 Application to the Galerkin reduction 310
E Spectral decomposition 311
E.I Derivation of an inversion formula 311
E.2 Three examples 313
F Shell equations on the complete sphere 316
F.I Tensor notations in curvilinear coordinates 316
F.2 Spherical harmonics 320
G Some properties of groups 324
G.I Naive definition of a group 324
G.2 Symmetry groups 325
G.3 Representation of groups by matrices 328
G.4 Transformation of functions and operators 329
G.5 Examples of invariant functions and operators 332
G.6 Abstract definition of a group 333
G.7 The orthogonal groups O(n) and SO(n) 335
G.8 The Euclidean group E(3) 338
H Stability boundaries in parameter space 340
I Differential equation of an elastic ring 344
1.1 Equilibrium equations and bending 344
1.2 Ring equations 345
J Shallow shell and plate equations 348
J.I Deformation of the shell 349
J.2 Constitutive law 351
J.3 Equations of equilibrium 352
J.4 Special cases 354
Contents xi
J.4.1 Plate 354
J.4.2 Sphere 354
J.4.3 Cylinder 355
K Shell equations for axisymmetric deformations 356
K.I Geometrical relations 356
K.2 Stress resultants, couples and equilibrium equations . . . 361
K.3 Stress strain relations 362
K.4 Spherical shell 363
L Equations of motion of a fluid conveying tube 366
L.I Geometry of tube deformation 367
L.2 Stress strain relationship 369
L.3 Linear and angular momentum 370
L.4 Tube equations and boundary conditions 372
M Various concepts of equivalences 376
M.I Right equivalence 377
M.2 Contact equivalence 379
M.3 Vector field equivalence 379
M.4 Bifurcation equivalence 382
M.5 Recognition problem 383
N Slowly varying parameter 384
O Transformation of dynamical systems into standard form386
O.I Power series expansion 386
O.2 Recursive calculation 388
Bibliography 389
Index 402
|
| any_adam_object | 1 |
| author | Troger, Hans Steindl, Alois |
| author_facet | Troger, Hans Steindl, Alois |
| author_role | aut aut |
| author_sort | Troger, Hans |
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| classification_tum | MAT 587f MAT 344f |
| ctrlnum | (OCoLC)630823680 (DE-599)BVBBV004802328 |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV004802328 |
| illustrated | Illustrated |
| indexdate | 2024-11-25T17:10:36Z |
| institution | BVB |
| isbn | 3211822925 0387822925 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-002953262 |
| oclc_num | 630823680 |
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| physical | XI, 407 S. graph. Darst. |
| publishDate | 1991 |
| publishDateSearch | 1991 |
| publishDateSort | 1991 |
| publisher | Springer |
| record_format | marc |
| spellingShingle | Troger, Hans Steindl, Alois Nonlinear stability and bifurcation theory Stabilität (DE-588)4056693-6 gnd Nichtlineare Stabilitätstheorie (DE-588)4171761-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4056693-6 (DE-588)4171761-2 (DE-588)4078889-1 |
| title | Nonlinear stability and bifurcation theory |
| title_auth | Nonlinear stability and bifurcation theory |
| title_exact_search | Nonlinear stability and bifurcation theory |
| title_full | Nonlinear stability and bifurcation theory Hans Troger ; Alois Steindl |
| title_fullStr | Nonlinear stability and bifurcation theory Hans Troger ; Alois Steindl |
| title_full_unstemmed | Nonlinear stability and bifurcation theory Hans Troger ; Alois Steindl |
| title_short | Nonlinear stability and bifurcation theory |
| title_sort | nonlinear stability and bifurcation theory |
| topic | Stabilität (DE-588)4056693-6 gnd Nichtlineare Stabilitätstheorie (DE-588)4171761-2 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Stabilität Nichtlineare Stabilitätstheorie Verzweigung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=002953262&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT trogerhans nonlinearstabilityandbifurcationtheory AT steindlalois nonlinearstabilityandbifurcationtheory |