Mathematics of models: continuous and discrete dynamical systems
Gespeichert in:
| Beteiligte Personen: | , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
New York u.a.
Horwood
1993
|
| Ausgabe: | 1. publ. |
| Schriftenreihe: | Statistics, operational research and computational mathematics
Ellis Horwood series in mathematics and its applications |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005291810&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIII, 435 S. graph. Darst. |
| ISBN: | 0135638003 0135637929 |
Internformat
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| 100 | 1 | |a Griffiths, Hubert Brian |e Verfasser |0 (DE-588)109065042 |4 aut | |
| 245 | 1 | 0 | |a Mathematics of models |b continuous and discrete dynamical systems |c H. Brian Griffiths ; Adrian Oldknow |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a New York u.a. |b Horwood |c 1993 | |
| 300 | |a XIII, 435 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 0 | |a Statistics, operational research and computational mathematics | |
| 490 | 0 | |a Ellis Horwood series in mathematics and its applications | |
| 650 | 4 | |a Mathematisches Modell | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Mathematical models | |
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| 689 | 0 | 0 | |a Mathematisches Modell |0 (DE-588)4114528-8 |D s |
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| 700 | 1 | |a Oldknow, Adrian J. |e Verfasser |4 aut | |
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Datensatz im Suchindex
| DE-BY-TUM_call_number | 0001 94 A 1068 |
|---|---|
| DE-BY-TUM_katkey | 596923 |
| DE-BY-TUM_location | Mag |
| DE-BY-TUM_media_number | 040000795973 |
| _version_ | 1821938810693353472 |
| adam_text | Table of contents
Preface xi
1 Introduction: how to use this book 1
1.1 Views of mathematics 1
1.2 Why models? 2
1.3 Three themes 4
1.4 Activity and joy 5
1.5 Computing 7
2 Discrete dynamics 9
2.1 The Malthus model 9
2.2 The Verhulst model 15
2.3 Second order linear difference equations 18
2.4 Pairs of populations 21
2.5 The battle problem 24
2.6 Linear dynamical systems on the plane 29
3 Growth and decay; some calculus models 33
3.1 The Malthus model again 33
3.2 The battle problem 37
3.3 Some portraits of systems 42
3.4 Uncoupled systems 44
3.5 Summary: the algorithm 49
4 Calculus and some classical models 51
4.1 Introduction: the derivative 51
4.2 What is a tangent to a graph? 52
4.3 Local maxima and minima 55
4.4 The mean value theorem 60
4.5 Integration 61
viii Table of contents
4.6 Some first order differential equations 65
4.7 Simple harmonic motion: planetary orbits 72
4.8 Linear differential equations of the second order 79
4.9 Proof of the algorithm 84
4.10 Modelling mathematics within mathematics 87
4.11 The exponential of a matrix 89
5 Plane curves 94
5.1 What is a plane curve? 94
5.2 Regularity 97
5.3 Parallel curves 98
5.4 Arc length 100
5.5 Curvature 102
5.6 Envelopes 106
5.7 Caustics 113
5.8 Polar coordinates 118
5.9 Complex numbers: the Argand diagram 121
6 Control space and phase space: polynomial functions 126
6.1 Quadratic functions: the control plane 126
6.2 Cubic functions 129
6.3 Construction of galleries 131
6.4 Formulae for the roots of cubic equations 138
6.5 Discriminants 142
6.6 Polynomial functions 147
6.7 Variation of roots with coefficients 149
7 Quadratics, conies, and quadrics 151
7.1 Introduction 151
7.2 Conies 151
7.3 Quadratic forms 158
7.4 Eigenvalues and eigenvectors 161
7.5 The standard form of a matrix 167
7.6 Lagrange s reduction process 169
7.7 Quadrics 173
8 Contours: the implicit function theorem 177
8.1 Contours and the implicit function theorem 177
8.2 Three variables 184
8.3 Constructing the portrait of a function 187
8.4 Cubic polynomials 191
9 Non linear models 200
9.1 Two population extensions of Verhulst s equation 200
9.2 Existence of trajectories: Hamiltonians 207
9.3 The three rules 211
9.4 The trajectory routine 215
9.5 Closed trajectories 218
Table of contents ix
9.6 The Poincare Bendixson theorem 223
9.7 Sketching the Lotka Volterra system 224
9.8 The van der Pol and Lienard equations 229
9.9 The Hopf bifurcation 235
10 Discrete non linear dynamics 238
10.1 Staircases and cobwebs 238
10.2 The Verhulst equation 242
10.3 The Maynard Smith equation 249
10.4 Change of stability 256
10.5 Periodic orbits 259
10.6 Cantor sets and the shift mapping 260
10.7 Differential equations and chaos 263
10.8 Fixed points: local theory 267
10.9 The Mandelbrot set 270
10.10 Self similar fractals 275
11 Catastrophe sets in modelling 277
11.1 Introduction 277
11.2 The cusp catastrophe 278
11.3 Potentials 282
11.4 The catastrophe machine 284
11.5 Decay of civilizations 288
11.6 Compromise 290
11.7 The umbilics 292
11.8 The elliptic umbilic 298
11.9 Thorn s theorem 301
A Using technology to help gain mathematical insights 303
A.I Introduction 303
A.2 Techniques for iteration 304
A.3 Converting programs between versions of BASIC 307
A.4 Other software for mathematics 312
B Linear mathematics in R3 314
B.I Matrices and equations in the plane 314
B.2 The space R3 316
B.3 The scalar product 319
B.4 Matrix multiplication in R2 320
B.5 Lines in space 323
B.6 The vector product 324
B.7 Determinants 325
B.8 Intersections 327
B.9 Linear transformations 328
B.10 Matrix multiplication 331
C The exponential function and its relatives 335
C.I The logarithm and exponential functions 335
x Table of contents
C.2 The Taylor Maclaurin expansion 339
C.3 The function cis 342
C.4 Continuous functions 347
C.5 Monotonicity 349
D Functions of several variables 358
D.I Partial differentiation 358
D.2 Continuity 361
D.3 The formula for small increments 362
D.4 The chain rule 364
D.5 The Taylor Maclaurin expansion in several variables 370
D.6 Maxima and minima 373
D.7 Three variables 378
D.8 When is F independent of x? 380
E Smooth mappings, diffeomorphisms, and integration 382
E.I Smooth mappings 382
E.2 Bijections 383
E.3 Local diffeomorphisms 385
E.4 Saddles and centres revisited 389
E.5 Modelling resemblances by diffeomorphisms 390
E.6 How can we integrate F(x, y)l 393
E.7 Line integrals 396
E.8 The winding number 398
E.9 Vector fields 401
F Norms, sequences, and contracting mappings 407
F.I Norms 407
F.2 Convergence 408
F.3 The contracting mapping theorem in U2 411
F.4 Metric spaces 413
F.5 Application to the construction of fractals 417
Appendix: Mathematical notation 419
References 422
List of programs 426
Index 428
|
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| discipline | Informatik Mathematik Wirtschaftswissenschaften |
| edition | 1. publ. |
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| id | DE-604.BV008042910 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T09:15:28Z |
| institution | BVB |
| isbn | 0135638003 0135637929 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-005291810 |
| oclc_num | 28182300 |
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| physical | XIII, 435 S. graph. Darst. |
| publishDate | 1993 |
| publishDateSearch | 1993 |
| publishDateSort | 1993 |
| publisher | Horwood |
| record_format | marc |
| series2 | Statistics, operational research and computational mathematics Ellis Horwood series in mathematics and its applications |
| spellingShingle | Griffiths, Hubert Brian Oldknow, Adrian J. Mathematics of models continuous and discrete dynamical systems Mathematisches Modell Differentiable dynamical systems Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd |
| subject_GND | (DE-588)4114528-8 |
| title | Mathematics of models continuous and discrete dynamical systems |
| title_auth | Mathematics of models continuous and discrete dynamical systems |
| title_exact_search | Mathematics of models continuous and discrete dynamical systems |
| title_full | Mathematics of models continuous and discrete dynamical systems H. Brian Griffiths ; Adrian Oldknow |
| title_fullStr | Mathematics of models continuous and discrete dynamical systems H. Brian Griffiths ; Adrian Oldknow |
| title_full_unstemmed | Mathematics of models continuous and discrete dynamical systems H. Brian Griffiths ; Adrian Oldknow |
| title_short | Mathematics of models |
| title_sort | mathematics of models continuous and discrete dynamical systems |
| title_sub | continuous and discrete dynamical systems |
| topic | Mathematisches Modell Differentiable dynamical systems Mathematical models Mathematisches Modell (DE-588)4114528-8 gnd |
| topic_facet | Mathematisches Modell Differentiable dynamical systems Mathematical models |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=005291810&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT griffithshubertbrian mathematicsofmodelscontinuousanddiscretedynamicalsystems AT oldknowadrianj mathematicsofmodelscontinuousanddiscretedynamicalsystems |
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