Dynamical Systems (eBook)

Proceedings of a Symposium Held at the University of Bahia, Salvador, Brasil, July 26-August 14, 1971

M. M. Peixoto (Herausgeber)

eBook Download: PDF

2014 | 1. Auflage
764 Seiten
Elsevier Science (Verlag)
978-1-4832-6910-8 (ISBN)

Infos

Dynamical Systems is a collection of papers that deals with the generic theory of dynamical systems, in which structural stability becomes associated with a generic property. Some papers describe structural stability in terms of mappings of one manifold into another, as well as their singularities. One paper examines the theory of polyhedral catastrophes, particularly, the analogues of each of the four basic differentiable catastrophes which map the line. Other papers discuss isolating blocks, the exponential rate conditions for dynamical systems, bifurcation, catastrophe, and a nondensity theorem. One paper reviews the results of functional differential equations which show that a qualitative theory will emerge despite the presence of an infinite dimensionality or of a semigroup property. Another paper discusses a class of quasi-periodic solutions for Hamiltonian systems of differential equations. These equations generalize a well-known result of Kolmogorov and Arnold on perturbations of n-dimensional invariant tori for Hamiltonian systems of n degrees of freedom. The researcher can derive mathematical models based on qualitative mathematical argument by using as 'axioms' three dynamic qualities found in heart muscle fibers and nerve axons. The collection can prove useful for mathematicians, students and professors of advanced mathematics, topology or calculus.

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Dynamical Systems is a collection of papers that deals with the generic theory of dynamical systems, in which structural stability becomes associated with a generic property. Some papers describe structural stability in terms of mappings of one manifold into another, as well as their singularities. One paper examines the theory of polyhedral catastrophes, particularly, the analogues of each of the four basic differentiable catastrophes which map the line. Other papers discuss isolating blocks, the exponential rate conditions for dynamical systems, bifurcation, catastrophe, and a nondensity theorem. One paper reviews the results of functional differential equations which show that a qualitative theory will emerge despite the presence of an infinite dimensionality or of a semigroup property. Another paper discusses a class of quasi-periodic solutions for Hamiltonian systems of differential equations. These equations generalize a well-known result of Kolmogorov and Arnold on perturbations of n-dimensional invariant tori for Hamiltonian systems of n degrees of freedom. The researcher can derive mathematical models based on qualitative mathematical argument by using as "e;axioms"e; three dynamic qualities found in heart muscle fibers and nerve axons. The collection can prove useful for mathematicians, students and professors of advanced mathematics, topology or calculus.

Inhalt

Front Cover 1
Dynamical Systems 4
Copyright Page 5
Table of Contents 6
List of Contributors 10
Preface 14
Acknowledgments 16
Chapter 1. Note on Foliations 20
Introduction 20
1 Foliations of manifolds with boundary 21
2 Foliations of closed manifolds 23
References 25
Chapter 2. Polyhedral Catastrophe Theory I: Maps of the Line to the Line 26
1 Introduction: the quartic case 26
2 The cubic and quadratic cases 31
3 The quintic 33
4 The sextic 38
References 40
Chapter 3. On Rk x Zl-Actions 42
Introduction 42
1 Hyperbolic linear actions 46
Appendix to Section 1: actions on a vector space bundle 58
2 Invariant manifold theory 60
3 Proof of Theorem 3 63
4 Proofs of Theorems 4 and 5 70
5 Some examples 77
6 R2-actions on three-manifolds 81
References 88
Chapter 4. Morse-Smale R2-Actions on Two-Manifolds 90
References 93
Chapter 5. Isolating Blocks, Regularization, and the Three-Body Problem 94
1 Introduction 94
2 Regularization by surgery 96
3 The planar three-body problem 99
4 Regularization of the three-body problem 103
5 The topology of the integral surfaces of the three-body problem 105
References 108
Chapter 6. Exponential Rate Conditions for Dynamical Systems 110
References 113
Chapter 7. Bifurcation and Catastrophe 114
1 The mathematical framework 115
References 128
Chapter 8. One-Parameter Families of Vector Fields on Two-Manifolds: Another Nondensity Theorem 130
Introduction 130
1 Three definitions of stability 133
2 Phase diagrams and canonical regions 134
3 Stability of one-parameter families 136
4 Diff(S1) and a nondensity theorem 141
References 146
Chapter 9. Some Infinite-Dimensional Dynamical Systems 148
References 152
Chapter 10. Stability of Compact Leaves of Foliations 154
Introduction 154
1 Statement of results 155
2 Holonomy 160
3 Proofs of theorems 166
4 Remarks and questions 171
References 172
Chapter 11. Hyperbolic Closed Geodesics 174
References 183
Chapter 12. Generic Vector Bundle Maps 184
Introduction 184
1 Vector Bundle Maps 184
2 Blowups and resolutions 187
3 Characteristic classes 190
4 Special formulas 193
References 194
Chapter 13. Sur une Généralisation des Structures Feuilletées de Codimension Un 196
1 Introduction 196
2 198
3 Remarques générales 201
Références 203
Chapter 14. Solutions of Generic Linear Equations 204
1 Generic matrices 205
2 Proof of Theorem 2 205
3 Local equivalence of matrices 205
4 Reduction to the analytic case 207
5 Malgrange's theorem 208
6 Proof of Theorem 1 in the case p = q 209
7 Proof of Theorem 1 in the case p = q 210
8 Proof in the case p = q 211
9 Symmetric matrices 211
References 212
Chapter 15. Stratifications and Mappings 214
I. Stratifications 218
II. Systems of tubular neighborhoods 231
III. Stratifications of Mappings 243
References 250
Chapter 16. On Thom-Boardman Singularities 252
1 Jacobian extensions of ideals 253
2 The operators d and ß 255
3 Truncated power series algebras 258
4 The Thom-Boardman singularities (complex case) 259
5 The tangent space of S 264
6 Singularities of maps 265
References 266
Chapter 17. Parabolic Orbits in the Three-Body Problem 268
References 272
Chapter 18. Structural Stability on Two-Manifolds 274
References 276
Chapter 19. Symmetries and Integrals in Mechanics 278
1 Introduction 278
2 General background and main results 279
3 Characteristic multipliers 287
4 The reduced space 289
References 291
Chapter 20. On a Nonlinear Problem in Differential Geometry 292
References 298
Chapter 21. On a Class of Quasi-Periodic Solutions for Hamiltonian Systems 300
1 Problem 300
2 Results 302
3 Generalizations and remarks 306
Chapter 22. A Liapounov Unstability Theorem 308
English abstract 308
Chapter 23. Hyperbolic Nonwandering Sets on Two-Dimensional Manifolds 312
References 320
Chapter 24. Bifurcations of Morse-Smale Dynamical Systems 322
1 Introduction 322
2 323
3 331
4 339
5 355
6 365
7 376
References 385
Chapter 25. Factorization of Nonsingular Circle Endomorphisms 386
References 392
Chapter 26. Partitions for Circle Endomorphisms 394
1 395
2 395
3 397
4 398
5 401
6 405
7 406
References 407
Chapter 27. On the Classification of Flows on 2-Manifolds 408
1 Introduction 408
2 Structurally stable flows 409
3 The graph of a gradient-like flow 410
4 The distinguished graph 411
5 The theorem 417
6 Proof of the theorem 417
7 The nonorientable case, without closed orbit 432
8 The orientable case with closed orbits 435
9 The nonorientable case with closed orbits 438
Bibliography 438
Chapter 28. Indices for Foliations of the Two-Dimensional Torus 440
References 443
Chapter 29. Relative Equilibria in Mechanical Systems 444
Introduction 444
1 The equation of motion 445
2 Generalities on G-manifolds 447
3 Systems with symmetry 449
4 Relative equilibria 450
5 Critical points of E X J 452
6 The n-body problem 455
Appendix. The exterior differential calculus 459
References 460
Chapter 30. Cr Structural Stability Implies Kupka-Smale 462
References 468
Chapter 31. Finite Stability Is Not Generic 470
Introduction 470
Statement of results 470
The Example 472
Proof of Theorem 1 476
Proof of Theorems 3 and 4 477
Proof of Theorem 4 478
Proof of Theorem 3 480
References 480
Chapter 32. Some Remarks on Foliations 482
Introduction 482
II The Godbillon-Vey invariant 488
III 491
IV 492
References 497
Chapter 33. Singularities of Newtonian Gravitational Systems 498
1 Introduction 498
2 History of the problem 499
3 Noncollision singularities 501
4 A new sufficient condition 505
References 506
Chapter 34. Morse-Smale Diffeomorphisms Are Unipotent on Homology 508
References 510
Chapter 35. Stability and Genericity for Diffeomorphisms 512
References 532
Chapter 36. Bounded Orbits in Mechanical Systems with Two Degrees of Freedom and Symmetry 534
I Introduction 534
II Central force problem 535
III Geodesies on surfaces of revolution 541
References 544
Chapter 37. Stability and Isotopy in Discrete Dynamical Systems 546
References 549
Chapter 38. Global Analysis and Economics I: Pareto Optimum and a Generalization of Morse Theory 550
1 551
2 553
3 554
4 556
5 559
6 560
References 563
Chapter 39. The Godbillon-Vey Invariant of a Product Foliation is Zero 564
1 Definition of the Godbillon-Vey invariant 564
2 Statement of results 565
3 Proof of the theorem 565
References 566
Chapter 40. Structural Stability and Bifurcation Theory 568
Introduction 568
1 Higher-order structural stability 568
2 Structural stability of higher degree 570
3 Structural stability of maps 571
4 Report of results 573
5 Genericity and the reduction problem 578
References 579
Chapter 41. Generic Bifurcations of Dynamical Systems 580
1 Statement of the problem 580
2 Quasi-hyperbolic orbits 581
3 Bifurcation of quasi-hyperbolic orbits 587
4 A genericity theorem 598
References 600
Chapter 42. A Nonstabilizable Jet of a Singularity of a Vector Field 602
1 Introduction, definitions, and statement of the result 602
2 Blowing up vector fields 603
3 Normal forms for rotations 606
4 Proof of the theorem 609
Reference 616
Chapter 43. Integral Curves Near Mildly Degenerate Singular Points of Vector Fields 618
1 Introduction 618
2 The blowing up (and down) construction 620
3 A local diffeomorphism 622
4 Proof of the theorem 634
References 636
Chapter 44. Langage et Catastrophes: Eléments pour une Sémantique Topologique 638
1 Structures syntaxiques et catégories grammaticales 638
2 La structure syntaxique des phrases élémentaires 640
2 La théorie des catastrophes et la notion d'objet 643
3 La régulation 654
4 Théorie des fonctions grammaticales 665
Références 673
Chapter 45. Secondary Obstructions to IntegrabiIity 674
References 679
Chapter 46. On the Tolerance Stability Conjecture 682
Reference 684
Chapter 47. An Anosov Translation 686
References 689
Chapter 48. Foliated S1-Bundles and Diffeomorphisms of S 690
Introduction 690
1 Abelian actions on S 691
2 Proof of theorem 1 692
3 Representations and foliations 693
4 Counterexample in the C° case 695
5 Which bundles arise? 696
6 A nonabelian example 698
References 700
Chapter 49. Differential Equations for the Heartbeat and Nerve Impulse 702
1 Introduction 702
2 Three qualities 703
3 Dynamical systems on the plane R 707
4 The slow manifold and the fast foliation 709
5 Biological digression 711
6 Nonlinear examples on R 712
7 Threshold 715
8 Solution in the case of the jump return 716
9 Application to the heartbeat 718
10 Dynamical systems on R 720
11 The cusp catastrophe 723
12 Application to the nerve impulse 727
13 Tension and blood pressure 728
14 The pacemaker wave 734
15 Conclusion and experiments 736
16 Propagation and timing of the nerve impulse 740
17 Voltage clamp data 742
18 Electrical equation for ionic flow 745
19 Local nerve impulse equations 747
20 Propagation wave 749
21 Calculation of action potential 752
22 Conclusions and experiments 753
23 Appendix on Lamarckian evolution 757
References 759
Index 762

Produkt-Details

• Erscheint lt. Verlag: 10.5.2014
• Sprache: englisch
• Themenwelt: Mathematik / Informatik ► Mathematik ► Geometrie / Topologie
• Themenwelt: Technik ► Bauwesen
• ISBN-10: 1-4832-6910-8 / 1483269108
• ISBN-13: 978-1-4832-6910-8 / 9781483269108