Global Bifurcation Theory and Hilbert's Sixteenth Problem:
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| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
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New York
Springer Science+Business Media
2003
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| Schriftenreihe: | Mathematics and Its Applications
559 |
| Schlagworte: | |
| Online-Zugang: | Inhaltsverzeichnis |
| Beschreibung: | On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second International Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathematics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possible complete information on the qualitative behaviour of integral curves defined by this equation (176] |
| Beschreibung: | XVI, 182 Seiten |
| ISBN: | 9781461348191 |
Internformat
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| 500 | |a On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second International Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathematics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possible complete information on the qualitative behaviour of integral curves defined by this equation (176] | ||
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Datensatz im Suchindex
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|---|---|
| adam_text | Contents
List of Figures ix
Preface xi
Acknowledgments xix
Foreword xxi
1. GEOMETRIC METHODS 1
1 On global bifurcation theory 2
1.1 Local bifurcations of limit cycles 2
1.2 On methods of the study of limit cycles 5
1.3 Geometric methods of bifurcations theory 5
2 Erugin’s two-isocline method 7
2.1 On the method of isoclines 7
2.2 Isocline portraits 9
2.3 Classification of singular points 17
2.4 Other applications of the two-isocline method 18
3 The problem of center and focus 19
3.1 On the distinguishing problem 19
3.2 Lyapunov’s focus quantities 19
3.3 Classification of symmetric cases 21
3.4 Geometric interpretation of quadratic centers 28
4 Poincare’s topographical systems 30
4.1 Basic definitions 30
4.2 Construction of topographical systems 31
4.3 Topographical systems and limit cycles 34
4.4 On the control of semi-stable limit cycles 36
5 Canonical systems with limit cycles 38
5.1 Systems with field-rotation parameters 38
5.2 Reduction of quadratic systems 39
VI
GLOBAL BIFURCATION THEORY
2. ANDRONOV-HOPF BIFURCATION 43
1 On the Andronov-Hopf bifurcation 44
1.1 On the history of the bifurcation 44
1.2 Bautin’s result 45
1.3 The example by Shi Sonling 49
1.4 The example by E. A. Andronova 50
2 Construction of systems 52
2.1 Canonical systems with two singular points 52
2.2 Properties of the canonical systems 57
3 Bifurcations of limit cycles 60
3.1 Bifurcations of algebraic limit cycles 60
3.2 The case of a saddle and an antisaddle 62
3.3 A quadratic system with four limit cycles 63
3.4 Asymptotic behavior of limit cycles 64
4 Numerical results 67
3. CLASSIFICATION OF SEPARATRIX CYCLES 69
1 On separatrix cycles 70
1.1 Dulac’s theorem 70
1.2 Existential problem 74
1.3 On application of canonical systems 75
2 One saddle and three antisaddles (the first case) 78
3 One saddle and three antisaddles (the second case) 86
4 Classification of separatrix cycles 94
4.1 Other cases of singular points 94
4.2 The complete classification 95
4. MULTIPLE LIMIT CYCLES 103
1 On multiple limit cycles 104
2 Local analysis of one-parameter families 106
2.1 Basic notions 106
2.2 Multiple limit cycles and Puiseux series 109
2.3 Arches and paths of limit cycles 111
3 Global analysis of one-parameter families 117
3.1 Planar termination principle 118
3.2 The proof of the principle 120
4 Bifurcation surfaces of multiple limit cycles 125
4.1 Fold and cusp bifurcation surfaces 125
(yOTltSfltS
vu
4.2 A swallow-tail bifurcation surface 128
4.3 General bifurcation surfaces 130
5 Wintner-Perko termination principle 132
5.1 General discussion 132
5.2 Monotonic families and rotated vector fields 134
5.3 The limit cycle problem for quadratic systems 137
5. APPLICATIONS, OPEN PROBLEMS, ALTERNATIVES 143
1 On polynomial models of dynamical systems 143
2 A generalized Lotka-Volterra system 145
3 Some open problems of qualitative theory 152
4 Abelian integrals and limit cycles 155
5 On a work by N. P. Erugin 159
References 165
Index
181
|
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| isbn | 9781461348191 |
| language | English |
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| spelling | Gaiko, Valery A. Verfasser aut Global Bifurcation Theory and Hilbert's Sixteenth Problem Valery A. Gaiko New York Springer Science+Business Media 2003 XVI, 182 Seiten txt rdacontent n rdamedia nc rdacarrier Mathematics and Its Applications 559 On the 8th of August 1900 outstanding German mathematician David Hilbert delivered a talk "Mathematical problems" at the Second International Congress of Mathematicians in Paris. The talk covered practically all directions of mathematical thought of that time and contained a list of 23 problems which determined the further development of mathematics in many respects (1, 119]. Hilbert's Sixteenth Problem (the second part) was stated as follows: Problem. To find the maximum number and to determine the relative position of limit cycles of the equation dy Qn(X, y) -= dx Pn(x, y)' where Pn and Qn are polynomials of real variables x, y with real coeffi cients and not greater than n degree. The study of limit cycles is an interesting and very difficult problem of the qualitative theory of differential equations. This theory was origi nated at the end of the nineteenth century in the works of two geniuses of the world science: of the Russian mathematician A. M. Lyapunov and of the French mathematician Henri Poincare. A. M. Lyapunov set forth and solved completely in the very wide class of cases a special problem of the qualitative theory: the problem of motion stability (154]. In turn, H. Poincare stated a general problem of the qualitative analysis which was formulated as follows: not integrating the differential equation and using only the properties of its right-hand sides, to give as more as possible complete information on the qualitative behaviour of integral curves defined by this equation (176] Mathematics Global analysis Differential Equations Ordinary Differential Equations Global Analysis and Analysis on Manifolds Mathematical Modeling and Industrial Mathematics Applications of Mathematics Mathematical and Computational Biology Mathematik Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Hilbertsches Problem 16 (DE-588)4391597-8 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 s Hilbertsches Problem 16 (DE-588)4391597-8 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4419-9168-3 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=029315685&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Gaiko, Valery A. Global Bifurcation Theory and Hilbert's Sixteenth Problem Mathematics Global analysis Differential Equations Ordinary Differential Equations Global Analysis and Analysis on Manifolds Mathematical Modeling and Industrial Mathematics Applications of Mathematics Mathematical and Computational Biology Mathematik Verzweigung Mathematik (DE-588)4078889-1 gnd Hilbertsches Problem 16 (DE-588)4391597-8 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4391597-8 |
| title | Global Bifurcation Theory and Hilbert's Sixteenth Problem |
| title_auth | Global Bifurcation Theory and Hilbert's Sixteenth Problem |
| title_exact_search | Global Bifurcation Theory and Hilbert's Sixteenth Problem |
| title_full | Global Bifurcation Theory and Hilbert's Sixteenth Problem Valery A. Gaiko |
| title_fullStr | Global Bifurcation Theory and Hilbert's Sixteenth Problem Valery A. Gaiko |
| title_full_unstemmed | Global Bifurcation Theory and Hilbert's Sixteenth Problem Valery A. Gaiko |
| title_short | Global Bifurcation Theory and Hilbert's Sixteenth Problem |
| title_sort | global bifurcation theory and hilbert s sixteenth problem |
| topic | Mathematics Global analysis Differential Equations Ordinary Differential Equations Global Analysis and Analysis on Manifolds Mathematical Modeling and Industrial Mathematics Applications of Mathematics Mathematical and Computational Biology Mathematik Verzweigung Mathematik (DE-588)4078889-1 gnd Hilbertsches Problem 16 (DE-588)4391597-8 gnd |
| topic_facet | Mathematics Global analysis Differential Equations Ordinary Differential Equations Global Analysis and Analysis on Manifolds Mathematical Modeling and Industrial Mathematics Applications of Mathematics Mathematical and Computational Biology Mathematik Verzweigung Mathematik Hilbertsches Problem 16 |
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