Bifurcations of planar vector fields and Hilbert's sixteenth problem:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Basel [u.a.]
Birkhäuser
1998
|
| Series: | Progress in mathematics
164 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 193 - 199 |
| Physical Description: | XVII, 204 S. graph. Darst. |
| ISBN: | 3764359005 0817659005 |
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MARC
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| 100 | 1 | |a Roussarie, Robert |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Bifurcations of planar vector fields and Hilbert's sixteenth problem |c Robert Roussarie |
| 264 | 1 | |a Basel [u.a.] |b Birkhäuser |c 1998 | |
| 300 | |a XVII, 204 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in mathematics |v 164 | |
| 500 | |a Literaturverz. S. 193 - 199 | ||
| 650 | 7 | |a Bifurcatie |2 gtt | |
| 650 | 4 | |a Bifurcation, Théorie de la | |
| 650 | 7 | |a Bifurcation, Théorie de la |2 ram | |
| 650 | 4 | |a Champs vectoriels | |
| 650 | 7 | |a Champs vectoriels |2 ram | |
| 650 | 7 | |a Hilberts zestiende probleem |2 gtt | |
| 650 | 7 | |a Vectorvelden |2 gtt | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Vector fields | |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Hilbertsches Problem 16 |0 (DE-588)4391597-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Planares Vektorfeld |0 (DE-588)4261750-9 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-008047306 | ||
Record in the Search Index
| _version_ | 1804126494642929664 |
|---|---|
| adam_text | Contents
Preface
...................................... ix
1
Families
of Two-dimensional Vector Fields
1.1
Vector fields on surfaces of genus
0.................. 1
1.1.1
The phase space
........................ 1
1.1.2
The
Poincaré-Bendixson
property
............... 1
1.1.3
Phase portrait
......................... 3
1.2
A first approach to Bifurcation Theory
................ 6
1.2.1
General definitions
....................... 7
1.2.2
Singularities of finite codimension.
The saddle-node bifurcations
................. 9
1.2.3
Bifurcations of singular points versus bifurcations of
periodic orbits. The Bogdanov-Takens bifurcation
..... 10
2
Limit Periodic Sets
2.1
Organizing centers for bifurcations of limit cycles
.......... 18
2.1.1
Definition of limit periodic sets
................ 18
2.1.2
The structure of limit periodic sets
.............. 20
2.2
The cyclicity property
......................... 22
2.2.1
Definition of cyclicity for limit periodic sets
......... 22
2.2.2
The finite cyclicity conjecture.
Local reduction of Hubert s
Іб*11
problem
.......... 23
2.2.3
A program for solving the existential Hubert s problem
. . 25
3
The
О
-Parameter
Case
3.1
Blowing up of singularities of vector fields
.............. 33
3.1.1
Polar and directional blow-up
................. 33
3.1.2
Successive Blow-ups
...................... 35
3.1.3
Quasi-homogeneous blow-up and the Newton diagram
... 39
3.2
The finiteness result for analytic vector fields on S2
......... 41
3.3
The Dulac problem
...........................
Bifurcations of Regular Limit Periodic Sets
4.1
The return map
............................. 51
4.1.1
Return map for a periodic orbit
................ 51
4.1.2
Return map near an elliptic point
.............. 52
4.2
Regular limit periodic sets of finite codimension
........... 55
4.2.1
Periodic orbit
.......................... 55
4.2.2
Elliptic focus
.......................... 57
4.3
Regular limit periodic set of infinite codimension
.......... 59
4.3.1
The Bautin Ideal
........................ 60
4.3.2
Properties of the Bautin Ideal
................. 63
4.3.3
Finite cyclicity of regular limit periodic sets
......... 67
4.3.4
Melnikov functions
....................... 72
4.3.5
Application to quadratic vector fields
............ 75
4.3.6
Some properties of Abelian Integrals
............. 82
Bifurcations of Elementary Graphics
5.1
Transition map near a hyperbolic saddle point
........... 92
5.1.1
Normal form of X near the saddle point
.......... 92
5.1.2
The structure of the transition map
for the normal family
..................... 94
5.1.3
The structure of the transition map of X
.......... 104
5.2
Unfoldings of saddle connections in the finite codimension case
. . 109
5.2.1
The codimension
1
case
.................... 109
5.2.2
The fc-codimension case, fc
> 2................ 112
5.2.3
Bifurcation diagrams for generic unfoldings
......... 117
5.3
Unfoldings of saddle connections of infinite codimension
...... 125
5.3.1
Finite cyclicity property of analytic unfoldings
....... 125
5.3.2
An example in quadratic vector fields
............ 136
5.4
Unfoldings of elementary graphics
.................. 139
5.4.1
Hyperbolic graphic with
2
vertices
.............. 139
5.4.2
Generic unfoldings of hyperbolic /c-graphics
......... 142
5.4.3
Generic elementary polycycles
................. 146
Desingularization Theory and Bifurcation
of Non-elementary Limit Periodic Sets
6.1
The use of rescaling formulas
..................... 151
6.2
Desingularization of unfoldings of cuspidal loops
.......... 155
Contents
vii
6.2.1 Global
blowing-up of the cusp unfolding
........... 158
6.2.2
Asymptotic form for the shift map equation
......... 162
6.2.3
Properties of gv(u)
....................... 169
6.2.4
Monotonicity property for io(^) and tn(x)
.......... 171
6.2.5
Indications for the proof of Theorem
31........... 172
6.3
A method of desingularization for analytic vector fields
....... 181
6.3.1
Foliated local vector fields
................... 181
6.3.2
Operations of desingularization
................ 184
6.3.3
Conjectures
........................... 186
6.3.4
Final comments and perspectives
............... 188
Bibliography
................................... 193
Index
....................................... 201
|
| any_adam_object | 1 |
| author | Roussarie, Robert |
| author_facet | Roussarie, Robert |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.35 |
| dewey-search | 515/.35 |
| dewey-sort | 3515 235 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011907107 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T18:18:21Z |
| institution | BVB |
| isbn | 3764359005 0817659005 |
| language | English |
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| physical | XVII, 204 S. graph. Darst. |
| publishDate | 1998 |
| publishDateSearch | 1998 |
| publishDateSort | 1998 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in mathematics |
| series2 | Progress in mathematics |
| spelling | Roussarie, Robert Verfasser aut Bifurcations of planar vector fields and Hilbert's sixteenth problem Robert Roussarie Basel [u.a.] Birkhäuser 1998 XVII, 204 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in mathematics 164 Literaturverz. S. 193 - 199 Bifurcatie gtt Bifurcation, Théorie de la Bifurcation, Théorie de la ram Champs vectoriels Champs vectoriels ram Hilberts zestiende probleem gtt Vectorvelden gtt Bifurcation theory Vector fields Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Hilbertsches Problem 16 (DE-588)4391597-8 gnd rswk-swf Planares Vektorfeld (DE-588)4261750-9 gnd rswk-swf Planares Vektorfeld (DE-588)4261750-9 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Hilbertsches Problem 16 (DE-588)4391597-8 s Progress in mathematics 164 (DE-604)BV000004120 164 Digitalisierung UB Passau application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008047306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Roussarie, Robert Bifurcations of planar vector fields and Hilbert's sixteenth problem Progress in mathematics Bifurcatie gtt Bifurcation, Théorie de la Bifurcation, Théorie de la ram Champs vectoriels Champs vectoriels ram Hilberts zestiende probleem gtt Vectorvelden gtt Bifurcation theory Vector fields Verzweigung Mathematik (DE-588)4078889-1 gnd Hilbertsches Problem 16 (DE-588)4391597-8 gnd Planares Vektorfeld (DE-588)4261750-9 gnd |
| subject_GND | (DE-588)4078889-1 (DE-588)4391597-8 (DE-588)4261750-9 |
| title | Bifurcations of planar vector fields and Hilbert's sixteenth problem |
| title_auth | Bifurcations of planar vector fields and Hilbert's sixteenth problem |
| title_exact_search | Bifurcations of planar vector fields and Hilbert's sixteenth problem |
| title_full | Bifurcations of planar vector fields and Hilbert's sixteenth problem Robert Roussarie |
| title_fullStr | Bifurcations of planar vector fields and Hilbert's sixteenth problem Robert Roussarie |
| title_full_unstemmed | Bifurcations of planar vector fields and Hilbert's sixteenth problem Robert Roussarie |
| title_short | Bifurcations of planar vector fields and Hilbert's sixteenth problem |
| title_sort | bifurcations of planar vector fields and hilbert s sixteenth problem |
| topic | Bifurcatie gtt Bifurcation, Théorie de la Bifurcation, Théorie de la ram Champs vectoriels Champs vectoriels ram Hilberts zestiende probleem gtt Vectorvelden gtt Bifurcation theory Vector fields Verzweigung Mathematik (DE-588)4078889-1 gnd Hilbertsches Problem 16 (DE-588)4391597-8 gnd Planares Vektorfeld (DE-588)4261750-9 gnd |
| topic_facet | Bifurcatie Bifurcation, Théorie de la Champs vectoriels Hilberts zestiende probleem Vectorvelden Bifurcation theory Vector fields Verzweigung Mathematik Hilbertsches Problem 16 Planares Vektorfeld |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008047306&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000004120 |
| work_keys_str_mv | AT roussarierobert bifurcationsofplanarvectorfieldsandhilbertssixteenthproblem |