Planar dynamical systems: selected classical problems
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Berlin ; Boston, Mass.
De Gruyter
2014
Beijing Science Press |
| Subjects: | |
| Online Access: | Volltext Inhaltsverzeichnis |
| Item Description: | Literaturangaben |
| Physical Description: | XVIII, 371 S. graph. Darst. |
| ISBN: | 9783110298291 9783110298376 |
| DOI: | 10.1515/9783110298369 |
Staff View
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| 245 | 1 | 0 | |a Planar dynamical systems |b selected classical problems |c Yirong Liu ; Jibin Li ; Wentao Huang |
| 264 | 1 | |a Berlin ; Boston, Mass. |b De Gruyter |c 2014 | |
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CONTENTS
PREFACE V
1 BASIC CONCEPT AND LINEARIZED PROBLEM OF SYSTEMS 1
1.1 BASIC CONCEPT AND VARIABLE TRANSFORMATION 1
1.2 RESULTANT OF THE WEIERSTRASS POLYNOMIAL AND MULTIPLICITY OF
A SINGULAR POINT 3
1.3 QUASI-ALGEBRAIC INTEGRALS OF POLYNOMIAL SYSTEMS 10
1.4 CAUCHY MAJORANT AND ANALYTIC PROPERTIES IN A NEIGHBORHOOD OF
AN ORDINARY POINT 15
1.5 CLASSIFICATION OF ELEMENTARY SINGULAR POINTS AND LINEARIZED PROBLEM
* * * 24
1.6 NODE VALUE AND LINEARIZED PROBLEM OF THE INTEGER-RATIO NODE 30
1.7 LINEARIZED PROBLEM OF THE DEGENERATE NODE 35
1.8 INTEGRABILITY AND LINEARIZED PROBLEM OF WEAK CRITICAL SINGULAR POINT
* * 39
1.9 INTEGRABILITY AND LINEARIZED PROBLEM OF THE RESONANT SINGULAR POINT
* * * 58
2 FOCAL VALUES, SADDLE VALUES AND SINGULAR POINT VALUES 69
2.1 SUCCESSOR FUNCTIONS AND PROPERTIES OF FOCAL VALUES 69
2.2 POINCARE FORMAL SERIES AND ALGEBRAIC EQUIVALENCE 74
2.3 LINEAR RECURSIVE FORMULAS FOR THE COMPUTATION OF
SINGULAR POINT VALUES 78
2.4 THE ALGEBRAIC CONSTRUCTION OF SINGULAR VALUES 83
2.5 ELEMENTARY GENERALIZED ROTATION INVARIANTS OF THE CUBIC SYSTEMS 88
2.6 SINGULAR POINT VALUES AND INTEGRABILITY CONDITION
OF THE QUADRATIC SYSTEMS 90
2.7 SINGULAR POINT VALUES AND INTEGRABILITY CONDITION OF
THE CUBIC SYSTEMS HAVING HOMOGENEOUS NONLINEARITIES 93
3 MULTIPLE HOPF BIFURCATIONS 97
3.1 THE ZEROS OF SUCCESSOR FUNCTIONS IN THE POLAR COORDINATES 97
3.2 ANALYTIC EQUIVALENCE 100
3.3 QUASI SUCCESSOR FUNCTION 102
3.4 BIFURCATIONS OF LIMIT CIRCLE OF A CLASS OF QUADRATIC SYSTEMS 108
4 ISOCHRONOUS CENTER IN COMPLEX DOMAIN ILL
4.1 ISOCHRONOUS CENTERS AND PERIOD CONSTANTS ILL
HTTP://D-NB.INFO/1045094064
XVI
CONTENTS
4.2 LINEAR RECURSIVE FORMULAS TO COMPUTE PERIOD CONSTANTS 116
4.3 ISOCHRONOUS CENTER FOR A CLASS OF QUINTIC SYSTEM
IN THE COMPLEX DOMAIN 122
4.3.1 THE CONDITIONS OF ISOCHRONOUS CENTER UNDER CONDITION CI 123
4.3.2 THE CONDITIONS OF ISOCHRONOUS CENTER UNDER CONDITION C2 124
4.3.3 THE CONDITIONS OF ISOCHRONOUS CENTER UNDER CONDITION
C3
127
4.3.4 NON-ISOCHRONOUS CENTER UNDER CONDITION
C4
AND
C\
128
4.4 THE METHOD OF TIME-ANGLE DIFFERENCE 128
4.5 THE CONDITIONS OF ISOCHRONOUS CENTER OF THE ORIGIN FOR
A CUBIC SYSTEM 134
5 THEORY OF CENTER-FOCUS AND BIFURCATION OF LIMIT CYCLES
AT INFINITY OF A CLASS OF SYSTEMS 138
5.1 DEFINITION OF THE FOCAL VALUES OF INFINITY 138
5.2 CONVERSION OF QUESTIONS 141
5.3 METHOD OF FORMAL SERIES AND SINGULAR POINT VALUE OF INFINITY 144
5.4 THE ALGEBRAIC CONSTRUCTION OF SINGULAR POINT VALUES OF INFINITY 156
5.5 SINGULAR POINT VALUES AT INFINITY AND INTEGRABLE CONDITIONS FOR
A CLASS OF CUBIC SYSTEM 161
5.6 BIFURCATION OF LIMIT CYCLES AT INFINITY 168
5.7 ISOCHRONOUS CENTERS AT INFINITY OF A POLYNOMIAL SYSTEMS 172
5.7.1 CONDITIONS OF COMPLEX CENTER FOR SYSTEM (5.7.6) 173
5.7.2 CONDITIONS OF COMPLEX ISOCHRONOUS CENTER FOR SYSTEM (5.7.6) 176
6 THEORY OF CENTER-FOCUS AND BIFURCATIONS OF LIMIT CYCLES FOR
A CLASS OF MULTIPLE SINGULAR POINTS 180
6.1 SUCCESSION FUNCTION AND FOCAL VALUES FOR A CLASS OF
MULTIPLE SINGULAR POINTS 180
6.2 CONVERSION OF THE QUESTIONS 182
6.3 FORMAL SERIES, INTEGRAL FACTORS AND SINGULAR POINT VALUES FOR
A CLASS OF MULTIPLE SINGULAR POINTS 184
6.4 THE ALGEBRAIC STRUCTURE OF SINGULAR POINT VALUES OF A CLASS OF
MULTIPLE SINGULAR POINTS 196
6.5 BIFURCATION OF LIMIT CYCLES FROM A CLASS OF
MULTIPLE SINGULAR POINTS 198
6.6 BIFURCATION OF LIMIT CYCLES CREATED FROM A MULTIPLE SINGULAR POINT
FOR A CLASS OF QUARTIC SYSTEM 199
CONTENTS XVII
6.7 QUASI ISOCHRONOUS CENTER OF MULTIPLE SINGULAR POINT FOR
A CLASS OF ANALYTIC SYSTEM 202
7 ON QUASI ANALYTIC SYSTEMS 205
7.1 PRELIMINARY 205
7.2 REDUCTION OF THE PROBLEMS 208
7.3 FOCAL VALUES, PERIODIC CONSTANTS AND FIRST INTEGRALS OF (7.2.3) 210
7.4 SINGULAR POINT VALUES AND BIFURCATIONS OF LIMIT CYCLES OF
QUASI-QUADRATIC SYSTEMS 214
7.5 INTEGRABILITY OF QUASI-QUADRATIC SYSTEMS 217
7.6 ISOCHRONOUS CENTER OF QUASI-QUADRATIC SYSTEMS 219
7.6.1 THE PROBLEM OF COMPLEX ISOCHRONOUS CENTERS UNDER
THE CONDITION OF
C\
219
7.6.2 THE PROBLEM OF COMPLEX ISOCHRONOUS CENTERS UNDER
THE CONDITION OF C2 222
7.6.3 THE PROBLEM OF COMPLEX ISOCHRONOUS CENTERS UNDER
THE OTHER CONDITIONS 225
7.7 SINGULAR POINT VALUES AND CENTER CONDITIONS FOR A CLASS
OF QUASI-CUBIC SYSTEMS 228
8 LOCAL AND NON-LOCAL BIFURCATIONS OF PERTURBED Z
G
-EQUI VARIANT
HAMILTONIAN VECTOR FIELDS 232
8.1 -EQUIVARIANT PLANAR VECTOR FIELDS AND AN EXAMPLE 232
8.2 THE METHOD OF DETECTION FUNCTIONS: ROUGH PERTURBATIONS OF
Z
Q
- EQUIVARIANT HAMILTONIAN VECTOR FIELDS 242
8.3 BIFURCATIONS OF LIMIT CYCLES OF A Z2- EQUIVARIANT PERTURBED
HAMILTONIAN VECTOR FIELDS 244
8.3.1 HOPF BIFURCATION PARAMETER VALUES 246
8.3.2 BIFURCATIONS FROM HETEROCLINIC OR HOMOCLINIC LOOPS 247
8.3.3 THE VALUES OF BIFURCATION DIRECTIONS OF HETEROCLINIC
AND HOMOCLINIC LOOPS 252
8.3.4 ANALYSIS AND CONCLUSIONS 255
8.4 THE RATE OF GROWTH OF HILBERT NUMBER H(N)
WITH N 258
8.4.1 PRELIMINARY LEMMAS 259
8.4.2 A CORRECTION TO THE LOWER BOUNDS OF H(2
K
* 1) GIVEN
IN [CHRISTOPHER AND LLOYD, 1995] 262
8.4.3 A NEW LOWER BOUND FOR H(2
H
* 1) 265
XVIII
CONTENTS
8.4.4 LOWER
BOUND FOR
H(
3 X 2
FC_1
* 1) 267
9 CENTER-FOCUS PROBLEM AND BIFURCATIONS OF LIMIT CYCLES
FOR A
Z2
-EQUIVARIANT CUBIC SYSTEM 272
9.1 STANDARD FORM OF A CLASS OF SYSTEM (E.F
2
) 272
9.2 LIAPUNOV CONSTANTS, INVARIANT INTEGRALS AND THE NECESSARY AND
SUFFICIENT CONDITIONS OF THE EXISTENCE FOR THE BI-CENTER 274
9.3 THE CONDITIONS OF SIX-ORDER WEAK FOCUS AND BIFURCATIONS
OF LIMIT CYCLES 286
9.4 A CLASS OF (ESYSTEM WITH 13 LIMIT CYCLES 290
9.5 PROOFS OF LEMMA 9.4.1 AND THEOREM 9.4.1 294
9.6 THE PROOFS OF LEMMA 9.4.2 AND LEMMA 9.4.3 300
10 CENTER-FOCUS PROBLEM AND BIFURCATIONS OF LIMIT CYCLES
FOR THREE-MULTIPLE NILPOTENT SINGULAR POINTS 308
10.1 CRITERIA OF CENTER-FOCUS FOR A NILPOTENT SINGULAR POINT 308
10.2 SUCCESSOR FUNCTIONS AND FOCUS VALUE OF THREE-MULTIPLE
NILPOTENT SINGULAR POINT 311
10.3 BIFURCATION OF LIMIT CYCLES CREATED FROM THREE-MULTIPLE
NILPOTENT SINGULAR POINT 314
10.4 THE CLASSIFICATION OF THREE-MULTIPLE NILPOTENT SINGULAR POINTS
AND INVERSE INTEGRAL FACTOR 322
10.5 QUASI-LYAPUNOV CONSTANTS FOR THE THREE-MULTIPLE
NILPOTENT SINGULAR POINT 326
10.6 PROOF OF THEOREM 10.5.2 329
10.7 ON THE COMPUTATION OF QUASI-LYAPUNOV CONSTANTS 334
10.8 BIFURCATIONS OF LIMIT CYCLES CREATED FROM A THREE-MULTIPLE
NILPOTENT SINGULAR POINT OF A CUBIC SYSTEM 336
BIBLIOGRAPHY 342
INDEX 369 |
| any_adam_object | 1 |
| author | Liu, Yirong Li, Jibin Huang, Wentao |
| author_facet | Liu, Yirong Li, Jibin Huang, Wentao |
| author_role | aut aut aut |
| author_sort | Liu, Yirong |
| author_variant | y l yl j l jl w h wh |
| building | Verbundindex |
| bvnumber | BV042152982 |
| classification_rvk | SK 810 |
| collection | ZDB-23-DGG ZDB-23-GOA |
| ctrlnum | (OCoLC)865734159 (DE-599)DNB1045094064 |
| dewey-full | 515.39 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.39 |
| dewey-search | 515.39 |
| dewey-sort | 3515.39 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| doi_str_mv | 10.1515/9783110298369 |
| format | Book |
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| id | DE-604.BV042152982 |
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| indexdate | 2025-02-20T07:08:48Z |
| institution | BVB |
| isbn | 9783110298291 9783110298376 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-027592773 |
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| physical | XVIII, 371 S. graph. Darst. |
| psigel | ZDB-23-DGG ZDB-23-GOA |
| publishDate | 2014 |
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| publisher | De Gruyter Science Press |
| record_format | marc |
| spellingShingle | Liu, Yirong Li, Jibin Huang, Wentao Planar dynamical systems selected classical problems Planares Vektorfeld (DE-588)4261750-9 gnd Qualitative Theorie (DE-588)4571165-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| subject_GND | (DE-588)4261750-9 (DE-588)4571165-3 (DE-588)4013396-5 |
| title | Planar dynamical systems selected classical problems |
| title_auth | Planar dynamical systems selected classical problems |
| title_exact_search | Planar dynamical systems selected classical problems |
| title_full | Planar dynamical systems selected classical problems Yirong Liu ; Jibin Li ; Wentao Huang |
| title_fullStr | Planar dynamical systems selected classical problems Yirong Liu ; Jibin Li ; Wentao Huang |
| title_full_unstemmed | Planar dynamical systems selected classical problems Yirong Liu ; Jibin Li ; Wentao Huang |
| title_short | Planar dynamical systems |
| title_sort | planar dynamical systems selected classical problems |
| title_sub | selected classical problems |
| topic | Planares Vektorfeld (DE-588)4261750-9 gnd Qualitative Theorie (DE-588)4571165-3 gnd Dynamisches System (DE-588)4013396-5 gnd |
| topic_facet | Planares Vektorfeld Qualitative Theorie Dynamisches System |
| url | https://doi.org/10.1515/9783110298369 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=027592773&sequence=000001&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT liuyirong planardynamicalsystemsselectedclassicalproblems AT lijibin planardynamicalsystemsselectedclassicalproblems AT huangwentao planardynamicalsystemsselectedclassicalproblems |