2-D quadratic maps and 3-D ODE systems: a rigorous approach
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Hackensack, NJ [u.a.]
World Scientific
c2010
|
| Series: | World Scientific series on nonlinear science
Series A, Monographs and treatises ; 73 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Includes bibliographical references and index |
| Physical Description: | XIII, 342 S. Ill., graph. Darst. 24 cm |
| ISBN: | 9789814307741 9814307742 |
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MARC
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| 245 | 1 | 0 | |a 2-D quadratic maps and 3-D ODE systems |b a rigorous approach |c Elhadj Zeraoulia ; Julien Clinton Sprott |
| 246 | 1 | 3 | |a Two-D quadratic maps and 3-D ODE systems |
| 264 | 1 | |a Hackensack, NJ [u.a.] |b World Scientific |c c2010 | |
| 300 | |a XIII, 342 S. |b Ill., graph. Darst. |c 24 cm | ||
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| 490 | 1 | |a World Scientific series on nonlinear science : Series A, Monographs and treatises |v 73 | |
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Record in the Search Index
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|---|---|
| adam_text | Titel: 2-D quadratic maps and 3-D ODE systems
Autor: Zeraoulia, Elhadj
Jahr: 2010
Contents
Preface vii
Acknowledgements xiii
1. Tools for the rigorous proof of chaos and bifurcations 1
1.1 Introduction.......................... 1
1.2 A chain of rigorous proof of chaos.............. 3
1.3 Poincaré map technique................... 7
1.3.1 Characteristic multiplier.............. 7
1.3.2 The generalized Poincaré map........... 8
1.3.3 Interval methods................... 10
1.3.4 Mean value form.................. 13
1-4 The method of fixed point index.............. 14
1.4.1 Periodic points of the TS-map........... 16
1.4.2 Existence of semiconjugacy............. 17
1.5 Smale s horseshoe map.................... 19
1.5.1 Some basic properties of Smale s horseshoe map . 20
1.5.2 Dynamics of the horseshoe map.......... 22
1.5.3 Symbolic dynamics................. 23
1.6 The Sil nikov criterion for the existence of chaos..... 26
1.6.1 Sil nikov criterion for smooth systems....... 26
1.6.2 Sil nikov criterion for continuous piecewise linear
systems........................ 27
1.7 The Marotto theorem.................... 28
1.8 The verified optimization technique............. 30
1.8.1 The checking routine algorithm.......... 30
1.8.2 Efficacy of the checking routine algorithm..... 31
? 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach
1.9 Shadowing lemma ...................... 33
1.9.1 Shadowing lemmas for ODE systems and discrete
mappings....................... 35
1.9.2 Homoclinic orbit shadowing ............ 36
1.10 Method based on the second-derivative test and bounds for
Lyapunov exponents..................... 38
1.11 The Wiener and Hammerstein cascade models....... 39
1.11.1 Algorithm based on the Wiener model ...... 39
1.11.2 Algorithm based on the Hammerstein model ... 42
1.12 Methods based on time series analysis........... 43
1.13 A new chaos detector..................... 46
1.14 Exercises ........................... 47
2. 2-D quadratic maps: The invertible case 49
2.1 Introduction.......................... 49
2.2 Equivalences in the general 2-D quadratic maps...... 50
2.3 Invertibility of the map ................... 59
2.4 The Hénon map ....................... 63
2.5 Methods for locating chaotic regions in the Hénon map . 64
2.5.1 Finding Smale s horseshoe maps.......... 64
2.5.2 Topological entropy................. 65
2.5.3 The verified optimization technique........ 68
2.5.4 The Wiener and Hammerstein cascade models . . 69
2.5.5 Methods based on time series analysis....... 70
2.5.6 The validated shadowing.............. 71
2.5.7 The method of fixed point index.......... 72
2.5.8 A new chaos detector................ 72
2.6 Bifurcation analysis ..................... 73
2.6.1 Existence and bifurcations of periodic orbits ... 73
2.6.2 Recent bifurcation phenomena........... 74
2.6.3 Existence of transversal homoclinic points .... 76
2.6.4 Classification of homoclinic bifurcations...... 94
2.6.5 Basins of attraction................. 99
2.6.6 Structure of the parameter space.......... 100
2.7 Exercises ........................... 103
3. Classification of chaotic orbits of the general 2-D
quadratic map 105
Contents xi
3.1 Analytical prediction of system orbits........... 105
3.1.1 Existence of unbounded orbits........... 105
3.1.2 Existence of bounded orbits ............ 107
3.2 A zone of possible chaotic orbits.............. 109
3.2.1 Zones of stable fixed points............. Ill
3.3 Boundary between different attractors........... 112
3.4 Finding chaotic and nonchaotic attractors......... 123
3.5 Finding hyperchaotic attractors............... 131
3.6 Some criteria for finding chaotic orbits........... 139
3.7 2-D quadratic maps with one nonlinearity......... 140
3.8 2-D quadratic maps with two nonlinearities........ 148
3.9 2-D quadratic maps with three nonlinearities....... 149
3.10 2-D quadratic maps with four nonlinearities........ 151
3.11 2-D quadratic maps with five nonlinearities........ 153
3.12 2-D quadratic maps with six nonlinearities......... 153
3.13 Numerical analysis...................... 154
3.13.1 Some observed catastrophic solutions in the dy-
namics of the map.................. 155
4. Rigorous proof of chaos in the double-scroll system 159
4.1 Introduction.......................... 159
4.2 Piecewise linear geometry and its real Jordan form .... 164
4.2.1 Geometry of a piecewise linear vector field in M3 . 164
4.2.2 Straight line tangency property .......... 166
4.2.3 The real Jordan form................ 168
4.2.4 Canonical piecewise linear normal form...... 171
4.2.5 Poincaré and half-return maps........... 175
4.3 The dynamics of an orbit in the double-scroll....... 176
4.3.1 The half-return map p0............... 177
4.3.2 Half-return map p?................. 185
4.3.3 Connection map F.................. 192
4.4 Poincaré map p........................ 194
4.4.1 Vi portrait of Vb................... 195
4.4.2 Spiral image property................ 196
4.5 Method 1: Sil nikov criteria................. 197
4.5.1 Homoclinic orbits.................. 197
4.5.2 Examination of the loci of points ......... 202
4.5.3 Heteroclinic orbits.................. 210
4.5.4 Geometrical explanation.............. 214
xii 2-D Quadratic Maps and 3-D ODE Systems: A Rigorous Approach
4.5.5 Dynamics near homoclinic and heteroclinic orbits 215
4.6 Subfamilies of the double-scroll family........... 219
4.7 The geometric model..................... 220
4.8 Method 2: The computer-assisted proof.......... 229
4.8.1 Estimating topological entropy........... 230
4.8.2 Formula for the topological entropy in terms of the
Poincaré map.................... 236
4.9 Exercises ........................... 238
5. Rigorous analysis of bifurcation phenomena 239
5.1 Introduction.......................... 239
5.2 Asymptotic stability of equilibria.............. 240
5.3 Types of chaotic attractors in the double-scroll...... 244
5.4 Method 1: Rigorous mathematical analysis........ 245
5.4.1 The pull-up map................... 246
5.4.2 Construction of the trapping region for the double-
scroll ......................... 247
5.4.3 Finding trapping regions using confinors theory . 252
5.4.4 Construction of the trapping region for the
Rössler-type attractor................ 257
5.4.5 Macroscopic structure of an attractor for the
double-scroll system................. 265
5.4.6 Collision process................... 268
5.4.7 Bifurcation diagram................. 279
5.5 Method 2: One-dimensional Poincaré map......... 281
5.5.1 Introduction..................... 281
5.5.2 Construction of the 1-D Poincaré map ...... 281
5.5.3 Properties of the 1-D Poincaré map p*...... 289
5.5.4 Numerical examples for the 1-D Poincaré map p* 291
5.5.5 Periodic points of the 1-D Poincaré map tt* . . . . 292
5.5.6 Bifurcation diagrams using confinors theory . . . 307
5.6 Exercises ........................... 312
Bibliography 315
Index 337
|
| any_adam_object | 1 |
| author | Zeraoulia, Elhadj 1976- Sprott, Julien C. 1942- |
| author_GND | (DE-588)142963844 (DE-588)142964077 |
| author_facet | Zeraoulia, Elhadj 1976- Sprott, Julien C. 1942- |
| author_role | aut aut |
| author_sort | Zeraoulia, Elhadj 1976- |
| author_variant | e z ez j c s jc jcs |
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| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.352 |
| dewey-search | 515.352 |
| dewey-sort | 3515.352 |
| dewey-tens | 510 - Mathematics |
| discipline | Physik Mathematik |
| format | Book |
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| illustrated | Illustrated |
| indexdate | 2024-07-09T22:48:43Z |
| institution | BVB |
| isbn | 9789814307741 9814307742 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-020724983 |
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| physical | XIII, 342 S. Ill., graph. Darst. 24 cm |
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| publisher | World Scientific |
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| series | World Scientific series on nonlinear science |
| series2 | World Scientific series on nonlinear science : Series A, Monographs and treatises |
| spelling | Zeraoulia, Elhadj 1976- Verfasser (DE-588)142963844 aut 2-D quadratic maps and 3-D ODE systems a rigorous approach Elhadj Zeraoulia ; Julien Clinton Sprott Two-D quadratic maps and 3-D ODE systems Hackensack, NJ [u.a.] World Scientific c2010 XIII, 342 S. Ill., graph. Darst. 24 cm txt rdacontent n rdamedia nc rdacarrier World Scientific series on nonlinear science : Series A, Monographs and treatises 73 Includes bibliographical references and index Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Chaotisches System (DE-588)4316104-2 s Gewöhnliche Differentialgleichung (DE-588)4020929-5 s DE-604 Sprott, Julien C. 1942- Verfasser (DE-588)142964077 aut World Scientific series on nonlinear science Series A, Monographs and treatises ; 73 (DE-604)BV009051753 73 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724983&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Zeraoulia, Elhadj 1976- Sprott, Julien C. 1942- 2-D quadratic maps and 3-D ODE systems a rigorous approach World Scientific series on nonlinear science Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| subject_GND | (DE-588)4020929-5 (DE-588)4316104-2 |
| title | 2-D quadratic maps and 3-D ODE systems a rigorous approach |
| title_alt | Two-D quadratic maps and 3-D ODE systems |
| title_auth | 2-D quadratic maps and 3-D ODE systems a rigorous approach |
| title_exact_search | 2-D quadratic maps and 3-D ODE systems a rigorous approach |
| title_full | 2-D quadratic maps and 3-D ODE systems a rigorous approach Elhadj Zeraoulia ; Julien Clinton Sprott |
| title_fullStr | 2-D quadratic maps and 3-D ODE systems a rigorous approach Elhadj Zeraoulia ; Julien Clinton Sprott |
| title_full_unstemmed | 2-D quadratic maps and 3-D ODE systems a rigorous approach Elhadj Zeraoulia ; Julien Clinton Sprott |
| title_short | 2-D quadratic maps and 3-D ODE systems |
| title_sort | 2 d quadratic maps and 3 d ode systems a rigorous approach |
| title_sub | a rigorous approach |
| topic | Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd Chaotisches System (DE-588)4316104-2 gnd |
| topic_facet | Gewöhnliche Differentialgleichung Chaotisches System |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=020724983&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV009051753 |
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