2-D quadratic maps and 3-D ODE systems: a rigorous approach
Gespeichert in:
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| Format: | Elektronisch E-Book |
| Sprache: | English |
| Veröffentlicht: |
Singapore
World Scientific
©2010
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| Schriftenreihe: | World Scientific series on nonlinear science
v. 73 |
| Schlagworte: | |
| Online-Zugang: | FAW01 FAW02 Volltext |
| Beschreibung: | Includes bibliographical references and index 1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises -- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises -- - 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis -- - 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises -- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hňon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hňon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods |
| Beschreibung: | 1 Online-Ressource (xiii, 342 pages) |
| ISBN: | 9789814307741 9789814307758 9814307742 9814307750 |
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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 |
| 264 | 1 | |a Singapore |b World Scientific |c ©2010 | |
| 300 | |a 1 Online-Ressource (xiii, 342 pages) | ||
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| 490 | 0 | |a World Scientific series on nonlinear science |v v. 73 | |
| 500 | |a Includes bibliographical references and index | ||
| 500 | |a 1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises -- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises -- | ||
| 500 | |a - 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis -- | ||
| 500 | |a - 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises -- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises | ||
| 500 | |a This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hňon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hňon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods | ||
| 650 | 4 | |a Mathematics | |
| 650 | 7 | |a MATHEMATICS / Differential Equations / Ordinary |2 bisacsh | |
| 650 | 4 | |a Mathematik | |
| 650 | 4 | |a Forms, Quadratic | |
| 650 | 4 | |a Differential equations, Linear | |
| 650 | 4 | |a Bifurcation theory | |
| 650 | 4 | |a Differentiable dynamical systems | |
| 650 | 4 | |a Proof theory | |
| 650 | 0 | 7 | |a Chaotisches System |0 (DE-588)4316104-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |2 gnd |9 rswk-swf |
| 689 | 0 | 0 | |a Chaotisches System |0 (DE-588)4316104-2 |D s |
| 689 | 0 | 1 | |a Gewöhnliche Differentialgleichung |0 (DE-588)4020929-5 |D s |
| 689 | 0 | |8 1\p |5 DE-604 | |
| 700 | 1 | |a Sprott, Julien C. |e Sonstige |4 oth | |
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Datensatz im Suchindex
| _version_ | 1804175649354547200 |
|---|---|
| any_adam_object | |
| author | Zeraoulia, Elhadj |
| author_facet | Zeraoulia, Elhadj |
| author_role | aut |
| author_sort | Zeraoulia, Elhadj |
| author_variant | e z ez |
| building | Verbundindex |
| bvnumber | BV043170075 |
| collection | ZDB-4-EBA |
| ctrlnum | (OCoLC)740446113 (DE-599)BVBBV043170075 |
| dewey-full | 515.352 |
| 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 | Mathematik |
| format | Electronic eBook |
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| id | DE-604.BV043170075 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-10T07:19:39Z |
| institution | BVB |
| isbn | 9789814307741 9789814307758 9814307742 9814307750 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-028594266 |
| oclc_num | 740446113 |
| open_access_boolean | |
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| owner_facet | DE-1046 DE-1047 |
| physical | 1 Online-Ressource (xiii, 342 pages) |
| psigel | ZDB-4-EBA ZDB-4-EBA FAW_PDA_EBA |
| publishDate | 2010 |
| publishDateSearch | 2010 |
| publishDateSort | 2010 |
| publisher | World Scientific |
| record_format | marc |
| series2 | World Scientific series on nonlinear science |
| spelling | Zeraoulia, Elhadj Verfasser aut 2-D quadratic maps and 3-D ODE systems a rigorous approach Elhadj Zeraoulia, Julien Clinton Sprott Singapore World Scientific ©2010 1 Online-Ressource (xiii, 342 pages) txt rdacontent c rdamedia cr rdacarrier World Scientific series on nonlinear science v. 73 Includes bibliographical references and index 1. Tools for the rigorous proof of chaos and bifurcations. 1.1. Introduction. 1.2. A chain of rigorous proof of chaos. 1.3. Poincare map technique. 1.4. The method of fixed point index. 1.5. Smale's horseshoe map. 1.6. The Sil'nikov criterion for the existence of chaos. 1.7. The Marotto theorem. 1.8. The verified optimization technique. 1.9. Shadowing lemma. 1.10. Method based on the second-derivative test and bounds for Lyapunov exponents. 1.11. The Wiener and Hammerstein cascade models. 1.12. Methods based on time series analysis. 1.13. A new chaos detector. 1.14. Exercises -- 2. 2-D quadratic maps : The invertible case. 2.1. Introduction. 2.2. Equivalences in the general 2-D quadratic maps. 2.3. Invertibility of the map. 2.4. The Henon map. 2.5. Methods for locating chaotic regions in the Henon map. 2.6. Bifurcation analysis. 2.7. Exercises -- - 3. Classification of chaotic orbits of the general 2-D quadratic map. 3.1. Analytical prediction of system orbits. 3.2. A zone of possible chaotic orbits. 3.3. Boundary between different attractors. 3.4. Finding chaotic and nonchaotic attractors. 3.5. Finding hyperchaotic attractors. 3.6. Some criteria for finding chaotic orbits. 3.7. 2-D quadratic maps with one nonlinearity. 3.8. 2-D quadratic maps with two nonlinearities. 3.9. 2-D quadratic maps with three nonlinearities. 3.10. 2-D quadratic maps with four nonlinearities. 3.11. 2-D quadratic maps with five nonlinearities. 3.12. 2-D quadratic maps with six nonlinearities. 3.13. Numerical analysis -- - 4. Rigorous proof of chaos in the double-scroll system. 4.1. Introduction. 4.2. Piecewise linear geometry and its real Jordan form. 4.3. The dynamics of an orbit in the double-scroll. 4.4. Poincare map [symbol]. 4.5. Method 1 : Sil'nikov criteria. 4.6. Subfamilies of the double-scroll family. 4.7. The geometric model. 4.8. Method 2 : The computer-assisted proof. 4.9. Exercises -- 5. Rigorous analysis of bifurcation phenomena. 5.1. Introduction. 5.2. Asymptotic stability of equilibria. 5.3. Types of chaotic attractors in the double-scroll. 5.4. Method 1 : Rigorous mathematical analysis. 5.5. Method 2 : One-dimensional Poincare map. 5.6. Exercises This book is based on research on the rigorous proof of chaos and bifurcations in 2-D quadratic maps, especially the invertible case such as the Hňon map, and in 3-D ODE's, especially piecewise linear systems such as the Chua's circuit. In addition, the book covers some recent works in the field of general 2-D quadratic maps, especially their classification into equivalence classes, and finding regions for chaos, hyperchaos, and non-chaos in the space of bifurcation parameters. Following the main introduction to the rigorous tools used to prove chaos and bifurcations in the two representative systems, is the study of the invertible case of the 2-D quadratic map, where previous works are oriented toward Hňon mapping. 2-D quadratic maps are then classified into 30 maps with well-known formulas. Two proofs on the regions for chaos, hyperchaos, and non-chaos in the space of the bifurcation parameters are presented using a technique based on the second-derivative test and bounds for Lyapunov exponents. Also included is the proof of chaos in the piecewise linear Chua's system using two methods, the first of which is based on the construction of Poincare map, and the second is based on a computer-assisted proof. Finally, a rigorous analysis is provided on the bifurcational phenomena in the piecewise linear Chua's system using both an analytical 2-D mapping and a 1-D approximated Poincare mapping in addition to other analytical methods Mathematics MATHEMATICS / Differential Equations / Ordinary bisacsh Mathematik Forms, Quadratic Differential equations, Linear Bifurcation theory Differentiable dynamical systems Proof theory Chaotisches System (DE-588)4316104-2 gnd rswk-swf Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd rswk-swf Chaotisches System (DE-588)4316104-2 s Gewöhnliche Differentialgleichung (DE-588)4020929-5 s 1\p DE-604 Sprott, Julien C. Sonstige oth http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=374914 Aggregator Volltext 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Zeraoulia, Elhadj 2-D quadratic maps and 3-D ODE systems a rigorous approach Mathematics MATHEMATICS / Differential Equations / Ordinary bisacsh Mathematik Forms, Quadratic Differential equations, Linear Bifurcation theory Differentiable dynamical systems Proof theory Chaotisches System (DE-588)4316104-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| subject_GND | (DE-588)4316104-2 (DE-588)4020929-5 |
| title | 2-D quadratic maps and 3-D ODE systems a rigorous approach |
| 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 | Mathematics MATHEMATICS / Differential Equations / Ordinary bisacsh Mathematik Forms, Quadratic Differential equations, Linear Bifurcation theory Differentiable dynamical systems Proof theory Chaotisches System (DE-588)4316104-2 gnd Gewöhnliche Differentialgleichung (DE-588)4020929-5 gnd |
| topic_facet | Mathematics MATHEMATICS / Differential Equations / Ordinary Mathematik Forms, Quadratic Differential equations, Linear Bifurcation theory Differentiable dynamical systems Proof theory Chaotisches System Gewöhnliche Differentialgleichung |
| url | http://search.ebscohost.com/login.aspx?direct=true&scope=site&db=nlebk&db=nlabk&AN=374914 |
| work_keys_str_mv | AT zeraouliaelhadj 2dquadraticmapsand3dodesystemsarigorousapproach AT sprottjulienc 2dquadraticmapsand3dodesystemsarigorousapproach |