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2-d Quadratic Maps And 3-d Ode Systems : A Rigorous Approach - Julien Clinton Sprott

2-d Quadratic Maps And 3-d Ode Systems

A Rigorous Approach


Published: 8th July 2010
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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 Henon 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 Henon 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.

As a reader in favor of tradition in mathematics, I find attractive the approach based on examples. -- Mathematical Reviews "Mathematical Reviews"
The book is generously illustrated; each chapter contains a short section with exercises for independent study. The monograph concludes with an exhaustive list of references and a concise index. It is a useful source for information specialists and graduate students working with the theory and applications of dynamical systems. -- Zentralblatt MATH "Zentralblatt MATH"

Prefacep. vii
Acknowledgementsp. xiii
Tools for the rigorous proof of chaos and bifurcationsp. 1
Introductionp. 1
A chain of rigorous proof of chaosp. 3
Poincaré map techniquep. 7
Characteristic multiplierp. 7
The generalized Poincaré mapp. 8
Interval methodsp. 10
Mean value formp. 13
The method of fixed point indexp. 14
Periodic points of the TS-mapp. 16
Existence of semiconjugacyp. 17
Smale's horseshoe mapp. 19
Some Basic properties of Smale's horseshoe mapp. 20
Dynamics of the horseshoe mapp. 22
Symbolic dynamicsp. 23
The Sil'nikov criterion for the existence of chaosp. 26
Sil'nikov criterion for smooth systemsp. 26
Sil'nikov criterion for continuous piecewise linear systemsp. 27
The Marotto theoremp. 28
The verified optimization techniquep. 30
The checking routine algorithmp. 30
Efficacy of the checking routine algorithmp. 31
Shadowing lemmap. 33
Shadowing lemmas for ODE systems and discrete mappingsp. 35
Homoclinic orbit shadowingp. 36
Method based on the second-derivative test and bounds for Lyapunov exponentsp. 38
The Wiener and Hammerstein cascade modelsp. 39
Algorithm based on the Wiener modelp. 39
Algorithm based on the Hammerstein modelp. 42
Methods based on time series analysisp. 43
A new chaos detectorp. 46
Exercisesp. 47
2-D quadratic maps: The invertible casep. 49
Introductionp. 49
Equivalences in the general 2-D quadratic mapsp. 50
Invertibility of the mapp. 59
The Hénon mapp. 63
Methods for locating chaotic regions in the Hénon mapp. 64
Finding Smale's horseshoe mapsp. 64
Topological entropyp. 65
The verified optimization techniquep. 68
The Wiener and Hammerstein cascade modelsp. 69
Methods based on time series analysisp. 70
The validated shadowingp. 71
The method of fixed point indexp. 72
A new chaos detectorp. 72
Bifurcation analysisp. 73
Existence and bifurcations of periodic orbitsp. 73
Recent bifurcation phenomenap. 74
Existence of transversal homoclinic pointsp. 76
Classification of homoclinic bifurcationsp. 94
Basins of attractionp. 99
Structure of the parameter spacep. 100
Exercisesp. 103
Classification of chaotic orbits of the general 2-D quadratic mapp. 105
Analytical prediction of system orbitsp. 105
Existence of unbounded orbitsp. 105
Existence of bounded orbitsp. 107
A zone of possible chaotic orbitsp. 109
Zones of stable fixed pointsp. 111
Boundary between different attractorsp. 112
Finding chaotic and nonchaotic attractorsp. 123
Finding hyperchaotic attractorsp. 131
Some criteria for finding chaotic orbitsp. 139
2-D quadratic maps with one nonlinearityp. 140
2-D quadratic maps with two nonlinearitiesp. 148
2-D quadratic maps with three nonlinearitiesp. 149
2-D quadratic maps with four nonlinearitiesp. 151
2-D quadratic maps with five nonlinearitiesp. 153
2-D quadratic maps with six nonlinearitiesp. 153
Numerical analysisp. 154
Some observed catastrophic solutions in the dynamics of the mapp. 155
Rigorous proof of chaos in the double-scroll systemp. 159
Introductionp. 159
Piecewise linear geometry and its real Jordan formp. 164
Geometry of a piecewise linear vector field in R3p. 164
Straight line tangency propertyp. 166
The real Jordan formp. 168
Canonical piecewise linear normal formp. 171
Poincaré and half-return mapsp. 175
The dynamics of an orbit in the double-scrollp. 176
The half-return map ¿0p. 177
Half-return map ¿1p. 185
Connection map ¿p. 192
Poincaré map ¿p. 194
V1 portrait of V0p. 195
Spiral image propertyp. 196
Method 1: Sil'nikov criteriap. 197
Homoclinic orbitsp. 197
Examination of the loci of pointsp. 202
Heteroclinic orbitsp. 210
Geometrical explanationp. 214
Dynamics near homoclinic and heteroclinic orbitsp. 215
Subfamilies of the double-scroll familyp. 219
The geometric modelp. 220
Method 2: The computer-assisted proofp. 229
Estimating topological entropyp. 230
Formula for the topological entropy in terms of the Poincaré mapp. 236
Exercisesp. 238
Rigorous analysis of bifurcation phenomenap. 239
Introductionp. 239
Asymptotic stability of equilibriap. 240
Types of chaotic attractors in the double-scrollp. 244
Method 1: Rigorous mathematical analysisp. 245
The pull-up mapp. 246
Construction of the trapping region for the double-scrollp. 247
Finding trapping regions using confinors theoryp. 252
Construction of the trapping region for the Rössler-type attractorp. 257
Macroscopic structure of an attractor for the double-scroll systemp. 265
Collision processp. 268
Bifurcation diagramp. 279
Method 2: One-dimensional Poincaré mapp. 281
Introductionp. 281
Construction of the 1-D Poincaré mapp. 281
Properties of the 1-D Poincaré map ¿*p. 289
Numerical examples for the 1-D Poincaré map ¿*p. 291
Periodic points of the 1-D Poincaré map ¿*p. 292
Bifurcation diagrams using confinors theoryp. 307
Exercisesp. 312
Bibliographyp. 315
Indexp. 337
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9789814307741
ISBN-10: 9814307742
Series: World Scientific Series on Nonlinear Science Series A
Audience: Professional
Format: Hardcover
Language: English
Number Of Pages: 356
Published: 8th July 2010
Publisher: World Scientific Publishing Co Pte Ltd
Country of Publication: SG
Dimensions (cm): 24.89 x 16.51  x 1.78
Weight (kg): 0.62