2-d Quadratic Maps And 3-d Ode Systems

A Rigorous Approach

By: Julien Clinton Sprott, Zeraoulia Elhadj

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Hardcover | 9 July 2010

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358 Pages

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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.

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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"

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Non-Fiction Mathematics Calculus & Mathematical Analysis Differential Calculus & Equations Science Physics Quantum Physics & Quantum Mechanics & Quantum Field Theory Engineering & Technology Energy Technology & Engineering Electrical Engineering Science in General Applied Mathematics

Chaos Theory Booktopia Publisher Services World Scientific Publishing

Preface	p. vii
Acknowledgements	p. xiii
Tools for the rigorous proof of chaos and bifurcations	p. 1
Introduction	p. 1
A chain of rigorous proof of chaos	p. 3
Poincaré map technique	p. 7
Characteristic multiplier	p. 7
The generalized Poincaré map	p. 8
Interval methods	p. 10
Mean value form	p. 13
The method of fixed point index	p. 14
Periodic points of the TS-map	p. 16
Existence of semiconjugacy	p. 17
Smale's horseshoe map	p. 19
Some Basic properties of Smale's horseshoe map	p. 20
Dynamics of the horseshoe map	p. 22
Symbolic dynamics	p. 23
The Sil'nikov criterion for the existence of chaos	p. 26
Sil'nikov criterion for smooth systems	p. 26
Sil'nikov criterion for continuous piecewise linear systems	p. 27
The Marotto theorem	p. 28
The verified optimization technique	p. 30
The checking routine algorithm	p. 30
Efficacy of the checking routine algorithm	p. 31
Shadowing lemma	p. 33
Shadowing lemmas for ODE systems and discrete mappings	p. 35
Homoclinic orbit shadowing	p. 36
Method based on the second-derivative test and bounds for Lyapunov exponents	p. 38
The Wiener and Hammerstein cascade models	p. 39
Algorithm based on the Wiener model	p. 39
Algorithm based on the Hammerstein model	p. 42
Methods based on time series analysis	p. 43
A new chaos detector	p. 46
Exercises	p. 47
2-D quadratic maps: The invertible case	p. 49
Introduction	p. 49
Equivalences in the general 2-D quadratic maps	p. 50
Invertibility of the map	p. 59
The Hénon map	p. 63
Methods for locating chaotic regions in the Hénon map	p. 64
Finding Smale's horseshoe maps	p. 64
Topological entropy	p. 65
The verified optimization technique	p. 68
The Wiener and Hammerstein cascade models	p. 69
Methods based on time series analysis	p. 70
The validated shadowing	p. 71
The method of fixed point index	p. 72
A new chaos detector	p. 72
Bifurcation analysis	p. 73
Existence and bifurcations of periodic orbits	p. 73
Recent bifurcation phenomena	p. 74
Existence of transversal homoclinic points	p. 76
Classification of homoclinic bifurcations	p. 94
Basins of attraction	p. 99
Structure of the parameter space	p. 100
Exercises	p. 103
Classification of chaotic orbits of the general 2-D quadratic map	p. 105
Analytical prediction of system orbits	p. 105
Existence of unbounded orbits	p. 105
Existence of bounded orbits	p. 107
A zone of possible chaotic orbits	p. 109
Zones of stable fixed points	p. 111
Boundary between different attractors	p. 112
Finding chaotic and nonchaotic attractors	p. 123
Finding hyperchaotic attractors	p. 131
Some criteria for finding chaotic orbits	p. 139
2-D quadratic maps with one nonlinearity	p. 140
2-D quadratic maps with two nonlinearities	p. 148
2-D quadratic maps with three nonlinearities	p. 149
2-D quadratic maps with four nonlinearities	p. 151
2-D quadratic maps with five nonlinearities	p. 153
2-D quadratic maps with six nonlinearities	p. 153
Numerical analysis	p. 154
Some observed catastrophic solutions in the dynamics of the map	p. 155
Rigorous proof of chaos in the double-scroll system	p. 159
Introduction	p. 159
Piecewise linear geometry and its real Jordan form	p. 164
Geometry of a piecewise linear vector field in R³	p. 164
Straight line tangency property	p. 166
The real Jordan form	p. 168
Canonical piecewise linear normal form	p. 171
Poincaré and half-return maps	p. 175
The dynamics of an orbit in the double-scroll	p. 176
The half-return map ¿₀	p. 177
Half-return map ¿₁	p. 185
Connection map ¿	p. 192
Poincaré map ¿	p. 194
V₁ portrait of V₀	p. 195
Spiral image property	p. 196
Method 1: Sil'nikov criteria	p. 197
Homoclinic orbits	p. 197
Examination of the loci of points	p. 202
Heteroclinic orbits	p. 210
Geometrical explanation	p. 214
Dynamics near homoclinic and heteroclinic orbits	p. 215
Subfamilies of the double-scroll family	p. 219
The geometric model	p. 220
Method 2: The computer-assisted proof	p. 229
Estimating topological entropy	p. 230
Formula for the topological entropy in terms of the Poincaré map	p. 236
Exercises	p. 238
Rigorous analysis of bifurcation phenomena	p. 239
Introduction	p. 239
Asymptotic stability of equilibria	p. 240
Types of chaotic attractors in the double-scroll	p. 244
Method 1: Rigorous mathematical analysis	p. 245
The pull-up map	p. 246
Construction of the trapping region for the double-scroll	p. 247
Finding trapping regions using confinors theory	p. 252
Construction of the trapping region for the Rössler-type attractor	p. 257
Macroscopic structure of an attractor for the double-scroll system	p. 265
Collision process	p. 268
Bifurcation diagram	p. 279
Method 2: One-dimensional Poincaré map	p. 281
Introduction	p. 281
Construction of the 1-D Poincaré map	p. 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 theory	p. 307
Exercises	p. 312
Bibliography	p. 315
Index	p. 337
Table of Contents provided by Ingram. All Rights Reserved.

2-d Quadratic Maps And 3-d Ode Systems

A Rigorous Approach

At a Glance

Hardcover

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