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E-grāmata: Robust Chaos and Its Applications [World Scientific e-book]

(Univ Of Tebessa, Algeria), (Univ Of Wisconsin-madison, Usa)
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Robust Chaos and Its ApplicationsPalielināt
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Rokasgrāmatas mājas lapa: http://www.worldscientific.com/worldscibooks/10.1142/8296#t=toc
Robust chaos is defined by the absence of periodic windows and coexisting attractors in some neighborhoods in the parameter space of a dynamical system. This unique book explores the definition, sources, and roles of robust chaos. The book is written in a reasonably self-contained manner and aims to provide students and researchers with the necessary understanding of the subject. Most of the known results, experiments, and conjectures about chaos in general and about robust chaos in particular are collected here in a pedagogical form. Many examples of dynamical systems, ranging from purely mathematical to natural and social processes displaying robust chaos, are discussed in detail. At the end of each chapter is a set of exercises and open problems (more than 260 in the whole book) intended to reinforce the ideas and provide additional experiences for both readers and researchers in nonlinear science in general, and chaos theory in particular.
Preface vii
1 Poincare Map Technique, Smale Horseshoe, and Symbolic Dynamics
1(26)
1.1 Poincare and generalized Poincare mappings
1(3)
1.2 Interval methods for calculating Poincare mappings
4(6)
1.2.1 Existence of periodic orbits
6(1)
1.2.2 Interval arithmetic
7(3)
1.3 Smale horseshoe
10(7)
1.3.1 Dynamics of the horseshoe map
12(5)
1.4 Symbolic dynamics
17(7)
1.4.1 The method of fixed point index
19(5)
1.5 Exercises
24(3)
2 Robustness of Chaos
27(12)
2.1 Strange attractors
27(7)
2.1.1 Concepts and definitions
27(2)
2.1.2 Robust chaos
29(3)
2.1.3 Domains of attraction
32(2)
2.2 Density and robustness of chaos
34(1)
2.3 Persistence and robustness of chaos
35(1)
2.4 Exercises
36(3)
3 Statistical Properties of Chaotic Attractors
39(22)
3.1 Entropies
39(11)
3.1.1 Lebesgue (volume) measure
39(1)
3.1.2 Physical (or Sinai--Ruelle--Bowen) measure
40(2)
3.1.3 Hausdorff dimension
42(1)
3.1.4 The topological entropy
43(4)
3.1.5 Lyapunov exponent
47(3)
3.2 Ergodic theory
50(1)
3.3 Statistical properties of chaotic attractors
51(6)
3.3.1 Autocorrelation function (ACF)
51(2)
3.3.2 Correlations
53(4)
3.4 Exercises
57(4)
4 Structural Stability
61(19)
4.1 The concept of structural stability
61(16)
4.1.1 Conditions for structural stability
62(2)
4.1.2 A proof of Anosov's theorem on structural stability of diffeomorphisms
64(13)
4.2 Exercises
77(3)
5 Transversality, Invariant Foliation, and the Shadowing Lemma
80(15)
5.1 Transversality
80(2)
5.2 Invariant foliation
82(2)
5.3 Shadowing lemma
84(9)
5.3.1 Homoclinic orbits and shadowing
88(2)
5.3.2 Shilnikov criterion for the existence of chaos
90(3)
5.4 Exercises
93(2)
6 Chaotic Attractors with Hyperbolic Structure
95(72)
6.1 Hyperbolic dynamics
96(6)
6.1.1 Concepts and definitions
96(1)
6.1.2 Anosov diffeomorphisms and Anosov flows
97(5)
6.2 Anosov diffeomorphisms on the torus Tn
102(10)
6.2.1 Anosov automorphisms
102(1)
6.2.2 Structure of Anosov diffeomorphisms
103(1)
6.2.3 Anosov torus Tn with a hyperbolic structure
104(1)
6.2.4 Expanding maps
105(2)
6.2.5 The Blaschke product
107(1)
6.2.6 The Bernoulli map
108(3)
6.2.7 The Arnold cat map
111(1)
6.3 Classification of strange attractors of dynamical systems
112(2)
6.4 Properties of hyperbolic chaotic attractors
114(7)
6.4.1 Geodesic flows on compact smooth manifolds
115(2)
6.4.2 The solenoid attractor
117(2)
6.4.3 The Smale--Williams solenoid
119(1)
6.4.4 Plykin attractor
120(1)
6.5 Proof of the hyperbolicity of the logistic map for μ > 4
121(5)
6.6 Generalized hyperbolic attractors
126(8)
6.7 Generating hyperbolic attractors
134(4)
6.8 Density of hyperbolicity and homoclinic bifurcations in arbitrary dimension
138(1)
6.9 Hyperbolicity tests
139(15)
6.9.1 Numerical procedure
141(3)
6.9.2 Testing hyperbolicity of the Henon map
144(9)
6.9.3 Testing hyperbolicity of the forced damped pendulum
153(1)
6.10 Uniform hyperbolicity test
154(4)
6.11 Exercises
158(9)
7 Robust Chaos in Hyperbolic Systems
167(41)
7.1 Modeling hyperbolic attractors
167(37)
7.1.1 Modeling the Smale--Williams attractor
168(8)
7.1.2 Testing hyperbolicity of system (7.1)
176(7)
7.1.3 Numerical verification of the hyperbolicity of system (7.1)
183(4)
7.1.4 Modeling the Arnold cat map
187(7)
7.1.5 Modeling the Bernoulli map
194(8)
7.1.6 Modeling Plykin's attractor
202(2)
7.2 Exercises
204(4)
8 Lorenz-Type Systems
208(50)
8.1 Lorenz-type attractors
208(2)
8.2 The Lorenz system
210(29)
8.2.1 Existence of Lorenz-type attractors
211(9)
8.2.2 Geometric models of the Lorenz equation
220(10)
8.2.3 Structure of the Lorenz attractor
230(9)
8.3 Expanding and contracting Lorenz attractors
239(2)
8.4 Wild strange attractors and pseudo-hyperbolicity
241(12)
8.5 Lorenz-type attractors realized in two-dimensional maps
253(2)
8.6 Exercises
255(3)
9 Robust Chaos in the Lorenz-Type Systems
258(14)
9.1 Robust chaos in the Lorenz-type attractors
258(2)
9.2 Robust chaos in Lorenz system
260(7)
9.3 Robust chaos in 2-D Lorenz-type attractors
267(2)
9.4 Exercises
269(3)
10 No Robust Chaos in Quasi-Attractors
272(31)
10.1 Quasi-attractors, concepts, and properties
273(2)
10.2 The Henon map
275(11)
10.2.1 Uniform hyperbolicity of the Henon map
276(5)
10.2.2 Hyperbolicity of Henon-like maps
281(1)
10.2.3 Henon attractor is a quasi-attractor
282(4)
10.3 The Strelkova--Anishchenko map
286(1)
10.4 The Anishchenko--Astakhov oscillator
286(2)
10.5 Chua's circuit
288(11)
10.5.1 Homoclinic and heteroclinic orbits
294(4)
10.5.2 The geometric model
298(1)
10.6 Exercises
299(4)
11 Robust Chaos in One-Dimensional Maps
303(55)
11.1 Unimodal maps
303(14)
11.1.1 S-unimodal maps
305(2)
11.1.2 Relation between unimodality and hyperbolicity
307(2)
11.1.3 Classification of unimodal maps of the interval
309(3)
11.1.4 Collet-Eckmann maps
312(2)
11.1.5 Statistical properties of unimodal maps
314(3)
11.2 The Barreto--Hunt--Grebogi--Yorke conjecture
317(25)
11.2.1 Counter-examples to the Barreto--Hunt--Grebogi--Yorke conjecture
324(8)
11.2.2 Robust chaos without the period-n-tupling scenario
332(3)
11.2.3 The B-exponential map
335(7)
11.3 Border-collision bifurcation and robust chaos
342(10)
11.3.1 Normal form for piecewise-smooth one-dimensional maps
342(1)
11.3.2 Border-collision bifurcation scenarios
343(6)
11.3.3 Robust chaos in one-dimensional singular maps
349(3)
11.4 Exercises
352(6)
12 Robust Chaos in 2-D Piecewise-Smooth Maps
358(43)
12.1 Robust chaos in 2-D piecewise-smooth maps
358(20)
12.1.1 Normal form for 2-D piecewise-smooth maps
359(1)
12.1.2 Border-collision bifurcations and robust chaos
360(2)
12.1.3 Regions for nonrobust chaos
362(5)
12.1.4 Regions for robust chaos: undesirable and dangerous bifurcations
367(3)
12.1.5 Proof of unicity of orbits
370(1)
12.1.6 Proof of robust chaos
371(7)
12.2 Robust chaos in noninvertible piecewise-linear maps
378(17)
12.2.1 Normal forms for two-dimensional noninvertible maps
380(2)
12.2.2 Onset of chaos: Proof of robust chaos
382(13)
12.3 Exercises
395(6)
Bibliography 401(50)
Index 451
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