Lectures on chaotic dynamical systems / Valentin Afraimovich and Sze-Bi Hsu.

Format:

Book

Author/Creator:

Afraĭmovich, V. S. (Valentin Senderovich)

Contributor:

Hsu, Sze-Bi, 1948-

Series:

AMS/IP studies in advanced mathematics 1089-3288 ; v. 28.

AMS/IP studies in advanced mathematics, 1089-3288 ; v. 28

Language:

English

Subjects (All):

Differentiable dynamical systems.

Chaotic behavior in systems.

Physical Description:

ix, 353 pages : illustrations ; 26 cm.

Place of Publication:

Providence, R.I. : American Mathematical Society : International Press, [2003]

Contents:

1.1 The World of the Observables 1

1.2 Dynamical Systems 3

1.3 Dynamical Chaos. Some Definitions 6

1.4 Systems with Dissipation 9

1.5 Strange Attractors: First Encounter 26

1.6 Characteristics of Complexity of Attractors 37

1.7 Mane Theorem and Takens Definition 41

2 Zero-Dimensional Dynamics 51

2.1 Symbolic Dynamics 51

2.2 Applications of the Bernoulli Scheme 55

2.3 "Two-sided" Bernoulli Shift 62

2.4 Topological Markov Chains 71

2.5 Topological Pressure, Hausdorff And Box Dimension 81

3 One-Dimensional Dynamics 95

3.1 Lorenz-type Maps 95

3.2 Continuous and Smooth Maps of the Interval 111

3.3 Ergodic Properties and Invariant Measures 134

4 Two-Dimensional Dynamics 153

4.1 Henon-type Maps 153

4.2 The Notion of Hyperbolicity 157

4.3 Sufficient Conditions for Hyperbolicity 161

4.4 Poincare-Birkhoff Problem 169

4.5 Homoclinic Bifurcations 195

4.6 Strange Attractors of Some Maps of the Plane 210

5 Systems with 1.5 Degrees of Freedom 217

5.1 Small Periodic Perturbation of Morse-Smale Systems 217

5.2 Bifurcations Of Codimension One Subjected to Periodic Perturbations 220

5.3 The Melnikov Function 229

5.4 Routes to Chaos. Period-Doubling Cascade 242

5.5 Critical Saddle-Node Bifurcations and Destruction of Tori 247

6 Generated by 3-D Vector Fields 257

6.1 Homoclinic Bifurcations in Systems 257

6.2 Two Homoclinic Orbits 266

6.3 The Geometric Lorenz Attractor 274

6.4 Saddle-Focus Homoclinic Bifurcations 286

7 Lyapunov Exponents 295

.1 Proof of the Annulus Principle 311

.2 Normal Form for the Andronov-Hopf-Naimark-Sacker Bifurcation 317

.3 Dissipative "Separatrix Map" 320

.4 Derivation of the Zaslavsky map [Z] 322

.5 Concluding Remarks on Symbolic Dynamics 325

.6 Hyperbolicity Conditions 329.

Notes:

Includes bibliographical references (pages 339-350) and index.

ISBN:

0821831682

OCLC:

49959409