Instabilities, chaos and turbulence : an introduction to nonlinear dynamics and complex systems / Paul Manneville.

Format:

Book

Author/Creator:

Manneville, P. (Paul), 1946-

Language:

English

Subjects (All):

Chaotic behavior in systems.

Dynamics.

Nonlinear theories.

Differentiable dynamical systems.

Physical Description:

xiv, 391 pages : illustrations ; 23 cm

Place of Publication:

London : Imperial College Press ; Hackensack, NJ : Distributed by World Scientific Pub. Co., [2004]

Summary:

This book is an introduction to the application of nonlinear dynamics to problems of stability, chaos and turbulence arising in continuous media and their connection to dynamical systems. With an emphasis on the understanding of basic concepts, it should be of interest to nearly any science-oriented undergraduate and potentially to anyone who wants to learn about recent advances in the field of applied nonlinear dynamics. Technicalities are, however, not completely avoided. They are instead explained as simply as possible using heuristic arguments and specific worked examples.

Contents:

1.1 Dynamical Systems as a Context 2

1.2 Continuous Media as a Subject 5

1.3 From Simple to Complex 9

1.3.1 Thermal convection: the instability mechanism 10

1.3.2 Nonlinear convection and dynamical systems 12

1.3.3 Stability and instability of open flows 15

1.3.4 Beyond the transition: fully developed turbulence 17

2 First Steps in Nonlinear Dynamics 25

2.1 From Oscillators to Dynamical Systems 25

2.1.1 First definitions 25

2.1.2 Formalism of analytical mechanics 32

2.1.3 Gradient systems 33

2.2 Stability and Linear Dynamics 35

2.2.1 Formulation of the linear stability problem 36

2.2.2 Two-dimensional linear systems 37

2.2.3 Stability of a time-independent regime 41

2.3 Two-dimensional Nonlinear Systems 43

2.3.1 Two examples of oscillators 44

2.3.2 Amplitude and phase of nonlinear oscillators 50

2.4 What Next? 60

3 Life and Death of Dissipative Structures 71

3.1 Emergence of Dissipative Structures 71

3.1.1 Qualitative analysis of the instability mechanism 71

3.1.2 Simplified model 73

3.1.3 Normal mode analysis, general perspective 75

3.1.4 Back to the model 78

3.1.5 Vicinity of the threshold: linear stage 82

3.1.6 Classification of unstable modes 84

3.2 Disintegration of Dissipative Structures 86

3.2.1 Simplified model of nonlinear convection 86

3.2.2 Transition to turbulence of convection cells 90

3.2.3 Transition toward chaos in confined systems 94

3.2.4 Dynamics of "textures" in extended systems 97

3.2.5 Turbulent convection 101

4 Nonlinear Dynamics: from Simple to Complex 115

4.1 Reduction of the Number of Degrees of Freedom 116

4.1.1 Role of aspect ratios 116

4.1.2 Low dimensional effective dynamics 118

4.1.3 Center manifolds and normal forms 121

4.2 Transition to Chaos 125

4.2.1 First steps: time-independent and periodic regimes 126

4.2.2 Quasi-periodicity and resonances 130

4.2.3 Quasi-periodicity and lockings 136

4.3 Characterization of Chaotic Regimes 147

4.3.1 Instability of trajectories and Lyapunov exponents 148

4.3.2 Fractal aspects 150

4.4 Empirical Approach of Chaotic Systems 154

4.4.1 Standard analysis by means of Fourier transform 156

4.4.2 Reconstruction by the method of delays 157

4.4.3 Sampling frequency and embedding dimension 160

4.4.4 Application 165

5 Nonlinear Dynamics of Patterns 181

5.1 Quasi-one-dimensional Cellular Structures 182

5.1.1 Steady states 182

5.1.2 Amplitude equation 185

5.2 Dissipative Crystals 187

5.3 Short Term Selection of Patterns 190

5.4 Modulations and Envelope Equations 191

5.4.1 Quasi-one-dimensional cellular patterns 191

5.4.2 2D modulations of quasi-1D cellular patterns 193

5.4.3 Quasi-two-dimensional cellular patterns 195

5.4.4 Oscillatory patterns and dissipative waves 197

5.4.5 Universal long-wavelength instabilities 199

5.5 What Lies Beyond? 204

6 Open Flows: Instability and Transition 211

6.1 Base Flow Profiles 213

6.1.1 Strictly one-dimensional flows 213

6.1.2 More general velocity profiles 215

6.1.3 Extension to arbitrary profiles 218

6.2 Linear Stability 219

6.2.1 General framework 219

6.2.2 Inviscid flows 223

6.2.3 Viscous flows 231

6.2.4 Instability and downstream transport 234

6.3 Transition to Turbulence 241

6.3.1 Nonlinear development of instabilities 241

6.3.2 Inviscidly unstable flows 244

6.3.3 Inviscidly stable flows 249

6.3.4 Turbulent spots and intermittency 253

7 Developed Turbulence 271

7.1 Scales in Developed Turbulence 272

7.1.1 Production scale 272

7.1.2 Inertial scales and Kolmogorov spectrum 274

7.1.3 Dissipation scales 276

7.1.4 Remarks 278

7.2 Mean Flow and Fluctuations 279

7.2.1 Statistical approach 279

7.2.2 Reynolds averaged Navier-Stokes (RANS) equations 280

7.2.3 Energy exchanges in a turbulent flow 283

7.3 Mean Flow and Effective Diffusion 284

7.3.1 Mixing length and eddy viscosity 284

7.3.2 Application to the determination of the mean flow 286

7.4 Beyond the Elementary Approach 290

7.4.1 Turbulence modeling 292

7.4.2 Large eddy simulations 294

8.1 Dynamics, Stability, and Chaos 304

8.2 Continuous Media, Instabilities, and Turbulence 308

8.3 Approach to a Complex System: the Earth's Climate 312

8.4 Exercise: Ice ages as catastrophes 328

Appendix A Linear Algebra 331

A.1 Vector Spaces, Bases, and Linear Operators 331

A.2 Structure of a Linear Operator 334

A.3 Metric Properties of Linear Operators 343

Appendix B Numerical Approach 351

B.1 Treatment of the Time Dependence 352

B.2 Treatment of Space Dependence in PDEs 358

B.2.1 Finite difference methods 358

B.2.2 Spectral methods 363

B.4.1 ODEs 1: Forced pendulum 371

B.4.2 ODEs 2: Lorenz model 373

B.4.3 ODEs 3: Rossler and Chua models 376

B.4.4 PDEs 1: SH model, finite differences 378

B.4.5 PDEs 2: SH model, pseudo-spectral method 380.

Notes:

Includes bibliographical references and index.

ISBN:

1860944833

1860944914

OCLC:

57486539