Instabilities, chaos and turbulence (Book, 2010) [WorldCat.org]
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Instabilities, chaos and turbulence

Author: P Manneville
Publisher: London : Imperial College Press, 2010.
Series: ICP fluid mechanics, v. 1.
Edition/Format:   Print book : English : 2nd edView all editions and formats
Summary:

Offers an introduction to a wide body of knowledge on nonlinear dynamics and chaos. This book emphasises the understanding of basic concepts and the nontrivial character of nonlinear response,  Read more...

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Details

Document Type: Book
All Authors / Contributors: P Manneville
ISBN: 1848163932 9781848163935 9781848163928 1848163924
OCLC Number: 595136733
Description: xv, 439 pages : illustrations ; 24 cm.
Contents: Machine generated contents note: 1. Introduction and Overview --
1.1. Dynamical Systems as a Context --
1.2. Continuous Media as a Subject --
1.3. From Simple to Complex --
1.3.1. Thermal convection: the instability mechanism --
1.3.2. Nonlinear convection and dynamical systems --
1.3.3. Complexity in dynamical systems --
1.3.4. Stability and instability of open flows --
1.3.5. Beyond the transition: fully developed turbulence --
1.4. More on Complexity and its Modelling --
1.4.1. Reacting systems, from chemistry to ecology --
1.4.2. Modelling distributed systems --
1.4.3. Phase transitions and critical phenomena --
1.4.4. Concluding remarks --
1.5. Exercises --
2. First Steps in Nonlinear Dynamics --
2.1. From Oscillators to Dynamical Systems --
2.1.1. First definitions --
2.1.2. Formalism of analytical mechanics --
2.1.3. Gradient systems --
2.2. Stability and Linear Dynamics --
2.2.1. Formulation of the linear stability problem --
2.2.2. Two-dimensional linear systems --
2.2.3. Stability of a time-independent regime --
2.3. Two-dimensional Nonlinear Systems --
2.3.1. Two examples of oscillators --
2.3.2. Amplitude and phase of nonlinear oscillators --
2.4. What Next? --
2.5. Exercises --
3. Life and Death of Dissipative Structures --
3.1. Emergence of Dissipative Structures --
3.1.1. Qualitative analysis of the instability mechanism --
3.1.2. Simplified model --
3.1.3. Normal mode analysis, general perspective --
3.1.4. Back to the model --
3.1.5. Vicinity of the threshold: linear stage --
3.1.6. Classification of unstable modes --
3.2. Disintegration of Dissipative Structures --
3.2.1. Simplified model of nonlinear convection --
3.2.2. Transition to turbulence of convection cells --
3.2.3. Transition toward chaos in confined systems --
3.2.4. Dynamics of textures in extended systems --
3.2.5. Turbulent convection --
3.3. Exercises --
4. Nonlinear Dynamics: from Simple to Complex --
4.1. Reduction of the Number of Degrees of Freedom --
4.1.1. Role of aspect ratios --
4.1.2. Low-dimensional effective dynamics --
4.1.3. Centre manifolds and normal forms --
4.2. Transition to Chaos --
4.2.1. Time-independent and periodic regimes --
4.2.2. Quasi-periodicity and resonance --
4.2.3. Quasi-periodicity and locking --
4.3. Beyond Quasi-Periodicity --
4.3.1. Dynamics on stable and unstable manifolds --
4.3.2. Nature of chaotic regimes --
4.3.3. Two classical scenarios of transition to chaos --
4.4. Exercises --
5. Characterising and Using Chaos --
5.1. Quantifying Chaos --
5.1.1. Instability of trajectories and Lyapunov exponents --
5.1.2. Fractal features --
5.2. Empirical Approach to Chaotic Systems --
5.2.1. Standard analysis using Fourier transforms --
5.2.2. Reconstruction using delays --
5.2.3. Sampling frequency and embedding dimension --
5.2.4. Application --
5.3. Mixing by Stirring --
5.3.1. Mixing: diffusion or/and advection? --
5.3.2. Time-periodic two-dimensional configurations --
5.3.3. Time-dependent two-dimensional open flows --
5.3.4. Time-independent three-dimensional open flows --
5.4. Control of Chaos --
5.4.1. Synchronisation of dynamical systems --
5.4.2. Delay systems and chaos control using delays --
5.4.3. OGY method --
5.5. Conclusions --
5.6. Exercises --
6. Nonlinear Dynamics of Patterns --
6.1. Quasi-one-dimensional Cellular Structures --
6.1.1. Steady states --
6.1.2. Amplitude equations --
6.2. Dissipative Crystals --
6.3. Short Term Selection of Patterns --
6.4. Modulations and Envelope Equations --
6.4.1. Quasi-one-dimensional cellular patterns --
6.4.2. Two-dimensional modulations of quasi-1D patterns --
6.4.3. Quasi-two-dimensional cellular patterns --
6.4.4. Oscillatory patterns and dissipative waves --
6.4.5. Universal long-wavelength instabilities --
6.5. What Lies Beyond? --
6.6. Exercises --
7. Open Flows: Instability and Transition --
7.1. Base Flow Profiles --
7.1.1. Strictly one-dimensional flows --
7.1.2. More general velocity profiles --
7.1.3. Extension to arbitrary profiles --
7.2. Linear Stability --
7.2.1. General framework --
7.2.2. Inviscid flows --
7.2.3. Viscous flows --
7.2.4. Instability and downstream transport --
7.3. Transition to Turbulence --
7.3.1. Nonlinear development of instabilities --
7.3.2. Inviscidly unstable flows --
7.3.3. Inviscidly stable flows --
7.3.4. Turbulent spots and intermittency --
7.4. Exercises --
8. Developed Turbulence --
8.1. Scales in Developed Turbulence --
8.1.1. Production scale --
8.1.2. Inertial scales and the Kolmogorov spectrum --
8.1.3. Dissipation scales --
8.1.4. Remarks --
8.2. Mean Flow and Fluctuations --
8.2.1. Statistical approach --
8.2.2. Reynolds averaged Navier-Stokes equation --
8.2.3. Energy exchanges in a turbulent flow --
8.3. Mean Flow and Effective Diffusion --
8.3.1. Mixing length and eddy viscosity --
8.3.2. Determination of the mean flow --
8.4. Beyond the Elementary Approach --
8.4.1. structure of turbulent flows --
8.4.2. Turbulence modelling --
8.4.3. Large eddy simulations --
8.5. Exercises --
9. Summary and Perspectives --
9.1. Dynamics, Stability and Chaos --
9.2. Continuous Media, Instabilities and Turbulence --
9.3. Approach to a Complex System: the Earth's Climate --
9.4. Exercise: Ice ages as catastrophes --
Appendix A Linear Algebra --
A.1. Vector Spaces, Bases, and Linear Operators --
A.2. Structure of a Linear Operator --
A.2.1. Jordan normal form --
A.2.2. Exponential of a matrix --
A.2.3. Perturbation of a linear problem --
A.3. Metric Properties of Linear Operators --
A.3.1. Scalar products, adjoints, non-normal operators --
A.3.2. Fredholm Alternative --
A.3.3. Boundary value problems and adjoint operators --
A.4. Application to linear control in state space --
A.5. Exercises --
Appendix B Numerical Approach --
B.1. Treatment of the Time Dependence --
B.2. Treatment of Space Dependence in PDEs --
B.2.1. Finite difference methods --
B.2.2. Spectral methods --
B.3. Exercises --
B.4. Case studies --
B.4.1. ODEs 1: Forced pendulum --
B.4.2. ODEs 2: Lorenz model --
B.4.3. ODEs 3: Rossler and Chua models --
B.4.4. PDEs 1: SH model, finite differences --
B.4.5. PDEs 2: SH model, pseudo-spectral method.
Series Title: ICP fluid mechanics, v. 1.
Responsibility: Paul Manneville.

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