Dissipative structures and weak turbulence. (2014)

• Record Type: Book
• Further Information: Note: Paul Manneville.
• Other Names: Manneville, P (Paul), 1946-
• Contents: Foreword; ; Chapter 1. Outlook; ; 1. Dissipative Structures; ; 2. Transition to Temporal Chaos; ; 3. Spatio-Temporal Chaos and Weak Turbulence; ; 4. Bibliographical Notes; ; Chapter 2. Evolution and Stability, Basic Concepts; ; 1. General Framework; ; 1.1. Discrete Versus Continuous Systems; ; 1.2. Autonomous Versus Time-Dependent Systems; ; 1.3. Deterministic Evolution; ; 1.4. Perturbations and Stability; ; 2. Global Stability; ; 2.1. General Viewpoint and Definitions; ; 2.2. Energy Method; ; 2.3. Different Concepts of Stability; ; 3· Normal Modes, Linear and Nonlinear Dynamics; ; 3.1. Normal Mode Analysis; ; 3.2. Weakly Nonlinear Dynamics; ; 4. Qualitative Dynamics; ; 4.1. Elements for a Phase Portrait; ; 4.2. Absorbing Zones and Limit Sets; ; 4.3. Limit Sets and Attractors; ; 4.4. Hyperbolicity, Structural Stability, and Bifurcations; ; 4.5. About More General Attractors; ; 5. Bibliographical Notes; ; Chapter 3. Instability Mechanisms; ; 1. Rayleigh-Bénard Convection; ; 1.1. Qualitative Analysis; ; 1.2. Simplified Model; ; 1.3. Normal Mode Analysis; ; 1.4. Linear Dynamics of Unstable Modes; ; 2. Convection in Binary Mixtures; ; 2.1. Stationary Mode; ; 2.2. Oscillatory Mode; ; 3. Thermal Convection in Nematic Liquid Crystals; ; 4. Electrohydrodynamic Instabilities in Nematics; ; 4.1. Carr-Helfrich Mechanism; ; 4.2. Two Instability Regimes; ; 5. Taylor-Couette instability; ; 5.1. Centrifugal Instabilities; ; 5.2. Rayleigh Instability Mechanism; ; 5.3. One-Dimensional Model; ; 6. Bénard-Marangoni Convection; ; 6.1. Mechanism and Simplified Model; ; 6.2. Marangoni Versus Buoyancy-Driven Convection; ; 7. Bibliographical Notes; ; Chapter 4. Thermal Convection; ; 1. Boussinesq Equations and Boundary Conditions; ; 1.1. Evolution Equations; ; 1.2. Boundary Conditions; ; 2. Normal Mode Analysis; ; 2.1. Stress-Free Solution, Rayleigh (1916); ; 2.2. No-Slip Solution, Pellew and Southwell (1940); ; 2.3. Vicinity of the Threshold (Linear Stage); ; 2.4. Approximate Determination of Threshold Curves; ; 2.5. Treatment of the Vertical Vorticity; ; 3. Phenomenology of Nonlinear Convection; ; 3.1. Introduction; ; 3.2. Universal Secondary Modes; ; 3.3. Specific Secondary Instabilities; ; 4. Phenomenology of the Transition to Turbulence; ; 4.1. "Old" Results; ; 4.2. "Recent" Results, Concrete Examples; ; 4.3. Concluding Remarks; ; 5. Bibliographical Notes; ; Chapter 5· Low-Dimensional Dynamical Systems; ; 1. Dimension Reduction: A Case Study; ; 1.1. The Model and Its Normal Modes; ; 1.2. Elimination of Slaved Modes: Heuristic Approach; ; 2. Center Manifold and Normal Forms; ; 2.1. Perturbative Approach to the Center Manifold; ; 2.2. Normal Forms of Dynamical Equations; ; 2.3. Normal Forms and Symmetries; ; 2.4. Slightly Off the Critical Point; ; 3. Dynamics and Bifurcations in One Dimension; ; 3.1. General Dynamics; ; 3.2. Normal/Inverse Bifurcations; ; 3.3. Conditional Stability, Hysteresis, and Turning Points; ; 3.4. Transcritical Bifurcations; ; 3.5. Imperfect Bifurcations; ; 3.6. Mathematical Context: Unfolding of Singularities; ; 4. Introduction to Higher Dimensional Problems; ; 5. Dynamics and Bifurcations in Two Dimensions; ; 5.1. Linear Dynamics; ; 5.2. Nonlinear Dynamics, an Example: The Pendulum; ; 5.3. General Dynamics in Two Dimensions; ; 5.4. Bifurcations in Two Dimensions; ; 6. Conclusion; ; 7. Bibliographical Notes; ; Chapter 6. Beyond Periodic Behavior; ; 1. Poincaré Maps; ; 1.1. Surface of Section and First Return Map; ; 1.2. Application to the Lorenz Model; ; 2. Stability of a Limit Cycle; ; 3. Bifurcations of a Limit Cycle; ; 3.1. Normal Form in the Complex Case; ; 3.2. Hopf Bifurcation for Maps; ; 3.3. Bifurcations at Strong Resonances 1/1 and 1/2; ; 4. Nature of Turbulence and Transition Scenarios; ; 5. Sub-Harmonic Route to Turbulence; ; 5.1. The Modeling Issue; ; 5.2. The Sub-Harmonic Cascade; ; 5.3. Universality and Renormalization; ; 5.4. Beyond the Accumulation Point; ; 5.5. Concluding Remark; ; 6. Temporal Intermittency; ; 6.1. Modeling of Intermittent Behavior; ; 6.2. Type I Intermittency; ; 6.3. Type III Intermittency; ; 6.4. Type II Intermittency; ; 7. Quasi-Periodicity; ; 7.1. Introduction to the Locking Phenomenon; ; 7.2. The Winding Number and the Structure of Lockings; ; 7.3. The Breakdown of a Two-Torus; ; 7.4. Ruelle-Takens Scenario and n-Periodicity; ; 8. Beyond "Classical" Scenarios; ; 9. Bibliographical Notes; ; Chapter 7. Characterization of Temporal Chaos; ; 1. Divergence of Trajectories and Lyapunov Exponents; ; 1.1. One-Dimensional Iterations; ; 1.2. Generalization to d-Dimensional Maps; ; 1.3. Generalization to Differential Systems; ; 1.4. Lyapunov Signature of Temporal Behavior; ; 2. Probabilistic Approach; ; 2.1. Entropy; ; 2.2. Invariant Measures; ; 2.3. Invariant Measures for One-Dimensional Maps; ; 2.4. Natural Measures; ; 3. Chaos and Dimensions; ; 3.1. Introduction; ; 3.2. Fractal Geometry; ; 3.3. Probabilistic Viewpoint; ; 3.4. Dimensions and Lyapunov Exponents; ; 4. Experimental Approach; ; 4.1. The Method of Time Delays; ; 4.2. Embedding; ; 4.3. Practical Problems; ; 4.4. Dimension Estimates; ; 4.5. Determination of Lyapunov Exponents; ; 4.6. Final Remark; ; 5. Bibliographical Notes; ; Chapter 8. Basics of Pattern Formation in Weakly Confined Systems; ; 1. Instabilities, Confinement, and Aspect Ratios; ; 2. Pattern Formation; ; 2.1. Modeling of Weakly Confined Systems; ; 2.2. Pattern Formation in Two Dimensions; ; 3. Uniform Nonlinear Solutions; ; 3.1. General Setting; ; 3.2. Steady Solutions of Two-Dimensional Models; ; 3.3. Rayleigh-Bénard Convection; ; 4. Modulated Structures; ; 4.1. Systematic Expansion; ; 4.2. Roll Modulations in a Rotationally Invariant System; ; 4.3. Extension to Two-Dimensional Patterns; ; 4.4. Short-Term Stability and Early Nonlinear Selection; ; 4.5. Phenomenological Extension: The Example of Waves; ; 5. Bibliographical Notes; ; Chapter 9. Applications of the Envelope Formalism; ; 1. Phase Winding Solutions; ; 2. Long Wavelength Instab
• Publisher Details: Place of publication not identified : Academic Press
• Publication Date: 2014
• Extent: 1 online resource (448 pages)
• Subjects: 531/.11; Dynamics; Nonlinear theories; Chaotic behavior in systems; Turbulence; Dynamique; Théories non linéaires; Chaos; Turbulence; Structures dissipatives; Chaotic behavior in systems; Dynamics; Nonlinear theories; Turbulence; Structures dissipatives; Turbulence en air limpide; Dissipative Struktur; Turbulente Strömung; Chaos; Instabilität; Physik; Nichtlineare Dynamik; Chaos; Dynamique; Structures dissipatives; Théories non linéaires; Turbulence; Chaos; Dissipative Struktur; Instabilität; Nichtlineare Dynamik; Physik; Turbulente Strömung
• Languages: English
• ISBNs: 9780080924458
• Related ISBNs: 008092445X
• Access Rights: Legal Deposit; Only available on premises controlled by the deposit library and to one user at any one time; The Legal Deposit Libraries (Non-Print Works) Regulations (UK).
• Access Usage: Restricted: Printing from this resource is governed by The Legal Deposit Libraries (Non-Print Works) Regulations (UK) and UK copyright law currently in force.