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  • Produktbild: Advanced Synergetics
  • Produktbild: Advanced Synergetics
Band 20

Advanced Synergetics Instability Hierarchies of Self-Organizing Systems and Devices

Aus der Reihe Springer Series in Synergetics

51,99 €

inkl. gesetzl. MwSt., Versandkostenfrei

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

03.03.2012

Verlag

Springer Berlin

Seitenzahl

356

Maße (L/B/H)

24,4/17/2,1 cm

Gewicht

648 g

Auflage

Softcover reprint of the original 1st ed. 1983

Sprache

Englisch

ISBN

978-3-642-45555-1

Beschreibung

Produktdetails

Einband

Taschenbuch

Erscheinungsdatum

03.03.2012

Verlag

Springer Berlin

Seitenzahl

356

Maße (L/B/H)

24,4/17/2,1 cm

Gewicht

648 g

Auflage

Softcover reprint of the original 1st ed. 1983

Sprache

Englisch

ISBN

978-3-642-45555-1

Herstelleradresse

Springer-Verlag KG
Sachsenplatz 4-6
1201 Wien
AT

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  • Produktbild: Advanced Synergetics
  • Produktbild: Advanced Synergetics
  • 1. Introduction.- 1.1 What is Synergetics About?.- 1.2 Physics.- 1.2.1 Fluids: Formation of Dynamic Patterns.- 1.2.2 Lasers: Coherent Oscillations.- 1.2.3 Plasmas: A Wealth of Instabilities.- 1.2.4 Solid-State Physics: Multistability, Pulses, Chaos.- 1.3 Engineering.- 1.3.1 Civil, Mechanical, and Aero-Space Engineering: Post-Buckling Patterns, Flutter, etc.- 1.3.2 Electrical Engineering and Electronics: Nonlinear Oscillations.- 1.4 Chemistry: Macroscopic Patterns.- 1.5 Biology.- 1.5.1 Some General Remarks.- 1.5.2 Morphogenesis.- 1.5.3 Population Dynamics.- 1.5.4 Evolution.- 1.5.5 Immune System.- 1.6 Computer Sciences.- 1.6.1 Self-Organization of Computers, in Particular Parallel Computing.- 1.6.2 Pattern Recognition by Machines.- 1.6.3 Reliable Systems from Unreliable Elements.- 1.7 Economy.- 1.8 Ecology.- 1.9 Sociology.- 1.10 What are the Common Features of the Above Examples?.- 1.11 The Kind of Equations We Want to Study.- 1.11.1 Differential Equations.- 1.11.2 First-Order Differential Equations.- 1.11.3 Nonlinearity.- 1.11.4 Control Parameters.- 1.11.5 Stochasticity.- 1.11.6 Many Components and the Mezoscopic Approach.- 1.12 How to Visualize the Solutions.- 1.13 Qualitative Changes: General Approach.- 1.14 Qualitative Changes: Typical Phenomena.- 1.14.1 Bifurcation from One Node (or Focus) into Two Nodes (or Foci).- 1.14.2 Bifurcation from a Focus into a Limit Cycle (Hopf Bifurcation).- 1.14.3 Bifurcations from a Limit Cycle.- 1.14.4 Bifurcations from a Torus to Other Tori.- 1.14.5 Chaotic Attractors.- 1.14.6 Lyapunov Exponents *.- 1.15 The Impact of Fluctuations (Noise). Nonequilibrium Phase Transitions.- 1.16 Evolution of Spatial Patterns.- 1.17 Discrete Maps. The Poincaré Map.- 1.18 Discrete Noisy Maps.- 1.19 Pathways to Self-Organization.- 1.19.1 Self-Organization Through Change of Control Parameters.- 1.19.2 Self-Organization Through Change of Number of Components.- 1.19.3 Self-Organization Through Transients.- 1.20 How We Shall Proceed.- 2. Linear Ordinary Differential Equations.- 2.1 Examples of Linear Differential Equations: The Case of a Single Variable.- 2.1.1 Linear Differential Equation with Constant Coefficient.- 2.1.2 Linear Differential Equation with Periodic Coefficient.- 2.1.3 Linear Differential Equation with Quasiperiodic Coefficient.- 2.1.4 Linear Differential Equation with Real Bounded Coefficient.- 2.2 Groups and Invariance.- 2.3 Driven Systems.- 2.4 General Theorems on Algebraic and Differential Equations.- 2.4.1 The Form of the Equations.- 2.4.2 Jordan’s Normal Form.- 2.4.3 Some General Theorems on Linear Differential Equations.- 2.4.4 Generalized Characteristic Exponents and Lyapunov Exponents.- 2.5 Forward and Backward Equations: Dual Solution Spaces.- 2.6 Linear Differential Equations with Constant Coefficients.- 2.7 Linear Differential Equations with Periodic Coefficients.- 2.8 Group Theoretical Interpretation.- 2.9 Perturbation Approach*.- 3. Linear Ordinary Differential Equations with Quasiperiodic Coefficients*.- 3.1 Formulation of the Problem and of Theorem 3.- 3.2 Auxiliary Theorems (Lemmas).- 3.3 Proof of Assertion (a) of Theorem 3.1.1: Construction of a Triangular Matrix: Example of a 2 × 2 Matrix.- 3.4 Proof that the Elements of the Triangular Matrix C are Quasiperiodic in ? (and Periodic in ?j and Ck with Respect to ?: Exampleof a 2 × 2 Matrix.- 3.5 Construction of the Triangular Matrix C and Proof that Its Elements are Quasiperiodic in ? (and Periodic in ?j and Ck with Respect to ?): The Case of an m × m Matrix, all A’s Different.- 3.6 Approximation Methods. Smoothing.- 3.6.1 A Variational Method.- 3.6.2 Smoothing.- 3.7 The Triangular Matrix C and Its Reduction.- 3.8 The General Case: Some of the Generalized Characteristic Exponents Coincide.- 3.9 Explicit Solution of (3.1.1) by an Iteration Procedure.- 4. Stochastic Nonlinear Differential Equations.- 4.1 An Example.- 4.2 The Îto Differential Equation and the Îto-Fokker-Planck Equation.- 4.3 The Stratonovich Calculus.- 4.4 Langevin Equations and Fokker-Planck Equation.- 5. The World of Coupled Nonlinear Oscillators.- 5.1 Linear Oscillators Coupled Together.- 5.1.1 Linear Oscillators with Linear Coupling.- 5.1.2 Linear Oscillators with Nonlinear Coupling. An Example. Frequency Shifts.- 5.2 Perturbations of Quasiperiodic Motion for Time-Independent Amplitudes (Quasiperiodic Motion Shall Persist).- 5.3 Some Considerations on the Convergence of the Procedure*.- 6. Nonlinear Coupling of Oscillators: The Case of Persistence of Quasiperiodic Motion.- 6.1 The Problem.- 6.2 Moser’s Theorem (Theorem 6.2.1).- 6.3 The Iteration Procedure*.- 7. Nonlinear Equations. The Slaving Principle.- 7.1 An Example.- 7.1.1 The Adiabatic Approximation.- 7.1.2 Exact Elimination Procedure.- 7.2 The General Form of the Slaving Principle. Basic Equations.- 7.3 Formal Relations.- 7.4 The Iteration Procedure.- 7.5 An Estimate of the Rest Term. The Question of Differentiability.- 7.6 Slaving Principle for Discrete Noisy Maps*.- 7.7 Formal Relations*.- 7.8 The Iteration Procedure for the Discrete Case*.- 7.9 Slaving Principle for Stochastic Differential Equations*.- 8. Nonlinear Equations. Qualitative Macroscopic Changes.- 8.1 Bifurcations from a Node or Focus. Basic Transformations.- 8.2 A Simple Real Eigenvalue Becomes Positive.- 8.3 Multiple Real Eigenvalues Become Positive.- 8.4 A Simple Complex Eigenvalue Crosses the Imaginary Axis. Hopf Bifurcation.- 8.5 Hopf Bifurcation, Continued.- 8.6 Frequency Locking Between Two Oscillators.- 8.7 Bifurcation from a Limit Cycle.- 8.8 Bifurcation from a Limit Cycle: Special Cases.- 8.8.1 Bifurcation into Two Limit Cycles.- 8.8.2 Period Doubling.- 8.8.3 Subharmonics.- 8.8.4 Bifurcation to a Torus.- 8.9 Bifurcation from a Torus (Quasiperiodic Motion).- 8.10 Bifurcation from a Torus: Special Cases.- 8.10.1 A Simple Real Eigenvalue Becomes Positive.- 8.10.2 A Complex Nondegenerate Eigenvalue Crosses the Imaginary Axis.- 8.11 Instability Hierarchies, Scenarios, and Routes to Turbulence.- 8.11.1 The Landau-Hopf Picture.- 8.11.2 The Ruelle and Takens Picture.- 8.11.3 Bifurcations of Tori. Quasiperiodic Motions.- 8.11.4 The Period-Doubling Route to Chaos. Feigenbaum Sequence.- 8.11.5 The Route via Intermittency.- 9. Spatial Patterns.- 9.1 The Basic Differential Equations.- 9.2 The General Method of Solution.- 9.3 Bifurcation Analysis for Finite Geometries.- 9.4 Generalized Ginzburg-Landau Equations.- 9.5 A Simplification of Generalized Ginzburg-Landau Equations. Pattern Formation in Bénard Convection.- 10. The Inclusion of Noise.- 10.1 The General Approach.- 10.2 A Simple Example.- 10.3 Computer Solution of a Fokker-Planck Equation for a Complex Order Parameter.- 10.4 Some Useful General Theorems on the Solutions of Fokker-Planck Equations.- 10.4.1 Time-Dependent and Time-Independent Solutions of the Fokker-Planck Equation, if the Drift Coefficients are Linear in the Coordinates and the Diffusion Coefficients Constant.- 10.4.2 Exact Stationary Solution of the Fokker-Planck Equation for Systems in Detailed Balance.- 10.4.3 AnExample.- 10.4.4 Useful Special Cases.- 10.5 Nonlinear Stochastic Systems Close to Critical Points: A Summary.- 11. Discrete Noisy Maps.- 11.1 Chapman-Kolmogorov Equation.- 11.2 The Effect of Boundaries. One-Dimensional Example.- 11.3 Joint Probability and Transition Probability. Forward and Backward Equation 305.- 11.4 Connection with Fredholm Integral Equation 306.- 11.5 Path Integral Solution 307.- 11.6 The Mean First Passage Time 308.- 11.7 Linear Dynamics and Gaussian Noise. Exact Time-Dependent Solution of the Chapman-Kolmogorov Equation 310.- 12. Example of an Unsolvable Problem in Dynamics.- 13. Some Comments on the Relation Between Synergetics and Other Sciences.- Appendix A: Moser’s Proof of His Theorem.- A.1 Convergence of the Fourier Series.- A.2 The Most General Solution to the Problem of Theorem.- A.3 Convergent Construction.- A.4 Proof of Theorem.- References.