Haken H. Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and Devices

Springer, 1983. — 371 p. — (Springer Series in Synergetics). — ISBN: 978-3-642-45555-1.This text on the interdisciplinary field of synergetics will be of interest to students and scientists in physics, chemistry, mathematics, biology, electrical, civil and mechanical engineering, and other fields. It continues the outline of basic con­ cepts and methods presented in my book Synergetics. An Introduction, which has by now appeared in English, Russian, J apanese, Chinese, and German. I have written the present book in such a way that most of it can be read in­ dependently of my previous book, though occasionally some knowledge of that book might be useful. But why do these books address such a wide audience? Why are instabilities such a common feature, and what do devices and self-organizing systems have in common? Self-organizing systems acquire their structures or functions without specific interference from outside. The differentiation of cells in biology, and the process of evolution are both examples of self-organization. Devices such as the electronic oscillators used in radio transmitters, on the other hand, are man­ made. But we often forget that in many cases devices function by means of pro­ cesses which are also based on self-organization. In an electronic oscillator the motion of electrons becomes coherent without any coherent driving force from the outside; the device is constructed in such a way as to permit specific collective motions of the electrons. Quite evidently the dividing line between self-organiz­ ing systems and man-made devices is not at all rigid.What is Synergetics About?
Physics
Engineering
Chemistry: Macroscopic Patterns
Biology
Computer Sciences
Economy
Ecology
Sociology
What are the Common Features of the Above Examples?
The Kind of Equations We Want to Study
How to Visualize the Solutions
Qualitative Changes: General Approach
Qualitative Changes: Typical Phenomena
The Impact of Fluctuations (Noise). Nonequilibrium Phase Transitions
Evolution of Spatial Patterns
Discrete Maps. The Poincare Map
Discrete Noisy Maps
Pathways to Self-Organization
How We Shall Proceed
Linear Ordinary Differential Equations
Examples of Linear Differential Equations: The Case of a Single Variable
Groups and Invariance
Driven Systems
General Theorems on Aigebraic and Differential Equations
Forward and Backward Equations: Dual Solution Spaces
Linear Differential Equations with Constant Coefficients
Linear Differential Equations with Periodic Coefficients
Group Theoretical Interpretation
Perturbation Approach
Linear Ordinary Differential Equations with Quasiperiodic Coefficients
Formulation ofthe Problem and of Theorem 3.1.1
Auxiliary Theorems (Lemmas)
Proof of Assertion (a) of Theorem 3.1.1: Construction of a Triangular Matrix: Example of a 2 x 2 Matrix
Proof that the Elements of the Triangular Matrix C are Quasiperiodie in τ (and Periodic in φ_j and C^k with Respect to φ): Example of a 2 x 2 Matrix
Construction of the Triangular Matrix C and Proof that Its Elements are Quasiperiodic in τ (and Periodic in φ_j and C^k with Respect to φ): The Case of an m × m Matrix, all λ's Different
Approximation Methods. Smoothing
The Triangular Matrix C and Its Reduction
The General Case: Some of the Generalized Characteristic Exponents Coincide
Explicit Solution of (3.1.1) by an Iteration Procedure
Stochastic Nonlinear Differential Equations
An Example
The Ito Differential Equation and the Ito-Fokker-Planck Equation
The Stratonovich Calculus
Langevin Equations and Fokker-Planck Equation
The World of Coupled Nonlinear Oscillators
Linear Oscillators Coupled Together
Perturbations of Quasiperiodic Motion for Time-Independent Amplitudes (Quasiperiodic Motion Shall Persist)
Some Considerations on the Convergence of the Procedure
Nonlinear Coupling of Oscillators: The Case of Persistence of Quasiperiodic Motion
The Problem
Moser'sTheorem (Theorem 6.2.1)
The Iteration Procedure
Nonlinear Equations. The Slaving Principle
An Example
The General Form of the Slaving Principle. Basic Equations Formal Relations
The Iteration Procedure
An Estimate of the Rest Term. The Question of Differentiability
Slaving Principle for Discrete Noisy Maps
Formal Relations
The Iteration Procedure for the Discrete Case
Slaving Principle for Stochastic Differential Equations
Nonlinear Equations. Qualitative Macroscopic Changes
Bifurcations from aNode or Focus. Basic Transformations
A Simple Real Eigenvalue Becomes Positive
Multiple Real Eigenvalues Become Positive
A Simple Complex Eigenvalue Crosses the Imaginary Axis. Hopf Bifurcation
Hopf Bifurcation, Continued
Frequency Locking Between Two Oscillators
Bifurcation from a Limit Cyde
Bifurcation from a Limit Cyde: Special Cases
Bifurcation from a Torus (Quasiperiodic Motion)
Bifurcation from a Torus: Special Cases
Instability Hierarchies, Scenarios, and Routes to Turbulence
Spatial Patterns
The Basic Differential Equations
The General Method of Solution
Bifurcation Analysis for Finite Geometries
Generalized Ginzburg-Landau Equations
A Simplification of Generalized Ginzburg-Landau Equations. Pattern Formation in Benard Convection
The Inclusion of Noise
The General Approach
A Simple Example
Computer Solution of a Fokker-Planck Equation for a Complex
Order Parameter
Some Useful General Theorems on the Solutions of Fokker-Planck Equations
Nonlinear Stochastic Systems Close to Critical Points: A Summary
Discrete Noisy Maps
Chapman-KolmogorovEquation
The Effect of Boundaries. One-Dimensional Example
Joint Probability and Transition Probability. Forward and Backward Equation
Connection with Fredholm Integral Equation
Path Integral Solution
The Mean First Passage Time
Linear Dynamics and Gaussian Noise. Exact Time-Dependent Solution of the Chapman-Kolmogorov Equation
Example of an Unsolvable Problem in Dynamies
Some Comments on tbe Relation Between Synergetics and Otber Sciences
Appendix A: Moser' s Proof of His Theorem
Convergence ofthe Fourier Series
The Most General Solution to the Problem of Theorem 6.2.1
Convergent Construction
Proof of Theorem 6.2.1