## Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and DevicesThis 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. |

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### Contents

1 | |

Linear Ordinary Differential Equations | 61 |

Equations | 77 |

Linear Ordinary Differential Equations with Quasiperiodic | 103 |

Stochastic Nonlinear Differential Equations | 143 |

The World of Coupled Nonlinear Oscillators | 154 |

The Case of Persistence | 172 |

Nonlinear Equations The Slaving Principle | 187 |

Spatial Patterns | 267 |

The Inclusion of Noise | 282 |

Discrete Noisy Maps | 303 |

Example of an Unsolvable Problem in Dynamics | 312 |

The Most General Solution to the Problem of Theorem 6 2 1 | 319 |

Proof of Theorem 6 2 1 | 331 |

350 | |

Nonlinear Equations Qualitative Macroscopic Changes | 222 |

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Advanced Synergetics: Instability Hierarchies of Self-Organizing Systems and ... Hermann Haken No preview available - 2012 |

### Common terms and phrases

analytic apply approximation assume attractor behavior bifurcation Chap Chapman-Kolmogorov equation characteristic exponents coefficients consider constant construct control parameter convergence coordinate corresponding defined denote depends derive described detailed balance diagonal differential equations discuss dynamics eigenvalues elements example explicitly exponential function expressed finite fluid Fokker-Planck equation formal Fourier series frequencies fulfilled given Haken hypothesis Inserting instability integral introduce iteration procedure Lemma lim sup limit cycle Lyapunov exponents macroscopic manifold maps mathematical means nonlinear occur operator order parameter equations oscillators patterns periodic function phase transitions present problem properties quasiperiodic function quasiperiodic motion readily obtain renormalized replaced respect rest term result Sect self-organization sequence slaving principle solution matrix solution vectors solve spatial stable stochastic stochastic differential equations Stratonovich synergetics theorem theory time-dependent tion torus trajectory transformation treat vanish variables