Abstract
In this and the subsequent chapter we deal with a problem central to synergetics, namely qualitative macroscopic changes of complex systems. Though it is possible to treat the various instabilities under the impact of noise by means of a single approach, for pedagogical reasons we shall deal with the special cases individually. For the same reasons we first start with equations which do not contain fluctuations (noise) and shall treat the corresponding problems only later. The general philosophy of our approach was outlined in Sect. 1.14.
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Bibliography
In this chapter I present an approach initiated in 1962 (H. Haken: Talk at the International Conference on Optical Pumping, Heidelberg 1962), and applied to laser theory including quasi-periodic motion, e.g. bifurcation to tori (see, e.g.
H. Haken, H. Sauermann: Z. Phys. 176, 47 (1963)
H. Haken: Laser Theory, in Encylopedia of Physics, Vol. XXV, 2c, Light and Matter Ic (Springer, Berlin, Heidelberg, New York 1970) and reprint edition Laser Theory (Springer, Berlin, Heidelberg, New York 1983)
This author’s approach is based on the slaving principle and represents, in modern language, “dynamic bifurcation theory” (which allows one to cope with transients and fluctuations). “Static” bifurcation theory was initiated in the classical papers by
H. Poincaré: Les méthodes nouvelles de la mécanique céleste T. 1 (Gauthier-Villars, Paris 1892) H. Poincaré: Acta Math. 7, 1 (1885)
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While this field seems to have been more or less dormant for a while (with the exception of bifurcation theory in fluid dynamics), the past decade has seen a considerable increase of interest as reflected by recent texts. We mention in particular
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These authors deal in an elegant fashion with “static” bifurcation theory.
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Haken, H. (1983). Nonlinear Equations. Qualitative Macroscopic Changes. In: Advanced Synergetics. Springer Series in Synergetics, vol 20. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-45553-7_8
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