By Jürgen Parisi, Stefan C. Müller, Walter Zimmermann

Innovations of nonlinear physics are utilized to a growing number of learn disciplines. With this quantity, the editors supply a range of articles on nonlinear issues in growth, starting from physics and chemistry to biology and a few functions of social technology. The booklet covers quantum optics, electron crystallization, mobile or circulate styles in fluids and in granular media, organic structures, and the regulate of mind constructions through neuronal excitation. Chemical styles are checked out either in bulk recommendations and on surfaces in heterogeneous structures. From ordinary constructions, the authors flip to the extra advanced habit in biology and physics, resembling hydrodynamical turbulence, low-dimensional dynamics in solid-state physics, and gravity.

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**Example text**

2) has infinitely many sinks (stable periodic points), and a number of other exotic properties. 4)), can be understood as the simple consequence that the only stable periodic solution for beAk has a period (2n(k) + 1)2n, and satisfies n(k + 1) = n(k) - 1. On the other hand, for bEBk there are two stable periods, (2n + 1)2n and (2n - 1)2n. Therefore, as b is increased from Ak to Ak+l, and then reduced again, there is a hysteresis effect entering Bk. This is illustrated in Fig. 23. It explains the hysteresis effect, obtained in the numerical calculations of Flaherty and Hoppensteadt (1978), to be discussed in the next section.

Therefore, by the Poincare-Bendixson theorem, every trajectory must approach (or be) a closed curve, proving the theorem. The closed curve may, of course, be a boundary of the annulus. 4) approach a closed curve as 4 -+ co, then that system must be dissipative. 4) are closed curves, the system is always conservative. This is because any dissipative effect can only be compensated by an external periodic force for a discrete (enumerable) set of periodic orbits - otherwise the force and the dynamics will not stay in a'compensating phase'.

Thus this set does not possess the feature that it is (exponentially) sensitive to initial conditions. Put another way, the set X is geometrically wild, but dynamically tame. It has been suggested by F. Albrecht that dynamically wild sets might be called `chaotic sets', to distinguish them from the purely geometric strangeness. Again in the case of the logistic map, when c = 4, most points in the interval (0, 1) belong to such a chaotic set, but they are not a strange attractor, or any type of attractor (see again the discussions in Chapters 4 and 5).