Topology long period

If we define systems as being complex or chaotic by their having only small numbers of cycles with long periods, then, for most of the small size systems studied here, it is upon the intermediate topologies that the most complex dynamics takes place when compared with the behaviors on surrounding graphs, dynamical complexity appears to be inhibited by r lattices. 

Figure 8.17. The topological information on the structure of a multiphase system that is related to one-dimensional projections / 1 (sj) in different directions. The demonstration shows two directions indicated by arrows and the related chords. From /jj ( ) the distributions of the chord segments between domain edges are retrieved. Long periods are indicated by broken lines...

After we have discussed the composition parameters of the SAXS of a multiphase material, we now start with the investigation of the topology. The most simple access to the arrangement of domains in the material is the discussion of long period... 

Figure 8.21. Features of a ID correlation function, yi (x/L) for perfect and disordered topologies. L is the number-average distance of the domains from each other (i.e., long period). Dotted Perfect lattice. Dashed and solid lines Paracrystalline stacks with increasing disorder. a = — v/j / (1 — v/j) with 0 < v/j < 0.5 is a measure of the linear volume crystallinity in the material, which is either or 1 —...

Intramolecular dynamics and chemical reactions have been studied for a long time in terms of classical models. However, many of the early studies were restricted by the complexities resulting from classical chaos, Tlie application of the new dynamical systems theory to classical models of reactions has very recently revealed the existence of general bifurcation scenarios at the origin of chaos. Moreover, it can be shown that the infinite number of classical periodic orbits characteristic of chaos are topological combinations of a finite number of fundamental periodic orbits as determined by a symbolic dynamics. These properties appear to be very general and characteristic of typical classical reaction dynamics. 

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