Farey tree

The convergence follows the Fibonacci fractions which appear in the Farey tree structure that develops between the limits and . 

The equivalence between Sk, the infinite Farey tree structure and the nuclide mapping is shown graphically in Figure 8.4. The stability of a nuclide depends on its neutron imbalance which is defined, either by the ratio Z/N or the relative neutron excess, (N — Z) jZ. When these factors are in balance, Z2 + NZ — N2 = 0, with the solution Z = N(—1 /5)/2 = tN. The minimum (Z/N) = r and hence all stable nuclides are mapped by fractions larger than the golden mean. 

Conveniently, MMOs are characterized by a symbolic notation where L denotes the number of large and S the number of small oscillations during one period. Thus, the MMOs depicted in Fig. 27 are designated as F , P, and 1 P states. In the notation of the latter state, it is indicated that one period is built up from concatenated principal states. In fact, in the simulations, many such concatenated states were found for example, between the P and the P state, P(P) states with n going from 1 to 10 were observed. These sequences are called Farey sequences because a one-to-one correspondence of successive MMO states and the ordering of the rational numbers, which is conveniently represented in a Farey tree (see Fig. 31), can be established. In general, at low values of the resistance, the sequences of MMOs obey an incomplete Farey arithmetic. 

A Farey tree arises in number theory as a scheme for the generation of all the rational numbers between a given pair of rationals. This proceeds by the so-called Farey addition of two rationals piq and ris which is equal io p + q)l (r + s). 

Another most remarkable experimental study in which the two types of mixed-mode sequences were also observed was carried out by Albahadily et al, who studied the electrodissolution of copper in phosphoric acid from a rotating disk. Figure 30 shows a series of Farey states observed in this system, and in Fig. 31, the experimentally observed mixed-mode states are listed in the structure of a Farey tree. On the high rotation-rate end of the 1° state, alternating periodic and chaotic behavior appeared. The first period-doubled oscillation arising from the P parent state is reproduced in Fig. 32 together with the P parent state. In Fig. 33, a two-parameter bifurcation diagram is depicted in which the succession... 

Figure 31. A portion of the Farey tree constructed from observed states.

Obviously, one cannot expect to observe an infinite number of generations on the Farey tree, but Maselko and Swiimey did find that when they were able to adjust their residence time with sufficient precision, they saw the intermediate states predicted by the Farey arithmetic, though after a few cycles the system would drift off to another, higher level state on the tree, presumably because their pump could not maintain the precise flow rate corresponding to the intermediate state. An even more complex and remarkable Farey arithmetic can be formulated for states consisting of sequences of three basic patterns (Maselko and Swinney, 1987). The fact that the mixed-mode oscillations in the BZ system form a Farey sequence places significant constraints on any molecular mechanism or dynamical model formulated to explain this behavior. 

Table 8.1 Farey Tree Arithmetic for the Sequence of States shown in Figure 8.11. States are Listed in Order of Decreasing Residence Time...

Figure 8.11 Several states in the Farey tree for a sequence of states observed as the residence time decreases in the Mn-catalyzed BZ reaction. Each state consists of a concatenation of the basic 1 and patterns. Firing numbers are given in brackets. The 4/3 and 17/23 states in Table 8.1 are not shown here. (Reprinted with permission from Maselko, J. Swinney, H.L. 1986. Complex Periodic Oscillations and Farey Arithmetic in the Belousov-Zhabotinskii Reaction, J. Chem. Phys., 85, 6430-6441. 1986 American Institute of Physics.)...

The most convincing derivation of periodic structure, using the concepts of number theory, comes from a comparison with Farey sequences. The Farey scheme is a device to arrange rational fractions in enumerable order. Starting from the end members of the interval [0,1] an infinite tree structure is generated by separate addition of numerators and denominators to produce the Farey sequences ni of order n, where n limits the values of denominators... 

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