in tabulating Z, there is a precise two-to-one relation between M_n and M_n+1. The initial maximum M₁=483 is shown as if it had followed a maximum M_o=385, since maxima near 385 are followed by close approaches to the origin, and then by exceptionally large maxima.
It follows that an investigator, unaware of the nature of the governing equations, could formulate an empirical prediction scheme from the "data" pictured in Figs. 2 and 4. From the value of the most recent maximum of Z, values at future maxima may be obtained by repeated applications of Fig. 4. Values of X, Y, and Z between maxima of Z may be found from Fig. 2, by interpolating between neighboring curves. Of course, the accuracy of predictions made by this method is limited by the exactness of Figs. 2 and 4, and, as we shall see, by the accuracy with which the initial values of X, Y, and Z are observed.
Some of the implications of Fig. 4 are revealed by considering an idealized two-to-one correspondence between successive members of sequences M_o, M₁, …, consisting of numbers between zero and one. These sequences satisfy the relations

The correspondence defined by (35) is shown in Fig. 5, which is an idealization of Fig. 4. It follows from repeated applications of (35) that in any particular sequence, 

where m_n is an even integer.
Consider first a sequence where M_o=u/2^p, where u is odd. In this case M_p-l = 1/2, and the sequence terminates. These sequences form a denumerable set, and correspond to the trajectories which score direct hits upon the state of no convection.
Next consider a sequence where M_o=u/2^pυ, where u and υ are relatively prime odd numbers. Then if k>0, M_p+1+k=u_k/υ, where u_k and υ are relatively prime and u_k is even. Since for any υ the number of proper fractions u_k/υ is finite, repetitions must occur, and the sequence is periodic. These sequences also form a denumerable set, and correspond to periodic trajectories.
The periodic sequences having a given number of distinct values, or phases, are readily tabulated. In particular there are a single one-phase, a single two-phase, and two three-phase sequences, namely,

2/3, …
2/5, 4/5, …
2/7, 4/7, 6/7, …
2/9, 4/9, 8/9, …

The two three-phase sequences differ qualitatively in that the former possesses two numbers, and the latter only one number, exceeding 1/2. Thus the trajectory corresponding to the former makes two circuits about C, followed by one about C' (or vice versa). The trajectory corresponding to the latter makes three circuits about C, followed by three about C', so that actually only Z varies in three phases, while X and Y vary in six.
Now consider a sequence where M_o is not a rational fraction. In this case (36) shows that M_n+k cannot equal

FIG. 4. Corresponding values of re1ative maximum of Z (abscissa) and subsequent relative maximum of Z (ordinate) occurring during the first 6000 iterations.

FIG. 5. The function M_n+1=2M_n if M_n<½. M_n+1=2-2M_n if M_n>½, serving as an idea1ization of the locus of points in Fig. 4.