VII—
Periodic Limits

For a large number of 3-dimensional systems, randomly chosen initial vectors iterated to a periodic limit, i.e., the limiting behavior was an oscillation of period 2 or 3 between fixed points. Twentyfour systems exhibited this behavior for three initial vectors, while six

others achieved a similar limiting configuration for at least one choice of initial vector (with no coordinates lying on the boundary). In most of these cases the final state was of the form:

i.e., a boundary double point. A few cases of a boundary triple point were also observed (cf. Table I). Such final states we call "trivial," for the reason that the algebraic structure of the transformation alone indicates that such a final state is at least possible. We may contrast this "trivial" type of oscillatory final state with those for which the oscillation takes place between two or three interior points. The latter we call an "interior double point" (i.d.p.) or "interior triple point" (i.t.p.).

For N = 3 we found just four examples of an i.d.p. and one of an i.t.p. There was also one case of a "non-trivial" b.d.p. (system I.2.e) for which the final state was oscillatory with period 2, but between two "non-trivial" boundary points, viz.:

The existence of an interior double (or triple) point means that the second (or third) power of the transformation possesses these limit values as fixed points. The algebraic difficulty of finding such points is in general prohibitive. For example, in the unique case of the interior triple point (system 1.3.g), if we let:

then one coordinate of the triple point is determined by the set of equations:

It may be verified that the successive values of xl given in Table II indeed satisfy this set of equations.

Although there are no oscillatory limiting configurations with periods greater than 3 for N = 3, one can, of course, find such by going to higher N. Indeed, we discovered, by chance, a particularly interesting case, viz.:

This can be considered as one particular generalization of the 3variable system I.5.j. This generalization-which we applied to several of our original systems (Table III)-consists in setting x4 = x3, replacing x3 in the original system by x4, and putting the new crossterms 2x ₄(il + x ₂+ X ₃) in the top line. When we generalized the triple periodic case, I.3.g, in this manner, the resulting limiting behavior (for 3 randomly chosen initial vectors) was still periodic with period 3, but the configuration was of the "trivial" sort, i.e., the b.t.p. (1,0, 0, 0), (0,0, 1,0), (0, 0,0, 1). However, in the case of I.5.j, whose resulting generalization is given above, the limiting configuration was oscillatory with period 12. (The values of the coordinates are given in Table III.) A further generalization to 5 variables, following the same prescription, yields the system:

In this case, 3 random initial vectors achieved an oscillatory limiting configuration of period 6. (See Table III for the numerical values.) In our opinion, it is not likely that the behavior observed in these two cases could be predicted by means of any simple criteria.