IV—
Modification of the Coefficients
1. In this section we present some results on the effect of modifying the original integer-valued coefficients of our cubic transformations in three variables. That is, we consider, as before, transformations of the form: 10 ' =dijMj, (1) j=l 3 d=ijl, (2) i=l
but we no longer require that the dij all be 1 or 0. As already remarked in the introduction, if we impose on the dij only the additional condition: 0 <dij< 1 (3)
then (1), (2), and (3) define a class of cubic transformations depending on 20 parameters, e.g., dlj, d2j (j = 1 to 10).** Since we are unable to formulate a complete theory for the finite subclass of transformations characterized by the restrictions: dij = 0 or 1, all i,j, it is clear a fortiori that we do not have a theory for the infinite class.
In this paper we limit ourselves to showing how an experimental study of some special cases can help to throw light on the properties of our original cubic transformations.
In effect, what we do is study certain transformations which are "close to" some particular transformations of our basic type. A natural
* This technique has actually been used for periods with orders k as large as 148. For very large k the method might fail owing to round-off errors or other numerical inaccuracies.
** For the definition of the cubic monomials Mj, see equation (4) of section II. The domain of this class of transformations is again the region 3 xi=- 1, 0 < Xi 1. i=l
way to define a transformation close to some given Tc,c2 would be to choose its coefficients as follows: dij-1 - eij,j C Ci , (4) dij = eij3, j Ci
where the dij must satisfy (2) and (3), and the eij are small. This class of transformations, defined with respect to some Tc,C2, is still too extensive to study, even if the various eij are restricted to a few discrete values. What we have actually done is to consider 20 such transformations, each of which depends on a single parameter e. We denote these by the symbol: T(r,s, 1< r < 10, 0 <e< , s = 0, 1. (5)
It is understood that these transformations are only defined relative to some TC 1c 2. For convenience we shall generally refer to the transformations T(r,s), defined relative to some Tc,c2 as associated transformations. The coefficients of the T(,,,), are specified as follows: For j ~ r: dij = 1, j c Ci, (6) dij = , j f Ci For j = r: dij - -e, rC 1, dir(1-s)e, r C1 , d 2r = 1-e, r C 2,d2r = se, r 02 C2
In words: T(r,s) is formed from Tc,c2 by the replacement Mr (1 -e)M wherever the term Mr occurs, and by adding eMr to one of the other two lines of the three-line schema. As an example, consider the transformation TA introduced in section III: TCA = {3, 5,8,9, 10}, C2 = {1,2,8}.
Relative to TA, T(5,1)E would read: x' = M3+ (1 - e)M5 + M7+ M9+ M 10, X2' = M1 + M2 + eMs + M8 (9) X3= M4 + M6,
while T(4 ,0) would take the form: x1=M3+ eM4 + Ms + M7 + M9+ Mo , 2 =M1+ M2+ M 8, (10) 3 = (1- C)M4+ M6,
In the S,a coordinates, the T(r,s), can be written: S' = F(S,a) + efrs(S,a), a' = G(S,a) + eg,,(S,a) ; (11)
the original Tc,c 2 is obtained from (11) by setting e = 0. For the two examples given above, we have: frs = f51-(S - a)(1 -S - a)2 e(5, 31)S : 2 (12) 951=-f51; frs = f40 = 12a2 (S - a) (13) T(4,0)e : ' (13) 940 = 0 -
It turns out that for these one-term modifications T(r,s)e we always have gr = ±frs or 0. frs can further be factored into a numerical coefficient crs and a function Mr(S,a); the Mr are of course just the original cubic monomials expressed in terms of S and a. The crs and g,s are determined as follows:
For r C C sgrs 0, Crs -1, rs -frs Crs -; ( for r C Cgrs -frs, Crs rs frs, Crs - ( for r t cl,r t C s rs O, Crs 1, s s frs Crs(
2. We have studied the modified transformations T(r,s)e for a variety of our original cubic Tc 1C 2 that happen to have infinite limit sets.
Our usual procedure has been to vary e in steps of 1/100 in the range 1/100 < e < 1/10 for a given T(,,s) relative to a given Tc,c 2, although on occasion intermediate values of e have been used. Only for the transformation TA [equation (8) above] have we looked at all 20 modified transformations. For a few other TC 1C2 we have limited ourselves to selecting certain of the associated T(r,s)e for detailed study. Since this selection has generally been made on intuitive grounds, we cannot claim that the most "interesting" modifications of the original transformations have always been considered. Nevertheless, this part of our study has proved most revealing, especially as regards the structure of class IV limit sets.
Before describing the results, we insert a few remarks on the difference between the two types of generalizations we have considered, the At-modification of section III and the associated transformations Tr,,s). The At-modification is essentially nothing but the application of a technique frequently employed in the practical solution of nonlinear equations by iterative methods; it is, in fact, one way-perhaps the simplest--of introducing a linear convergence factor. Apart from our use of this device for obtaining the coordinates of the fixed point, our principal interest is in small convergence factors (At close to unity)-too small, in fact, to produce convergence to the fixed point. In view of the fact that the At-modified transformation TC 1C 2(At) has precisely the same fixed point as the original transformation Tc1C 2, one might expect that there exists a close relationship between the corresponding limit sets Lp(At)(T) and Lp(T). In some sense this is true, as the examples given in section III show (see also below, subsection 4). We may express this more formally as follows:
We define a sequence of transformations Tc1c2(Ati)=TAt with corresponding limit sets Lp(Ati)(T) by some convenient rule: Ato = Atlim, Ati = - t (17)
The sequence Tt,, i = 1, 2, . . ., clearly converges to Tc,C2. We then formulate the following conjecture:
Given a Tc,c2and a 6 > 0, then, for all p in the triangle, there exists an N(p) such that, for i > N(p) and for all x C Lp(T), there exists a y C Lp(At )(T) satisfying I y - x < 6.
The modification of Tc1 c2 defined by the associated transformations T(r,s) differs from the At-modification in several respects. In the first place, the perturbation introduced is not linear. Furthermore,
the fixed points of T(r,s)c are in general not the same as those of TC0 C2 (fixed points on the boundary of the triangle may, of course, be common to T(,s),e and Tc,c2 for some pairs r, s).* Finally, each pair r, s must be treated separately; for fixed e, perturbations of different terms of TC1C2 may lead to quite different limit sets. Nevertheless, a conjecture analogous to that formulated for the sequence T xt would most probably turn out to be correct.
3. Limits Sets of the TransformationsT(r,s)e Associated withTA. Since we usually deal with values of e of the form: ei = 1 I, << 10, (18) we introduce a symbol to denote a set of such values: I(i,j)- {en}, i< n < j . (19) In addition, Rk will denote the closed interval of e: (r,s) Rk8 R-±k + k -Rk k(r,s) = [ (r,s) (rs)] ( (rs) < <+(rs) (20)
for which the limit sets L(r,,) of T(r,s), are periods of order k. The photographs illustrating the examples that follow will be found, suitably labelled, in subsection 2 of appendix II.
There is one significant feature common to all the T(,s)E associated with TA; for every pair r, s at least one periodic limit set-that is, a period of order k > 1- -was found in the range I(1, 10). The order of periodicity of most frequent occurrence was k = 7. Thus, for example, for (r,s) = (10,0), we found periodic limit sets with k = 7 over the range 1(3, 10), and the case (r, s) = (6, 1) behaves in the same fashion over the same range. For both series of associated transformations, the limit sets for e = 0.01 are of class IV type and closely resemble L(TA). At e = 0.02, bright spots show up in the pattern (figure A-1); this usually indicates that one is near a period, i.e., that a relatively small change in e will yield a transformation having a finite limit set. In the notation (20), this would be written: -R7 -0.02 < 1. In (r,s)
* The new fixed point S(r,s) S = S + As, a(r,s)e = a + Aa, calculated to first order in (AS, Aa), has both AS and Aa proportional to e. The ration AS/Aa in this order is therefore independent of e, though not of r, s. There are in fact, 6 possible directions of displacement, two for each of the three cases: grs = frs, grs =-frs, rs = 0, cf. equations (14), (15), and (16).
these two examples it happens that a period of order 7 is observed over the range 1(3, 10), that is: I(3, 10) cR L). This is not generally the case. Thus for the case (r, s) = (9, 0), I(2, 9) cR70, whereas L(9 ,o)0.01 and L(9 ,o)0.10 are of class IV type and are morphologically similar to L(TA). It may be recalled (section III) that an analogous behavior was observed for the At-modified transformations TA(at), namely that LAt(TA) was found to be a period of order 7 for a particular range of values of At (0.9930 < At < 0.9772), and different in character (actually, of class III type) outside the range on both sides.
Periodic limit sets of order 7 have been found for some range I(i, j) of e in 9 out of the 20 possible cases. For one of these, (r, s) = (2, 1), 1(4, 7) CR(2), while L(21)o0.io is periodic with k = 28 (figures A-2, A-3). In the transition region, i.e., for +R2, ) < e < -R(28 ) the -C) -cl 1 I ICIV(2,1)(2,1)2 limit sets are infinite. These are shown in figures A-4 and A-5 for the range 1(8,9). They look like pseudo-periods, but, when suitably enlarged, they are seen to be of class I type (figures A-6, A-7). In these pictures one clearly sees with increasing e the onset of instabilityto use an expression from mechanics-and the eventual attainment of a different stable state. The transition region at the lower end of the range also contains infinite limit sets. Figures A-8 and A-9 show (2,1)0.03, first to normal scale, then enlarged. It is manifestly a class III limit set.
For other T(r,s), periods of order k > 7 are found for certain ranges of the parameter e, viz.: k = 9, 16, 23, 30, 37, 46, 62, 148. In two cases, two periods of relatively prime order are found in different sub-ranges of I(1, 10). Thus T(5 ,l)e has two periodic limit sets, one with kl = 23 for e = 0.01 and one with k2 = 16 over 1(9, 10). Similarly, T(1,i1 ) has a periodic limit set with kl 16 for e = 0.01, and one with k2 = 9 for e = 1.10. In these cases the dependence of the limit set on e in the transition region +fRkl< e < Rk2 is more complicated than that t(r,s) (< < r,s) described above. For e-values in this region and sufficiently close to the end-points we observe the expected pseudo-periodic limit sets. For values of e not too close to either boundary the limit set may be either of class IV or of class I type. Figure A-10 shows L(5 ,1 )0.04 to normal scale; in figure A-11, a portion of the limit set is shown enlarged.
We conclude this subsection with two further examples. These illustrate a phenomenon previously mentioned in our general description of limit sets (section II), namely the coexistence of finite periods and class IV sets. Figures A-12 and A-13 show two distinct limit sets belonging to T(3 ,1 )0.01 . One is a period of order k = 23, while the other is a class IV set closely resembling L(TA). The same phenomenon
is perhaps more strikingly illustrated by the case of T(5 ,0)0.02. Here we find both a class IV limit set and a period of order k = 148 (figures A-14 and A-15). We can say virtually nothing in this case about the dependence of the limit set on the initial point. Current computing facilities and techniques are not sufficiently powerful to effect an acceptably accurate determination of the respective regions of convergence without using prohibitive amounts of computing time. We have, however, carried out a few numerical experiments, the results of which certainly confirm our first impression that the geometrical structure of these regions is immensely complicated.
4.Study of theAssociated Transformations for OtherTcIC 2. In this subsection we discuss a few additional examples to illustrate the dependence of infinite limit sets on the parameter e. The relevant photographs and tables will be found in appendix II.
For our first example, we choose the transformation: T Ci-{2,4,6, 7, 9}, TCI2=TBC2 = 5,8,10} (21)
The class IV limit set L(TB) belonging to this transformation is shown in figure B-1. As is evident, it consists of three separate pieces. Each of these is, of course, a limit set for T ). It is instructive to compare the limits sets LAt(TB) with those belonging to certain of the associated T(r,s),. In appendix II we list the results for only one case; (r, s) = (1,0). The limit sets LAt(TB) and L( 1,o), are described in table B. There are (at least) three ranges of At values for which LAt(TB) is periodic; for At close to unity the behavior of LAt(TB) as a function of At is rather wild. As At approaches Atlim = 0.854320 the (class I) limit set shrinks in a continuous manner. The behavior of L(1,0) as e is varied over 1(1, 10) is, if anything, more "pathological"; there are at least six different intervals R(l'°) for which the limit set is periodic, and each period has a different order. Note the similarity in appearance between the two class IV limit sets: LAt(TB) (At = 0.994) and L(1,o)o.oi.
The next two examples may be taken together: C: {2, 7,8,9,101 T C2 {4, 5, 6} (22) T C = {2,5,7,8,9} (23) C2={4,5,10}.
The basic class IV limit sets L(TD) and L(TE) are shown in figures D-l and E-l; their morphological resemblance is apparent. The behavior of the LAt and L(1,0)i for these two cases is set forth il the tables and photographs of appendix II. Detailed comment is perhaps superfluous at this state of our knowledge; we limit ourselves to drawing attention to the following comparisons:
1. Compare L Tt(TD) (At = 0.97) with L(l,o)o.lo(TD). 2. Compare LAt(TE) (At = 0.97) and At = 0.96) with L(1,o)o.o 9(TE) and L(1,o)o.lo(TE).
5. The original transformations TB, TD, TE are closely related from the point of view of formal structure. TD and TE differ by exchange of a single term between the defining sets C1 and C2, while each of these goes over into TB under the simultaneous interchange of two terms between C1 and C2. A comparison of the associated limit sets for TD and TE shows that the initial similarity of L(TD) to L(TE) is roughly preserved under perturbation. This suggests the possibility that some meaningful classification based on algebraic form might be devised.* Of even greater interest is the correspondence, in these examples, between the LAt and the L(1,0), for some ranges of the respective parameters. We are not at present in a position to draw any significant conclusions from the existence of this correspondence; it seems likely, however, that a closer study of these examples would yield criteria enabling one to predict such behavior.
6. There is one property of these transformations which may safely be inferred from the data, namely, that they are close to transformations having periodic limit sets (for some common set of initial points), where close is to be interpreted with reference to some appropriate parameter space---e.g., a range of e values of At values. Their limit sets are "close" to periods, not in the sense that pseudo-periods are, but rather by virtue of the fact that they contain points which lie closeperhaps arbitrarily close--to a set of algebraic solutions of Tk(p) = p. In other words, the Hausdorff distance between the set of period points and the limit set L is small. In this connection, the following piece
* The difference in behavior of Lat(TB) on the one hand and LAt(TD), LAt(TE) on the other is undoubtedly due in part to the fact that in the first case the jacobian matrix has complex eigenvalues at the fixed point, while for TD and TE the eigenvalues are real; this is probably sufficient to explain the qualitative difference of behavior of the corresponding LAt as At - Atlim for At - Atlim sufficiently small.
of evidence may be presented. Consider the transformation T(1.o)o.oi associated with TE, for which we have observed that the sequence T((,)ool(P ), n = 1,2,..., converges to a period of order k = 10 for almost all p. Let us choose a p close to the fixed point. If we then examine the sequence for n = 1, 2,..., N, where N is sufficiently large, we find that the images T() (p) of p have traced out a pattern which closely resembles the original class IV limit set L(TE) of figure E-1. This is shown in figure E-2. The bright spots are the points belonging to the periodic limit set L1 ,o)o.01 . Presumably this means that the effect of introducing a small perturbation into TE, of the form specified by T(.0o).o 01,* is to make the limit set L(TE) contract to 10 points. Alternatively, we could say that, as e - 0, the periodic limit set L(l,o)E(TE) spreads out until it becomes the class IV limit set L(TE).
This and other similar examples suggest that it might be useful to consider the periodic limit sets as fundamental, the hope being that one could develop an appropriate perturbation method, taking these periods as the unperturbed states. The effect of a small change of a parameter (in the direction of instability) is then simply to make the period non-attractive. This can in principle be studied by purely algebraic methods. Determining the structure of the resulting limit set-the perturbed state-is of course a more difficult matter.
In some cases this may amount to nothing more than the development of improved techniques for handling algebraic expressions of very high order. To clarify this statement, we offer one further example. Consider the Atmodified transformations TAt(TA), where TA is a transformation introduced in subsection 2 above. For At = 0.99300, LAt(TA) is a period of order 7. With a very small change in At- namely, for At = 0.99301-the limit set is of class III type, a pseudo-period. Rather than investigating TA(,t) let us turn our attention to the seventh power of the nmodified transformation, TA(at)(p) (At = 0.99301). If we choose our initial point p sufficiently close to one of the (repellent) fixed points of T(,t) ** we find that the first 516 iterated
* If TE is written in the form: S' =F(S, a), a' =G(S, a), then T(1,o), is S' = F(S, a) + 6(S - a)3 , a' = G(S, a). ** The actual values are not known: we have not yet developed good techniques for finding the coordinates of the points of a non-attractive period. The initial point for this example was taken as: S = 0.7034477, a = 0.1159449, chosen on the basis of some simple numerical experimentation. It is close to one of the periodic points belonging to the limit set LA(at) (At = 0.99300), viz.: S = 0.7037400, a = 0.1157123.
images of p, T(Z7t) (P), n = 1, 2,...,516, lie on an almost exact straight line in the S,a triangle. This is shown in figure A-16. The initial point p is at the lower right, and the successive images trace out the line continuously from right to left.* If we continue the iteration, we find that the later images deviate from the straight line, then oscillate in position, and finally settle down to generate another straight line with a different end-point--presumably very close to another fixed point of T(7) It is clear that if one had powerful enough algebraic tools, one could calculate this linear behavior.
7. We close this section with two remarks: 1) A study of the TAt and T(r,s)e associated with those Tc1C2 which have only class I limit sets indicates that the latter are much more stable with respect to these one-parameter modifications than are the limit sets discussed above. 2) Even these unstable limit sets appear to be stable with respect to some one-parameter perturbations of the corresponding transformations. Thus, the transformations T(3 ,0) associated with TE have limit sets visually identical with L(TE) over the whole range I(1, 10). Anomalies such as these make general pronouncements about absolute stability (or instability) impossible.
To illustrate: one rlight be tempted to explain the observed stability in this case as follows: Explicitly, T(3,0)e has the form: S' = F(S, a) +e(1- S-a)3 , a' = G(S,a) . (24)
Now the density of L(TE) is relatively large near the right-hand boundary of the triangle, S + a = 1. The perturbing term, however, vanishes on this line. Thus the transformation is on the average very little altered by the perturbation. But this "explanation" becomes less convincing when one looks at other transformations associated with TE. T(2 ,l1), for example has the form:
One would expect the same argument to apply here, but in fact the limit sets only resemble L(TE) over the two ranges 1(1, 2) and 1(6, 10). In between, we get the familiar periodic and pseudo-periodic behavior.
* The final point plotted has coordinates: S = 0.7030206, a = 0.11628136, so the slope of the line is roughly Aa/AS r -0.713. For this photograph, the scaling factor is approximately 2340.