The Nature of the Interior Fixed Points
1. Leaving aside for the moment the question of convergence, it is of interest to inquire into the nature of the various i.f.p. Since for a given N, there are a finite number of different systems, there are only a finite number of i.f.p. These are, of course, defined by the set of algebraic equations obtained on suppressing the primes on the left-hand side of the systems in question. As mentioned above, from Brouwer's theorem it follows that there exists at least one fixed point,
but it need not lie in the interior. Frequently these systems have no solutions such that 0 < xi< 1, all xi, which means that no i.f.p. exists. Consider, for example, system I.1.a. The set of equations defining the fixed point is: 2 2 2 X1 = C2 + X2 +-X2 X2 = 2x1X2+2x1x3x3 = 2X2 X3
Since, by definition of an i.f.p., x34 0, we must have x2= 1/2; the second equation then implies: 1/2 = 2x(1 - xl), i.e., xl = 1/2, so that xl+ x2 = 1, implying x3= 0. Thus no i.f.p. exists.
In general, to find the i.f.p. we must eliminate two of the variables. The resulting equation is then of 4th order in the remaining variable, say xi, although it may have factors corresponding to x 1 = 1, xi = 0,
or perhaps to extraneous roots like xi = -1. (In Table II the equation listed is always in "reduced" form, with these factors removed.) Occasionally the equation may have two real roots in the interval 0 to 1. For N = 3, in all such cases one of the roots proved to be spurious, i.e., it did not satisfy the original system. In fact, excepting the case II.l.d, mentioned above, which had a continuum of i.f.p., no system had more than one i.f.p. Although it is doubtless possible to give a complete theory of these equations for N = 3, a similar treatment for general N seems beyond reach. Here the elimination process can yield an (unreduced) equation of order 2N-1.
2. Bounds for the i.f.p. Consider an i.f.p. satisfying: I >X1>x 2 >X 3>...> > 0 (16)
Clearly, we lose no generality by specifying this ordering, since we can always carry out a permutation on the system so that (16) holds. For a given N, the "largest" i.f.p. will be defined as that i.f.p. for which xl< 1 has the largest numerical value as we range over all possible systems. (Since the number of systems is finite for finite N, there will always exist a largest i.f.p.) The question then arises: Given N, which system has the largest i.f.p., and what is the corresponding value of
xl? In view of the astronomical number of inequivalent systems (for even moderate values of N) it is of some interest that a partial answer can be given to this question.
For N = 3, our complete study reveals that the system possessing the largest i.f.p.-hereafter called the "maximal system"-is II.3.d, for which the defining equations are (we interchange x2 and x3 for convenience): X1 = X3 + 2x1x2+ 2x1x3+ 2x2x3X2 = X (17) X3 = X2 The (unreduced) equation is clearly (x = xi): x+x2 +x4 = 1 (18) which yields: x =.56984029 (19)
The natural generalization of this system to N dimensions is: N\2 N-1 XI=XN + ...= Xi - Xii=l i=l X2 = X12 X3 = X22 (20) 2 XN-XN-1 The root x = xl is then given as a root of the equation: p=N-1 fN(x)-- E x2 =1 (21) N=O This root converges very rapidly as N - oo; for example, N = 4: x = .566160865... N = 5: x = .566123797... (22) N = o: x = .566123792...
It is tempting to consider this last number as an N-independent bound for all binary reaction systems. Unfortunately, this is false, as will be shown below.
Consider a system with xl > 1/2 and satisfying the ordering (16). Let us assume the "skeleton": x1 = x1+ 2x1x2+ ... X2 = 2x lx 3 + ... X 3= 2X 1X 4+ ... (23)XN_1= 2XN+ ...XN
Clearly: 1-Xl = X2+X3+X4+ ...+XN<X2 [1+2+42+-8X3(22 (24)l 2xL 2 (2x,1) 1- xi But X2< 2 (25) i - Xl 1 - xl or >2>p 2 (2x)P (26) If we equate these bounds and set y = 2x1 , we obtain the equation: yN-1 - 2yN-2 + I = 0 (27) Calling the root of this equation yN, it is evident from (26) that: Xz< Y (28)
Clearly, yN - 2 as N - oo, i.e., the bound is N-dependent. One might suspect at first that this bound is a very weak one, and that the actual maximal system has a much lower i.f.p. However, that the bound is the best possible is proven by exhibiting a system for which yN/2 = x1 is actually obtained. In fact, such a system is:
X= X1+ 2x1x2 X2 = 2l1X3X3 = 2x1x4 (29) XXN_1 = 2xlx NN N = 1 2 x2 x j=(x .....X)2 =(1X)2xNx = E i + E2 E^ = Xj=(X2+• *.1) = (-Xl) i=2 i<j=2
It is easily verified that this system has (27) as its i.f.p. equation. For N = 4 x l= = .809016995 ( 4 (30) N = 5 x1 = .919643378
Experimentally (N = 4, 5, 6), this i.f.p. is not attained on iteration starting from a general point, but these converge to the n.f.p. XN = 1. This is to be contrasted with the behavior of the system (20), which actually attained its i.f.p. (N = 3, 4, 5). Indeed, for system (29) it turns out that the absolute value of the Jacobian at the i.f.p. is (y = 2x1): IJI = yN-2(2 - y)(31) > 1 which makes it reasonable that this i.f.p. is not attractive.
On the other hand, it is clear that for a "skeleton" of the form: Xl = ... X2 = X2+ ... X3 X2 + ... (31) xN-XN2 -_1 + -' we have:
2 x X2 >x2 X3> X2> xl (32) 2 >x 2>x2N-1N N-1 - 1 N-1 whence: 1 = xl+x2 ...+x>Zx = fN(Xl) p=O
Therefore, for such a skeleton, the root of fN(x) = 1 does indeed provide an upper bound (attained for the system (20)).
At this writing it has not yet been shown that the system (29) is actually maximal. However, a weak upper bound can be obtained for all systems such that x1 does not contain the term x2 (skeletons of the form (31) are a sub-class of these).
Namely, in this case xl<2xl (1 -xl ) + (1 -x)2 = 1--2 (33) Therefore, clearly x2 + Xl 1 or xI < .61803399 (34)
However, we can do much better for this case. In fact, we can show that for xl > 1/2, we must have, under the ordering (16): Xk > z-21 (35)
which then establishes the bound (21) by the previous argument.