[Reduced formulas in] continuous systems

The systems I discuss in the main text of this section are mostly discrete. But from experience with traditional mathematics one might have the impression that it would at some basic level be easier to get formulas for continuous systems. I believe, however, that this is not the case, and that the reason for the impression is just that it is usually so much more difficult even to represent the states of continuous systems that one normally tends to work only with ones that have comparatively simple overall behavior—and are therefore more readily described by formulas. (See also pages 167 and 729.)

As an example of what can happen in continuous systems consider iterated mappings x a x (1 - x) from page 920. Each successive step in such a mapping can in principle be represented by an algebraic formula. But the table below gives for example the actual algebraic formulas obtained in the case a = 4 after applying FullSimplify—and shows that these increase quite rapidly in complexity.

In the specific case a = 4, however, it turns out that by allowing more sophisticated mathematical functions one can get a complete formula: the result after any number of steps t can be written in any of the forms

Sin[2^{t} ArcSin[Sqrt[x]]]^{2}

(1 - Cos[2^{t} ArcCos[1 - 2 x]])/2

(1 - ChebyshevT[2^{t}, 1 - 2x])/2

where these follow from functional relations such as

Sin[2x]^{2} 4 Sin[x]^{2} (1 - Sin[x]^{2})

ChebyshevT[m n, x] ChebyshevT[m, ChebyshevT[n, x]]

For a = 2 it also turns out that there is a complete formula:

(1 - (1 - 2 x)^{2}^{t})/2

And the same is true for a = -2:

1/2 - Cos[(1/3) (π - (-2)^{t} (π - 3 ArcCos[1/2 - x]))]

In all these examples t enters essentially only in a^{t}. And if one assumes that this is a general feature then one can formally derive for any a the result

1/2 (1 - g[a^{t} InverseFunction[g] [1 - 2x]])

where g is a function that satisfies the functional equation

g[a x] 1 + (a/2) (g[x]^{2} - 1)

When a = 4, g[x] is Cosh[Sqrt[2 x]]. When a = 2 it is Exp[x] and when a = -2 it is 2 Cos[(1/3) (π - Sqrt[3] x)]. But in general for arbitrary a there is no standard mathematical function that seems to satisfy the functional equation. (It has long been known that only elliptic functions such as JacobiSN satisfy polynomial addition formulas—but there is no immediate analog of this for replication formulas.) Given the functional equation one can find a power series for g[x] for any a. The series has an accumulation of poles on the circle Abs[a]^{2} 1; the coefficient of x^{m} turns out to have denominator

2^(m - DigitCount[m, 2, 1]) Apply[Times, Table[Cyclotomic[s, a]^Floor[(m - 1)/s], {s, m - 1}]]

For other iterated maps general formulas also seem rare. But for example x a x + b and x 1/(a + b x) both give results just involving powers, while x Sqrt[a x + b] sometimes yields trigonometric functions, as on page 915. In addition, from a known replication formula for an elliptic or other function one can often construct an iterated map whose behavior can be expressed in terms of that function. (See also page 919.)