Origins of Chaos, Part II

Researchers:
In Part I we traced the origins of chaos to a Cartan-like set of matrices with (1,1,1,....) in the super and sub diagonals
(-1,0,0,0,...) in the main diagonal and stated that the cycle lengths for N using x^2 - 2 matched Euler's results for base 2 notation, 1/N.
Example: N = 7, cycle length = 3 since 3 is the "least exponent" a such that 2^a == 1 mod 7.; so a = 3 since 8 == 1 mod 7.
Similarly, 1/7 in base 2 = .001001001,... 3 cycle.

Similarly, we take X^3 - 2x^2 - x + 1 roots and using f(x), we find the 3 cycle; where for n-th degree equation, this corresponds to (2n+1) = N, so N = 7. Or, we can say given 2*Cos 2 Pi/7 = 1.24,....this has the 3 cycle using x^2 - 2. Thus

1. x^2 - 2 has a seed of 2*Cos 2Pi/N. and is cyclically chaotic. Using trig rules, we can manipulate x^2 - 2 and derive the other formulas for various trig values.

2. 2x^2 - 1 using seed Cos 2Pi/N.

3. 4x*(1-x); logistic map; using seed Sin^2 2Pi/N.

4. 2x / (1 - x^2) using Tan 2 Pi/N.

5. x^2 / (2 - x^2) using seed Sec 2 Pi/ N.

6. X^2 (x^2 - 2) using Csc. 2 Pi/N. and

7. (x^2 - 1 )/ 2x using seed Cot 2 Pi/N.

All 7 formulas are chaotic with (for example, N=7, the 3 cycle).

We can alternatively use Newton's method x1 = xo - f(xo)/f'x(xo) on x^2 = -1; resulting in #7, Cot.

PART III: Derivation of the M-set.

Bring up B. Mandelbrot's website, subsection: http://www.ncane.com/jm8d "Combinatorics in the M-Set - Lavaurs Algorithm" and notice the chart, where the 7-ths and 15-th are connected by "3" and ("4") respectively. The "3" is precisely the "3" cyclic period mentioned before for 7. Similarly, there's a "4" for the 15-th's since 2^4 == 16 == 1 mod 15.

For 31 we would have a "5" since referring to Table 48 in Beiler's book, for base 2, "5" is the "least exponent" mod 31. So Mandelbrot states "Similarly, the period 5 arcs connecting 5/31 to 6/31 and 7/31 to 8/31 collapse to the roots of 5-cycle midgets in the same limb".

Last, we note that according to Lavaur's algorithm, the M-set is precisely isomorphic with the logistic map; not suprising since we have the logistic map as #3 above.

We have thus traced the origins of Chaos to a particular variant of Pascal's triangle with repeated columns. Or, we can begin with the Cartan-like set of matrices as mentioned before : (1,1,1,...) in the super and subdiagonals and (-1,0,0,0,...) in the main diagonal. Alternatively, we can begin with using Newton's method on x^2 = (-1), then using trig, come up with the other 6 formulas. Chaos, at it's most fundamental level, originates with 2nd-degree equations and the results of our work bring us back to Euler with his observations regarding the cycle lengths of 1/N in base 2 notation. (generally, any base).; but the M-set and logistic maps incorporate the base 2.
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