{Chaotic Determinism (+ Extracts)}[3rd August 1988]

[Redbook5:248-262][19880803:1017b]{Chaotic Determinism (+ Extracts)}[3rd August 1988]

19880803.1017

[continued]

I’ve just this morning received James Gleick’s ‘Chaos: Making New Science’,** which could well complete the Circle’s structure – or destroy it.

.1253

Or just help to describe its workings. Already*** I have come across concepts from, or applicable to, Circles Analysis (a name which I doubt would be acceptable to Mathematicians) – 

for example, 

*the Lorenzian waterwheel, turning one way and another at certain water inputs;

*the Lorenz Attractor graph in 3 dimensions;****

Smale’s Horseshoe# (up to a point, anyway);

and most intriguingly so far,

*the ‘Point of Accumulation’#* where Periodicity gives way to Chaos, fluctuations that never settle down at all: in a series of bifurcations, 0, 1, 2, 4, 8, 16, 32 – the Point of Accumulation#** comes at 32,#*** in this particular example, but [is] not so far from the 16#**** (n=4)## after which the ordered development of the Circles turned into the variations of everyday experience, working down Pascal’s Triangle. And in the ensuing ‘chaos’, as the parameter##* (r) continues to rise, windows of periodicity open with different doubling sequences (i.e. starting with different numbers), e.g. 3, 6, 12, (24) or 7, 14, 28,##** then break off into ‘chaos’ again.

*Points of particular relevance to C[ircles] A[nalysis]. <880807>

**Gleick, ‘Chaos, Making a New Science’, Heinemann, London, 1988

***([Ibid ]to p.72)

****Ibid: p27,28

#Ibid p51

#*Ibid p72

#**or ‘Critical Amplitude’ per Pippard (T[imes] L[iterary] S[upplement], 22-28/7/88) <880805>

[See [Redbook5:218-239][19880722:2307]{The Sphere}[22nd July 1988]]

#***Is this general?

#****[[Redbook5:221][19880724:1443e]{The Sphere [continued (5)]}[24th July 1988]] p221,

[[Redbook5:223][19880724:1443h]{The Sphere [continued (8)]}[24th July 1988] (&/or f?)] 223,

[[Redbook5:228-229][19880724:2355g]{The Sphere [continued (19)]}[24th July 1988]? &/or 1/2 ant] 228 above.

(On Ibid p73 it is 16 after which chaos begins. Could this be linked with the Octet Rule?

[[Redbook5:222-223][19880724:1443g]{The Sphere [continued (7)]}[24th July 1988]])

##[ie for 2ⁿ]

##*eg in x_next = rx(1-x)

##**Prime numbers start them?

[continues]

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