When Healing Becomes Educating, Vol. 2 - Waldorf Research Institute

different, ranging from 0 to 1, and a minor change in the initial value for
x n
will result in a totally different sequence of figures after just a few stages.
Chaotic behavior is seen from a = 3.56994... onwards.
If a graph is plotted we get a bifurcating tree that shows how periods
double and then end in chaos (Fig. 1).
Oddly enough the chaotic range shows stable islands with bifurcations
that move in rhythms of three, six and twelve. Doubling periods leading to
chaos are one typical route seen with non-linear equations for evolution
dynamics, and the pattern is not limited to this kind of simple logistic
equation. The chaos is predetermined, which is why it is also called a
“deterministic” chaos. Eilenberger felt moved to paraphrase Hamlet, saying:
“Though this be chaos, yet there is method in’t.”
The logistic equation is the simplest kind of non-linear evolutional
equation with feedback, using merely one parameter (a) and one variable
(x n
). In reality, changes in rabbit populations are infinitely more complex.
Other animal populations share their food resources and the weather also
plays a role. The rate of reproduction is not constant, and the mortality rate
is subject to diseases and natural enemies, i.e., it is also influenced by other
animal populations, e.g., foxes, and the same holds true for their evolution.
An epoch-making discovery was made by Mitchell Feigenbaum, a young
physicist in the USA. He felt constrained by the thought structures of
classical physics and looked for different approaches, among other things
making an intense study of Goethe’s theory of color. The bifurcation
diagrams suggested to him that higher laws may pertain to the route to
chaos. He analyzed the diagrams and found that proportional laws applied.
The points of bifurcation come closer and closer together just like a line
of telegraph poles do with increasing distance when seen in perspective.
The factor for calculating subsequent bifurcation points is a constant:
d = 4.66920…, known as the Feigenbaum number.
Surprisingly, the Feigenbaum number is an irrational number, a
universal constant, rather like p.
In France, Libchaber, also looking for new approaches in physics—he
had studied Goethe’s scientific writings and particularly also Theodor
Schwenk’s work on turbulent flow—examined the behavior of fluids on the
way to turbulence. This also led to bifurcations and confirmation of the
Feigenbaum number.
The route to deterministic chaos could thus be defined. Once two
bifurcations are known, it is possible to use the Feigenbaum number and
determine all subsequent bifurcations up to the point of accumulation
where chaos begins, and this does not require an equation or individual
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