Deterministic Chaos
6.P. Renormalization
One way to visually understand the
perioddoubling sequence
is through a method called renormalization.
The sequence of pictures below shows graphs of L(x) and
of L^{2}(x).
To the graphs of L^{2}(x) are attached small squares
with corners at the nonzero fixed point of L(x) and with base length determined
by where a horizontal line from the fixed point next crosses the graph of
L^{2}(x).
These are the trapping squares.



Graphical iteration takes any point to the
fixed point at x = 0, the fixed point on a corner of the square.
Click the picture to see the iterates. 
Graphical iteration takes any point to the fixed point
at x = 0, the fixed point on a corner of the square.
Click the picture to see the iterates. 

The complete cascade of behaviors of L(x) inside the
unit square is repeated by the portions of L^{2}(x) in each of the two
trapping squares.
Similar results hold for all L^{n}(x). This explains the smaller
copies of the bifurcation diagram in the whole diagram.
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