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 most points to chaotic orbits.
Click the picture to see the iterates. 
Graphical iteration from most points in either
trapping square to chaotic orbits inside that 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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