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 generates a chaotic sequence.
Click the picture to see the iterates. 
What happens when the graph of L^{2}(x) extends outside the
trapping squares? Recall if the graph of the function
extends outside the unit square, most points iterate away to infinity. This cannot
happen here: even though the graph of L^{2}(x) extends outside the trapping
square, it remains in the unit square. Rather than running away to infinity, the
iterates simply exit the trapping squares for a while.
Click the picture to see the iterates. 

Another view of the relation of the iterates of L^{2} and of L can be
seen by looking at the corresponding bifurcation diagrams. 
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.
Return to Deterministic Chaos.