Deterministic Chaos

6.P. Renormalization

One way to visually understand the period-doubling sequence is through a method called renormalization.

The sequence of pictures below shows graphs of L(x) and of L2(x).

To the graphs of L2(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 L2(x).

These are the trapping squares.

Fixed point at 0 nonzero fixed point 2-Cycle Chaos Outside the trapping square
Graphical iteration takes almost any point to a 2-cycle. Click the picture to see the iterates. Graphical iteration from almost any point in either trapping square goes to a 2-cycle 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 L2(x) in each of the two trapping squares.

Similar results hold for all Ln(x). This explains the smaller copies of the bifurcation diagram in the whole diagram.

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