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 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 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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