Pictorial Renormalization Group - Demo 6

a=3.236 - Rescaling

Here is the orbit in f 2:

Now we focus on the region around x=0.5 by the following operation: This combination is obtained by setting nsc=1. With this combination the unstable fixed point that was at x=0.7 becomes the new origin, and the parabolic minimum becomes a quadratic maximum:

and the same orbit is plotted as in the first applet (the initial condition x=0.66 becomes x=0.1 after the rescaling operations).

The strong similarity of the scaled plot to demonstration 1 should be readily apparent.

Note that any initial condition within the plot leads to an orbit that remains within the truncated range of the plot. Initial conditions outside the range of the plot (but within the unit interval) lead, after a few transients, to orbits that are either within this plot, or correspond to the same orbit strobed with a different phase. Thus we may discard all the region outside this plot, and look for successive bifurcations purely within the range plotted.

Technical detail: the exact process of the rescaling is to construct a square on the horizontal base that extends from the unstable fixed point to the point on the curve at the same height and the opposite side of x=0.5. The square contains the stationary point at x=0.5. In the applet, for later convenience this is done in two steps: first we scale by the factor ||=2.50291..., and then fine tuning can be done if necessary through the multiplicative factor "scale".

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Last modified Sunday, December 12, 1999
Michael Cross