# Pictorial Renormalization Group - Demo 5

## a=3.236

*a=3.236* gives a superstable 2-cycle i.e. a 2-cycle with one point at the maximum of *f*. In the demonstration the transient behavior is eliminated by iterating 100 times before plotting.

In *f*^{ 2}(x) the two intersection points are at stationary points of *f*^{ 2} - the slopes at the two intersection points are necessarily equal by the chain rule giving the derivative of *f*^{ 2} as the product of the derivatives of *f* at the two orbit points *x*_{1}, x_{2}.

Notice that the region near *x=0.5*, if inverted, resembles the quadratic map, and the behavior of the orbit starting from *x=0.1* is reminiscent of the behavior in *f(x)* for *a=2* (demonstration 1).

Starting from a slightly different initial condition (*x=0.29*) leads to an orbit that converges to the other fixed point:

Although this looks like a quite different orbit in *f*^{ 2} this is just because we are "strobing" the orbit of *f*, looking at the value every other step. In fact this initial condition gives almost the same orbit in *f*, since *f(0.1)=0.29*.

The important point is that the behavior for *x>0.7* i.e. on the greater side of the unstable fixed point is just a reflection of the behavior for *x<0.7* (at least after a few transient steps to bring the orbit into this attracting region). This means *we can focus attention on the region around **x=0.5* (in fact at this value of *a* on the range *|x-0.5|<0.2*).

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Last modified Sunday, December 12, 1999

Michael Cross