# Bifurcations in Maps with a Quadratic Maximum - Demo 3

## Patterns for the Power Law map

The power law map has a variation *|x-0.5|*^{1+b} near *x=0.5*. For *0<b<1* the derivative of the curve is zero at *x=0.5*, so the curve still has a maximum, but is non analytic here. First we look at the map itself and the iterations, e.g. for *b=0.5, a=2.6*

Now look at the sequence of bifurcations for 1.93<a<2.33:

and the Lyapunov curve

The *qualitative* appearance is quite similar to the quadratic and sine maps, with a sequence of period doubling bifurcations accumulating at around *a=2.3*. However following the same procedure as before you can show that the values of and are in fact *different* than for the two examples of a quadratic maximum.

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Last modified Sunday, January 30, 2000

Michael Cross