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.