# Bifurcations in Maps with a Quadratic Maximum - Demo 2

## Patterns in the Sine Map

We can repeat the exercise of demonstration 1 for the Sine map.

Again follow the bifurcation to high order by enlarging the region around the bifurcations near *x=0.5* approaching *a=3.47*, increasing "*Transient*" and "*Points*" as necessary.

The Lyapunov exponent plot displays the values of *a* for the superstable cycles:

Using the two applets you can construct the table showing the geometric nature of the sequence of bifurcations. Notice that the *values* of *a* at which the bifurcations occur are different than for the quadratic map (demonstration 1), but the *geometric ratios* are the same. This is the "universality" and extends widely to maps with a quadratic maximum.

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Last modified Sunday, January 30, 2000

Michael Cross