One Dimensional Maps - Demo 3

Bifurcations of the Quadratic Map

The bifurcations as a changes are dramatically displayed by the bifurcation map, which shows the values of x on the attractor (i.e. visited after transients have been allowed to die out) for each value of a. (You will probably want to increase the Speed of the iteration. After the full range of a is spanned, the plot continues back close to the smallest a filling in the plot more densely.)

The point x=0 is the attracting fixed point for a<1. For a>1 this becomes unstable to a fixed point that moves continuously away from x=0. At a=3 the fixed point becomes unstable to a period 2 orbit as we saw in demonstration 2. Make sure you understand what the bifurcation map tells you compared with with iterations at specific values of a: e.g. a=3.5:
Change a and look at the behavior for other values that look interesting from the bifurcation map.

Most of the interesting action is for a>2 so we replot this region.

The following points should be apparent, and can be investigated in more detail by enlarging portions of the plot by dragging the mouse over a rectangle on the stopped plot, or by setting a_max and a_min:

(Technical points: The plot is constructed by plotting Points number of points at each value of a after iterating a number Transient times to allow the iteration time to converge to the attractor. Near bifurcation points the convergence becomes slow, and to get the correct, sharp, behavior here you might need to increase Transient. Clearly periodic orbits of order greater than Points will not be captured completely, so you may need to increase this value too.)

Return to discussion.

[Previous Demonstration] [Next Demonstration] [Introduction]
Last modified Tuesday, December 7, 1999
Michael Cross