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.)
- There is a sequence of period doubling bifurcations leading to period 2,4,8... orbits. The values of a at which these bifurcations occur get closer together.
- There is an accumulation point for the period doubling bifurcations, at which the period diverges, at around ac=3.57.
- Just beyond this value the orbit covers continuous bands of x and is chaotic.
- The bands successively merge until at a=4 the chaotic orbit visits every point in the unit interval 0<x<1.
- Even for a>3.57 there are windows in a for which periodic motion occurs, and then these too undergo their own sequence of period doubling bifurcations to chaos. Examples are the period 3 orbit near a=3.83 or the period 5 orbit near a=3.74. In fact orbits of any period can be found!
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Last modified Tuesday, December 7, 1999