Bifurcation

The animation in this page shows the changes of the attractor:

   x = x*y+a*x-y     
   y = x+y

as the parameter a varies from 0.01 to 0.35.

A bifurcation is a qualitative change in an attractor's structure as a control parameter is smoothly varied. For example a periodic attractor might become unstable and be replaced by a chaotic attractor.

This is what happens in a dripping faucet as the pressure changes.First drops come off the faucet with equal timing between them. As the pressure is increased the drops begin to fall with two drops falling close together, then a longer wait, then two drops falling close together again. In this case, a simple periodic process has given way to a periodic process with twice the period, a process described as "period doubling". If the flow rate of water through the faucet is increased further, often an irregular dripping is found and the behavior can become chaotic.

The same process of period doubling can be seen in the behaviour of the logistic map x=a*x*(1-x) as a changes from 0 to 4. From one value,we pass to periodic behaviour (2,4,etc) to chaos.This is shown in the applet nearby.