and asked you to find the fixed points of it. You would, quite rightly, tell me that I haven’t given you enough information. It should certainly depend on the value of
as to where the fixed points are...or indeed if there are any at all!
However, what you could do is study the system for different values of
and see if you can explore some patterns in the fixed points.
One way to think of this would be if you had an equation which described some physical system, and there was a free parameter (a growth rate, a carrying capacity, a friction term, an external magnetic field etc.) and you wanted to know how the system would change as you altered this parameter. A non free parameter would be something like Planck’s constant, or the speed of light, or anything else in your system which can’t take on different values.
It’s going to turn out that fixed points can change qualitatively. They can appear, or disappear, in a variety of interesting ways. And that is what the study of bifurcations is going to be. Bifurcation points are the points where the fixed points of a system alter in number, or in type.
An often-cited example is a vertical beam which can support some weight. If the weight is small, then the beam is stable. However, if you increase the weight, the beam will buckle. This will typically happen after some critical weight is added.
In fact, we can also exclude the arrows, and just put in the fixed points using solid lines for stable fixed points and dashed lines for unstable fixed points. Usually this is done by taking the above diagram and putting it on its side, so the
direction is vertical.
Clearly the fixed points are the values for which
=0 which in this case is just the roots of
. So we just have to solve for x as a function of r