Now look at the phase portraits and try and match up the different types of fixed points.
OK, let’s step back for a moment and try and get a bigger picture...this has gotten pretty messy!
We started here with a differential equation which may describe the dynamics of some physical system. We want to know some of the qualitative features of the behaviour of this system. One of the most important types of qualitative thing we can talk about are the fixed points, and their nature. We have a parameter,
and we know that for different values of this parameter there may be different numbers of fixed points. We figured out where the fixed points were, and you figured out what kinds of fixed points they were. Of course one can run the linear stability analysis on the fixed points and see whether they are exponential, or less than exponential in their behaviour.
Let’s ask now, not just about where the fixed points are, but how you might move around them. Let’s say you start your system off with an r value (which you can tune by a dial on your equipment) of
, and an x value of
. Close to
= 0 you can do a linear stability analysis and show that the equation looks like:
Which means that we will have a behaviour with these parameters of:
Because of our exponential decay, we are still at a positive (though small) value of x. It actually depends how quickly we move our dial and how long we’ve been waiting as to how small, but we know that we couldn’t have crossed over to a negative value of x. Let’s say that after waiting at r=-0.5 for 2 seconds, we suddenly altered the value of r to -0.01 and stay there for 2 seconds. We can first work out that after the first 2 seconds our x value would be:
Then our new r value is -0.01 and we wait there for 2 seconds:
Let’s think careful about where that 4-2 comes from. The solution to the new equation is:
but we started now not at t=0, but really at t=2, so everything should be measured from t=2. The amount of time that we have been sat there at t=4 is 2 seconds in the new r=-0.01 system, and the initial condition is the value that we were at the end of the r=-0.5 system, which was 0.0367879.
In terms of the trajectory, this looks like:
Note that in seconds 2 to 4, it looks flat but we are still decaying, just with a decay rate of only -0.01.
Now let's say that we move our dial to r=0.5. All of a sudden the value of x that we are sitting at is an unstable fixed point. In fact we will move away from it now with an exponential of the form:
This takes us to 14 seconds and to an x value of 1.16877
On the bifurcation diagram it looks like we have made this journey:
Dashed arrows are instantaneous jumps in r, the full red arrows are trajectories which take time to get to a new x value.
ok, so what happens if we now move the value of r back -0.01? We could do this by either slowly tuning r down, in which case it’s going to stick to the positive stable fixed point:
Then we find ourself at the upper fixed point again:
Let’s just add to this the values of r across time as well. These are in red:
Which corresponds to:
You could have done the above, but changed the value of r slowly, in which case you would have:
Here is a really strange example of hysteresis: If you take water at room temperature and freeze it, it will take a certain amount of time to freeze. However, if you boil water, and then freeze it, it actually freezes faster, despite the fact that in order to freeze, it has to pass through the room temperature state. There is information in the system about what happened to it in the past (you can think of it as remembering that it was in a high temperature state - though note very very importantly, this “memory” is not like the memory which people claim that water has in homeopathy. That isn’t memory, it’s wishful thinking). This phenomenon with water is called the Mpemba Effect, named after the Tanzanian schoolboy who discovered it. It’s actually not completely agreed upon what is the cause of this effect.
That was a lot! We’re going to look at something a bit different on the next page before returning to the study of these bifurcations.