Currently I'm reading a book on quantum physics ("In Search of Schrodinger's Cat" by John Gribbon) and I was struck by the similar history of development it has with chaos theory i.e. no-one wants to admit it's really true and almost no-one really understands it. It has taken almost 100 years for quantum theory to be discovered, accepted and still it is only fully taught at university level after student have already been ingrained into thinking that electrons whoosh around the nucleus. I remember each level of school (middle school, pre-GCSE, GCSE) I took chemistry, I was told that everything I had been taught previously was wrong and too simple and here is the real thing! It got to the point that when I was choosing my A levels I did not pick Chemistry. I figured they would just lie to me again and I was better off learning the truth out of university textbooks. So I took Philosophy instead. I wonder whether others felt the same. I just wished they could have taught it correctly from the beginning and get over the fact that most 10-16 year-olds might not get it. Some will! Maybe in another hundred years quantum will be understood by enough people *cough* politicians *cough* that it can be taught straight from the beginning and not require almost a reprogramming of minds. Perhaps the same progress will happen for chaos theory. One can only hope.
This week my paper is on interaction of populations of competing species. Available here. It's quite a nice paper (longer than the last one). The purpose was to analyse how three equally competing species interact to change each other and their own population size over time. The time series plots provide examples of how systems require a certain amount of time to settle down to the final state. The equilibria that I talk about in the paper are fixed points for the system. They are the stable behaviour of the species interactions. So the species may start at different values (with the same parameters) but, over time, they will settle down to a fixed point (or periodic behaviour, or chaotic behaviour).
Taking the example of the pendulum again, when you set the pendulum in motion it will oscillate a few times and then settle back down to the lower central point. This is because it is damped by drag. This behaviour can be seen mathematically as a perturbation of initial conditions away from the stable equilibrium but over time it will return to that stable equilibrium, that stable position where it can hang without oscillating quite happily until the end of time. This stable equilibrium attracts all motion of the pendulum towards it so it is called an attracting fixed point.
Now opposite that stable equilibrium there is an unstable equilibrium. Right at the top of the circle that the pendulum could swing through. At this point the complete opposite happens to the point at the bottom. It repels all motion from it. A pendulum cannot hold itself up until the end of time (not with gravity anyway). So at the top we have a repelling fixed point and at the bottom we have an attracting fixed point. Mathematically the system makes sense.
Right, so now the easy bit is done, I'll add something a little harder to the end. You might be about to comment, "So what is happening when the pendulum in a clock oscillates on and on". So I'll tell you...
That is when the system has a period two oscillation if you set drag to zero (it is not exactly possible in the real world because drag does affect it due to not quite being zero which is why you had to keep winding them up). This means that when you set the motion going, you can get the pendulum to swing for the rest of time side to side. The distance and time period of the swing will be exactly the same for every oscillation. This is a stable period two oscillation and has a pretty sine or cosine curve on the time series plot.
Pendulum's can display chaotic motion. Just check this out. And each time you set it in motion it will swing and flip in a completely different way thanks to the effect of chaos - sensitive dependence on initial conditions!
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