1) Scientists frequently rely on the assumption of stability. You do it all the time if you are taking physics. I'm sure you have problems like this:
A ball is thrown upwards at an initial velocity of 30 meters per second, from a height of 1 meter. Where will the ball be when the stopwatch shows that 3 seconds have elapsed?Sounds familiar, right? You can use this equation, which shows position as a function of time:
Current Position= 1/2gt2 + initial velocity*t + initial position.Use -10 as the acceleration due to gravity (g), and plug all the known values into the equation. Don't round anything off as you compute.
2) Now here is where the stability issue comes in. There is always the possibility of error in any scientific endeavor. In the problem above, perhaps the ball wasn't really at exactly one meter when it left the ground. Maybe it was at 1.01 meters. It would be hard to judge that one measly centimeter. Use that new value of initial position and recompute the height of the ball at 3 seconds. Don't round anything off.
3) The difference of .01 didn't seem to affect your final results too much, did it? And it's not supposed to, either. This is the whole idea of stability. I can make a slight disturbance in the initial conditions, and it won't have much effect on the outcome. Why do you suppose that idea is so important to scientists?
4) The word "tolerance" relates to how much error you can accept in your experiment. If I can tolerate an error of .01 in my final result, then it really doesn't matter that I made a measurement error at the beginning. And even if I had a real stickler for a physics teacher, I could use a better meter stick and keep the final result within a tighter tolerance (perhaps .001), if desired. That's the key to stability: I can control the error. I can't eliminate it, but I can make it as small as I need to. Call me over to establish that you understand this so far.
5) We can also see an example of stability in dynamical systems. Use the logistic equation, with parameter a set at 2.8. For your seed (P1), use a number that is extremely close, but not equal to what your partner chooses. The two seeds should differ by a hundredth (eg, 23.02 vs 23.03). Then use the graphing calculators and iterate at the same time, comparing your results for P2, P3, P4, etc.. Again it's a situation where there is a slight difference in initial conditions. Your seeds differed by a hundredth. Big deal. What effect did it have on the outcome?
6) Of course, 2.8 is a value of parameter a which happens to converge to equilibrium. Perhaps things won't be so stable if we try a= 3.3, where it won't converge. Do the same "dueling calculators" experiment with 3.3.
7) Now try again with a = 3.8. Start with seeds that differ by a hundredth. See how much the sequences differ by the time you get to P8. Keep going and check the difference by P20. What seems to be happening?
8) You should be shocked. You should be horrified. How could a difference in the hundredths place cause such a huge discrepancy after only 20 iterations? Try again. This time make the seeds differ only by a thousandth. See if that helps control the instability.
9) When a = 3.7, you get a dynamical system that is not stable. Rather, it exhibits a quality known as sensitivity, where even a tiny change in the initial conditions will have a tremendous effect on the outcome. That is exactly what Lorenz stumbled across by accident with his weather machine. Look at the graph on page 17 of Chaos, and read the narrative on pages 16 and 17.
10) In the computer room last week, a few of you wondered how we could tell the difference between something that is truly chaotic and something that does in fact have a cycle, only the cycle length is so long you wouldn't be able to see it start over. Excellent question. Now we have a technique, the sensitivity test, to help us see the difference. In fact, the existence of sensitivity is part of the definition of a chaotic dynamical system. Test the system with a = 3.5696. Is it stable or sensitive? If it is stable, can you find the length of its cycle?
11) Now check a = 3.572. Would you say that it's the same as a = 3.5696?
12) At the beginning of this unit, we looked at a dynamical system with the recursion rule:
Pn+1 = Pn2 - 2
Test some seeds and see if you think that system is stable or sensitive.| QR Home Page | Glossary | Previous Worksheet | Next Worksheet |