2/9/99

What I meant to say was that you should first follow the hint, i.e. assume that xn ~ a/n^b and determine the coefficients a and b. You will obtain xn ~ 1/n. This reasoning is heuristic because it assumes that xn has the desired form. If you do this, you will get partial credit. To get full credit, you have to go further and show that xn ~ 1/n without making an assumption as before. This part is more difficult. A hint I mentioned in class was to relate xn to yn, where yn given by the map y(n+1) = yn / ( 1 + yn).

(no, xn ~ eq means that lim xn/eq = 1, not lim eq*xn =1)

Yes, that is correct. To show that xn ~ 1/n, you could show lim n * xn = 1, or find two sequences yn, zn, for which you know that yn, zn ~ 1/n, and for which you can show zn < xn < yn. The yn I mentioned above will work, but the zn is a bit trickier.

1/25/99

An equilibrium point is a solution in which x(t) is constant. In both types of equations, x'=f(x) and x''=f(x), the condition that determines the equilbrium points is f(X)=0. For the 2nd order equation x''=f(x), you need to supply initial data x(0) and x'(0). Now suppose that x(0)=X. If x'(0)=0, then x(t)=X for all t. However, you might have x'(0) ne 0, in which case x(t) moves away from X.

1/25/99

The particle may be at rest at a point on the ring. In that sense, the particle may have zero kinetic energy with respect to its motion on the ring.

Yes.

1/23/99

Yes, if the answer is given in the back of the book, you should derive it, or justify it.

1/23/99

There are various ways to approach this. Using the formula for the roots of a cubic is a natural idea, but it doesn't help much in practice as you found. In this problem, you don't need to find X=f(a) explicitly, just the shape so that you can draw the bifurcation diagram. You know that a cubic function g(x)=F(a,x) may have 0, 1, or 2 critical points depending on the parameter a. My hint is to find these critical points and plot the graph of g(x) in each of the three generic cases.

Another possibility is to use the Matlab "contour" command to plot F(a,x)=0. You would still have to justify the answer on the homework, but at least you'd know what it looks like.