Note: this page has a couple links to ~gasp~ an outside page. This page is [Eric Weisstein's World of Mathematics], which contains references to just about anything mathematical you can think of, and also a [zillion] things you can't. Also, notice that I will set off these links with brackets.
So what is bifurcation, anyway? Well, let's analyze the word, Dave Barry-style: It comes from an old language, possibly Latin, from the words "bi," meaning "two," and "furcate," meaning "to split from one convergence point off to two attractors." Okay, so Dave Barry might've come up with something funnier than that, but his Supercomp grade doesn't depend on getting it somewhat right. So essentially you have bifurcation if at one value of a parameter of a function successive iterations seem to converge at a point, but as you change that parameter slightly it diverges. The main type we dealt with in our [logistic equation] is a more specific kind of bifurcation, called (appropriately) [pitchfork bifurcation]. For an even more rigorous definition of bifurcation, here's yet another bracketed link about (what else?) [bifurcation].
Now, enough of these links in brackets and definitions; what exactly did
I do for the assignment? I took the logistic equation y = kx(1-x)
(or, as I shall refer to it for he sake of clarity, the function
xn+1 = k xn(1-xn) and graphed
it. Well, that's not exactly very precise. First, my program asked for a
starting x-value, which it would then use every time it needed an
x0. The only requirement is that
x0 
[0, 1]. Then, my program
created a window which had along the typical x-axis the values of k
(k (0, 4)) and along the usual y-axis the
values of xn. Then for each k it simply found the
first hundred iterations of x and plotted them all on the
graph in blue. Then, it would take whichever is the selected iteration and
draw that path in red to make it clearly visible. Left-clicking increased
the number of the iteration by 1 and right-clicking decreased it by 1.
|
A sample screenshot, but made much, much smaller (this one has highlighted
the initial iteration). For more screenshots, both in animated and boring form, click here. |
And if you're one of those people that likes seeing code or if you're one that has to see it to grade it, it's all here for your viewing pleasure.