By now the bifurcation diagram for the function f(x) = x^2 + c is a familiar sight. It shows the transition in the dynamics of f(x) from a single attracting fixed point to attracting cycles as c moves along the real number line from 0.25 to -2.0. The first bifurcation takes place at c = -0.75, the point where the large disk is attached to the main cardiod, M0, of the Mandelbrot Set, M. As c passes through the point -0.75 from the main cardioid into that disk f(x) goes from having a single attracting fixed point to an attracting cycle of period 2. In this animation we look at what happens to the dynamics of the complex valued function f(z) = z^2 + c, where c is a complex-valued parameter. We let
c = r*exp(i*theta)/2 - r^2*exp(i*2*theta)/4. When r = 1, c is on the boundary of M0; when r < 1, c is in the interior of M0, and when r >1, c is outside of M0. In this animation we let r vary continuously from just less than 1 to a number greater than 1, and for each such r we plot 400 iterations of the orbit of the critical point, (0,0).

If we let theta = 2*pi*alpha, then for 0 <= alpha <1 we travel the boundary of M0. If alpha is a rational number, p/q in lowest terms, then the point c on the boundary of M0 corresponding to that alpha is the point where a bulb attaches to M0, and for c inside that bulb, f(z) has an attracting cycle of period q. So when alpha is chosen to be a rational number p/q in lowest terms, we will see a "q-furcation". In the animation we traverse the boundary of the cardioid, M0. You may select the type of number for alpha (rational, irrational, random or what I call "almost" rational). Of course in a computer all numbers are rational; howver we can get a pretty good approximation to an irrational number. If you choose rational alphas, of the form p/q, where q is not too large, the animation runs through alpha = 1/2, 1/3, 1/4, 1/5,... you will see a "q-furcation". I have included the type "almost" rational which means I add a tiny decimal, such as .00102137 to a rational such as 1/2, 1/3, 1/4,...because this category makes the prettiest pictures.If you choose irrational alpha we approximate an irrational number by a decimal which does not repeat or terminate.For these angles you may see an explosion into many points, or, if the angle is "close" to a rational multiple of 2*pi, you will see something "close" to a q-furcation.

If alpha is "badly" approximated by rational numbers then for c on the boundary of M0 there is a Siegel disk around the indifferent fixed point. Inside this disk there are circles which are invariant under f. For these values of alpha the orbits will be concentric circles spreading ever wider.

In the applet nfurc1 (Choose your own angle) you can experiment by typing in your own choice of alpha. Enter a value for alpha; it should be a real number between 0 and 1 in decimal form. Then click on Draw Picture. (Two values for alpha that give nice pictures are alpha = 0.66496 and alpha = 0.4287.) Just changing one digit in the decimal can give a completely different picture. To see a 3-furcation (tri-furcation?) let alpha be 0.3333333. Even though this number is not exacly 1/3, it seems to be close enough. For a 4-furcation try alpha = 0.25 or 0.75, for a 5-furcation, let alpha be 0.2, 0.4 or 0.6. For an irrational alpha pick a number that is not obviously near any rational number, for example alpha = 0.121131114.

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e-mail me aburns@liu.edu
Anne M. Burns, Department of Mathematics
Long Island University, C.W. Post Campus
Brookville, NY 11548
(516)-299-3061