What Are Quasi-stable Orbits?

Quasi-stable orbits occur along the boundaries of the famous Mandelbrot Set (M-Set), shown to the right. Each pixel in an M-set display is a representation of the result of an iteration in the complex plane. Complex numbers are represented as z = a + ib, with the *i* reprsenting the square root of negative one and *a* and *b* are real numbers, Such numbers are sometimes called *imaginary numbers*. The iteration is simply:

z_{n+1} = z_{n}^{2} + *c*.

Where *c* is a number in the complex plane and *n* = 1,2,3,... . Each M-set point is a different value of *c* and z is initially set to zero.

Most iterations do one of two things: (1) They head off to infinity, in this case if the magnitude of z exceeds 2 the iteration is stopped and the pixel is usually colored according to the number of iterations. These iterations can be considered *unstable*. (2) The iteration asymptotically approaches a number or numbers in the complex plane, or the iteration is periodic whereby z repeats itself in some kind of pattern over and over again. In either case the value of z never exceeds 2. Such iterations are usually stopped after some finite time and the pixel is colored black in M-set displays. These type of iterations can be considered *asymptotically stable* or *periodic *

Quasi-stable orbits fall on the boundary of stable and unstable. Where a typical unstable orbit may leave the complex plane in a few hundred iterations, a quasi-stable orbit may take millions of iterations before it finally heads off to infinity. One has to sift through the field of M-set points to find these gems. A result of one quasi-stable orbit is shown to the right. The image conveys the orbit's quasi-stable nature. There is an attractor at the center which the orbit is tending to collapse into. At the same time it is being pulled out of the complex plane by the attractor at infinity. Eventually you can see the spiral path it takes to infinity, at the top and bottom of the image.