Mathematical Glossary

The Mandelbrot Set Explorer

Mathematical Glossary.

Bifurcation. To bifurcate means to split apart. In dynamics, bifurcation often means a change in the structure of orbits. For example, two new fixed points may "bifurcate" away from a given fixed point. Or an attracting n-cycle may bifurcate away from a fixed point.

Complex Dynamics. The study of iteration of functions of a complex variable such as complex polynomials (especially x2 + c which is the only function discussed herein), rational functions (of the form P(x)/Q(x) where P and Q are complex polynomials), and other functions, including the complex exponential and trigonometric functions.

Cycle (of period n). An orbit that repeats itself every n iterations. For example, 0 lies on a cycle of period 2 for x2 - 1 since its orbit is

0 --> -1 --> 0 --> -1 ...

An n-cycle would assume the form

x0 -->x1 -->x2 --> ... --> xn-1 -->x0 --> x1 ...

Fate of the orbit of 0. The orbit of 0 is the collection of all points (complex numbers) under iteration of x2 + c starting at the seed 0. The fate of the orbit is its eventual behavior. For example, the orbit may tend to a cycle of some period---its fate would then be that cycle. Or it ---may tend to a fixed point. Its fate would then be that fixed --> --point. Or its fate could be that it tends to infinity.

Filled Julia Set. The set of all seeds whose orbits do not escape to infinity under iteration of x2 + c.

Iteration. To repeat a process or rule over and over. In the Explorer, the process that is repeated is the rule x --> x2 + c for some constant c. That is, starting with the seed x0, we compute in order

x1 = x02 + c
x2 = x12 + c
x3 = x22 + c
x4 = x32 + c

where the seed x0 is a real or complex number. We also write this as

x0 -->x1 -->x2 --> ... --> xn-1 -->xn --> ...

Julia set. The boundary of the filled Julia set. Points in the Julia set are on the edge between points whose orbits escape and points whose orbits do not escape. Orbits of points in the Julia set also lie in the filed Julia set.

Main Cardioid The main cardioid in the Mandelbrot set is the largest black region in the images of the Mandelbrot set. It consists of a cusp at c = 1/4 together with a smooth curve that winds around the origin meeting the real axis again at c = -3/4. All c-values strictly inside this cardioid have a single atracting fixed point toward which all orbits in the interior of the filled Julia set tend.

Mandelbrot set. The set of complex c-values for which the orbit of 0 does not escape under iteration of x2 + c. Equivalently, the Mandelbrot set is the set of c-values for which the filled Julia set of x2 + c is a connected set.

Orbit of 0 under x2 + c. The orbit of 0 is the list of complex numbers given by

0 --> c --> c2 + c --> (c2 + c)2 + c -->...

That is, successive complex numbers in this list are derived by squaring the previous entry and adding c.

Period of the Bulb. Each bulb in the Mandelbrot set consists of a set of c-values for which the corresponding quadratic function has an attracting cycle of a given period. That period is the period of the bulb.

Primary Bulb. A primary bulb is one of the infinitely many disk-like bulbs that are attached to the main cardioid in the Mandelbrot set. The bulb does not included the sub-bulbs attached to it, nor does it include the intricate antenna-like structures that protrude from the collections of secondary bulbs.

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