The Mandelbrot Set Explorer

** Complex Dynamics**. The study of iteration of functions of a
complex variable such as complex polynomials (especially
**x ^{2} + c** which is the only function discussed herein),
rational functions (of the form

An **n**-cycle would assume the form

** Fate of the orbit of 0**. The orbit of

** Filled Julia Set**. The set of all seeds whose orbits do not
escape to infinity under iteration of **x ^{2} +
c**.

**Iteration**. To repeat a process or rule over and over. In the
Explorer, the process that is repeated is the rule ** x -->
x ^{2} + c** for some constant

**
x _{1} = x_{0}^{2} + c**

x_{2} = x_{1}^{2} + c

x_{3} = x_{2}^{2} + c

x_{4} = x_{3}^{2} + c

where the seed **x _{0}** is a real or complex number. We
also write this as

**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 **x ^{2} +
c**. Equivalently, the Mandelbrot set is the set of

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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