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
An n-cycle would assume the form
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
where the seed x0 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 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.
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.
x2 = x12 + c
x3 = x22 + c
x4 = x32 + c
Orbit of 0 under x2 + c. The orbit of
0 is the list of complex numbers given by
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