Fractal Geometry of the Mandelbrot Set (Cover Page)

Fractal Geometry of the Mandelbrot Set (Previous Section)

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The Mandelbrot set is generated by iteration. Iteration means to
repeat a process over and over again. In mathematics this process is
most often the application of a mathematical function. For the
Mandelbrot set, the function involved is the simplest nonlinear
function imaginable, namely **x ^{2} + c**, where

To iterate **x ^{2} + c**, we begin with a

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

**
**

**
**

**
**

**
**

and so forth. The list of numbers ** x _{0}, x_{1}, x_{2},...** generated
by this iteration has a name: It is called
the

Let's begin with a few examples. Suppose we start with
the constant **c = 1**.
Then, if we choose the seed **0**, the orbit is

**
**

_{1} = 1 = 0^{2} + 1

_{2} = 2

_{3} = 5

_{4} = 26

_{5} = BIG

_{6} = BIGGER

As another example, for **c = 0**, the orbit of the seed **0** is quite
different: this orbit remains * fixed* for all iterations.

**
**

_{1} = 0

_{2} = 0

If we now choose **c = -1**, something else happens. For the seed **0**, the
orbit is

**
**

_{1} = -1

_{2} = 0

_{3} = -1

Here we see that the orbit bounces back and forth between **0** and **-1**,
a * cycle of period 2*.

To understand the fate of orbits, it is most often easiest to proceed
geometrically. Accordingly, a time series plot of the orbit often
gives more information about the fate of the orbits. In the plots
below, we have displayed the time series for ** x ^{2} + c** where

** Figure 1. Time series for **

** Figure 2. Histogram of the orbit of 0 under **

To see additional time series plots for other values of **c**, select a
**c** value from the options below:

**c = -0.65**(Tends to a fixed point)**c = -1.6**(Chaotic behavior)**c = -1.75**(Period 3)**c = -1.8**(Chaotic behavior close to 3-cycle, sometimes called intermittency)**c = -1.85**(Chaotic behavior)**c = 0.2**(Tends to a fixed point)

Before proceeding, let us make a seemingly obvious and uninspiring
observation. Under iteration of ** x ^{2} + c **,
either the orbit of

How then is the
Mandelbrot set a planar picture? The answer is, instead of considering
real values of **c**, we also allow **c** to be a complex number.
For example, the orbit of **0** under ** x ^{2} + i ** is given by

**
**

_{1} = i

_{2} = -1 + i

_{3} = -i

_{4} = -1 + i

_{5} = -i

_{6} = -1 + i

and we see that this orbit eventually cycles with period 2. If we
change **c** to **2i**, then the orbit behaves very differently

**
**

_{1} = 2i

_{2} = -4 + 2i

_{3} = 12 - 14i

_{4} = -52 - 334i

_{5} = BIG (meaning far from the origin)

_{6} = BIGGER

and we see that this orbit tends to infinity in the complex plane (the
numbers comprising the orbit recede further and further from the
origin). Again we make the fundamental observation either the orbit of
0 under ** x ^{2} + c** tends to infinity, or it does not.

Fractal Geometry of the Mandelbrot Set (Cover Page)

Fractal Geometry of the Mandelbrot Set (Previous Section)

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Prof. Robert L. Devaney (Boston University)