For the following information (paraphrased from Chapter 1, "A Mathematical and Historical Tour") and much more, see Robert Devaney, A First Course in Chaotic Dynamic Systems.
Chaos occurs in objects like quadratic equations when they are regarded as dynamical systems by treating simple mathematical operations like taking the square root, squaring, or cubing and repeating the same procedure over and over, using the output of the previous operation as the input for the next (iteration). This procedure generates a list of real or complex numbers that are changing as you proceed - a dynamic system.
For some types of functions, the set of numbers that yield chaotic or unpredictable behavior in the plane is called the Julia set after the French mathematician Gaston Julia, who first formulated many of the properties of these sets in the 1920s. These Julia sets are complicated even for quadratic equations. They are examples of fractals - sets which, when magnified over and over, always resemble the original image. The closer you look at a fractal, the more you see exactly the same object. Fractals naturally have a dimension that is not an integer - not 1 or 2, but often somewhere in between.
The black points in graphic representations of these sets are the non-chaotic points, representing values that under iteration eventually tend to cycle between three different points in the plane so that their dynamical behavior is predictable. Other points are points that "escape," tending to infinity under iteration. The boundary between these two points of behavior - the interface between the escaping and the cycling points - is the Julia set.
The totality of all possible Julia sets for quadratic functions is called the Mandelbrot set: a dictionary or picture book of all possible quadratic Julia sets. First viewed in 1980 by Benoit Mandelbrot and others, the Mandelbrot set completely characterizes the Julia sets of quadratic functions, and has been called one of the most intricate and beautiful objects in mathematics.
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