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Chaos theory (or chaology [1]) is an interdisciplinary area of scientific study and branch of mathematics. It focuses on underlying patterns and deterministic laws of dynamical systems that are highly sensitive to initial conditions. These were once thought to have completely random states of disorder and irregularities. [2]
With the aid of the "chaos game" a new fractal can be made and while making the new fractal some parameters can be obtained. These parameters are useful for applications of fractal theory such as classification and identification. [3] [4] The new fractal is self-similar to the original in some important features such as fractal dimension.
The book sparked widespread popular interest in fractals and contributed to chaos theory and other fields of science and mathematics. Mandelbrot also put his ideas to work in cosmology. He offered in 1974 a new explanation of Olbers' paradox (the "dark night sky" riddle), demonstrating the consequences of fractal theory as a sufficient, but not ...
The Origins of Chaos Theory. ... These attractors look different for different systems, but often take the form of recursive, fractal shapes. Sadly, finding an attractor for every type of chaotic ...
Devaney is known for formulating a simple and widely used definition of chaotic systems, one that does not need advanced concepts such as measure theory. [8] In his 1989 book An Introduction to Chaotic Dynamical Systems, Devaney defined a system to be chaotic if it has sensitive dependence on initial conditions, it is topologically transitive (for any two open sets, some points from one set ...
Sierpiński Carpet - Infinite perimeter and zero area Mandelbrot set at islands The Mandelbrot set: its boundary is a fractal curve with Hausdorff dimension 2. (Note that the colored sections of the image are not actually part of the Mandelbrot Set, but rather they are based on how quickly the function that produces it diverges.)
Animated creation of a Sierpiński triangle using the chaos game If one takes a point and applies each of the transformations d A , d B , and d C to it randomly, the resulting points will be dense in the Sierpiński triangle, so the following algorithm will again generate arbitrarily close approximations to it: [ 4 ]
The Beauty of Fractals is a 1986 book by Heinz-Otto Peitgen and Peter Richter which publicises the fields of complex dynamics, chaos theory and the concept of fractals. It is lavishly illustrated and as a mathematics book became an unusual success. The book includes a total of 184 illustrations, including 88 full-colour pictures of Julia sets.