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Bill M. Williams (1932–2019) [1] was an American trader and author of books on trading psychology, technical analysis, and chaos theory [2] in trading the stock, commodity, and foreign exchange (Forex) markets. His study of stock market data led him to develop a number of technical analyses that identify trends in the financial markets.
While an optimally packed fractal appears only for a defined value of r, i.e., r opt, it is possible to play the chaos game using other values as well.If r>1 (the point x k+1 jumps at a greater distance than the distance between the point x k and the vertex v), the generated figure extends outside the initial polygon. [5]
Chaotic maps and iterated functions often generate fractals. Some fractals are studied as objects themselves, as sets rather than in terms of the maps that generate them. This is often because there are several different iterative procedures that generate the same fractal. See also Universality (dynamical systems).
His current venture is "Fractal Market Cycles and Regimes" at www.edgarepeters.com. [2] His books include Chaos and Order in the Capital Markets (According to WorldCat, the book is held in 813 libraries, [3]) Fractal Market Analysis (held in 580 libraries [4]) and Patterns in the Dark: Understanding Risk and Financial Crisis with Complexity ...
In chaos theory, the correlation dimension (denoted by ν) is a measure of the dimensionality of the space occupied by a set of random points, often referred to as a type of fractal dimension. [1] [2] [3]
Chaos theory has been used for many years in cryptography. In the past few decades, chaos and nonlinear dynamics have been used in the design of hundreds of cryptographic primitives. These algorithms include image encryption algorithms, hash functions, secure pseudo-random number generators, stream ciphers, watermarking, and steganography. [123]
In fractal geometry, the generalized Hurst exponent has been denoted by H or H q in honor of both Harold Edwin Hurst and Ludwig Otto Hölder (1859–1937) by Benoît Mandelbrot (1924–2010). [3] H is directly related to fractal dimension, D, and is a measure of a data series' "mild" or "wild" randomness. [4]
A Poincaré plot, named after Henri Poincaré, is a graphical representation used to visualize the relationship between consecutive data points in time series to detect patterns and irregularities in the time series, revealing information about the stability of dynamical systems, providing insights into periodic orbits, chaotic motions, and bifurcations.