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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).
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]
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 ...
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.
These subregions are called bands. When there are multiple bands, the orbit moves through each band in a regular order, but the values within each band are irregular. Such chaotic orbits are called band chaos or periodic chaos, and chaos with k bands is called k -band chaos. Two-band chaos lies in the range 3.590 < r < 3.675, approximately.
In stochastic processes, chaos theory and time series analysis, detrended fluctuation analysis (DFA) is a method for determining the statistical self-affinity of a signal. It is useful for analysing time series that appear to be long-memory processes (diverging correlation time, e.g. power-law decaying autocorrelation function) or 1/f noise.
The Hénon attractor is a fractal, smooth in one direction and a Cantor set in another. Numerical estimates yield a correlation dimension of 1.21 ± 0.01 or 1.25 ± 0.02 [ 2 ] (depending on the dimension of the embedding space) and a Box Counting dimension of 1.261 ± 0.003 [ 3 ] for the attractor of the classical map.