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The main catalyst for the development of chaos theory was the electronic computer. Much of the mathematics of chaos theory involves the repeated iteration of simple mathematical formulas, which would be impractical to do by hand. Electronic computers made these repeated calculations practical, while figures and images made it possible to ...
An example is the well-studied logistic map, + = (), whose basins of attraction for various values of the parameter are shown in the figure. If r = 2.6 {\displaystyle r=2.6} , all starting x {\displaystyle x} values of x < 0 {\displaystyle x<0} will rapidly lead to function values that go to negative infinity; starting x {\displaystyle x ...
Orbit diagram for the Hénon map with b=0.3.Higher density (darker) indicates increased probability of the variable x acquiring that value for the given value of a.Notice the satellite regions of chaos and periodicity around a=1.075-- these can arise depending upon initial conditions for x and y.
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.
For example, even the small flap of a butterfly's wings could set the earth's atmosphere on a vastly different trajectory, in which for example a hurricane occurs where it otherwise would have not (see Saddle points). The shape of the Lorenz attractor itself, when plotted in phase space, may also be seen to resemble a butterfly.
An example would be plotting the , value every time it passes through the = plane where is changing from negative to positive, commonly done when studying the Lorenz attractor. In the case of the Rössler attractor, the x = 0 {\displaystyle x=0} plane is uninteresting, as the map always crosses the x = 0 {\displaystyle x=0} plane at z = 0 ...
Thankfully, the Chaotic Good subreddit is changing that. This online group shares good intentions manifested through unorthodox methods, to say the least. ... for example. #10 Civil Disobedience ...
In chaos theory, Wada basins arise very frequently. Usually, the Wada property can be seen in the basin of attraction of dissipative dynamical systems. But the exit basins of Hamiltonian systems can also show the Wada property. In the context of the chaotic scattering of systems with multiple exits, basins of exits show the Wada property. M. A. F.