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English: The attractor shown here is known as the Poisson Saturne attractor. It is a set in three-dimensional space and this video aims to give the viewer a fuller understanding of the set than what can be gained from one 2D image. The set consists of two separate parts; one is here colored in yellow/green and one in blue/magenta.
Visual representation of a strange attractor. [1] Another visualization of the same 3D attractor is this video. Code capable of rendering this is available. In the mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, [2] for a wide variety of starting conditions of the system. System ...
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 attractor is a stable point which is also called a "sink". The repeller is considered as an unstable point, which is also known as a "source". A phase portrait graph of a dynamical system depicts the system's trajectories (with arrows) and stable steady states (with dots) and unstable steady states (with circles) in a phase space.
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 ...
An easy way to visualize a chaotic attractor is to start with a point in the basin of attraction of the attractor, and then simply plot its subsequent orbit. Because of the topological transitivity condition, this is likely to produce a picture of the entire final attractor, and indeed both orbits shown in the figure on the right give a picture ...
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.
In mathematics, a chaotic map is a map (an evolution function) that exhibits some sort of chaotic behavior.Maps may be parameterized by a discrete-time or a continuous-time parameter.
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