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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.
David Ruelle and Floris Takens later predicted, against Landau, that fluid turbulence could develop through a strange attractor, a main concept of chaos theory. Edward Lorenz was an early pioneer of the theory. His interest in chaos came about accidentally through his work on weather prediction in 1961. [14]
Almost all initial points will tend to an invariant set – the Lorenz attractor – a strange attractor, a fractal, and a self-excited attractor with respect to all three equilibria. Its Hausdorff dimension is estimated from above by the Lyapunov dimension (Kaplan-Yorke dimension) as 2.06 ± 0.01 , [ 20 ] and the correlation dimension is ...
A double-scroll Chen attractor from a simulation. In the mathematics of dynamical systems, the double-scroll attractor (sometimes known as Chua's attractor) is a strange attractor observed from a physical electronic chaotic circuit (generally, Chua's circuit) with a single nonlinear resistor (see Chua's diode).
Thomas' cyclically symmetric attractor. In the dynamical systems theory , Thomas' cyclically symmetric attractor is a 3D strange attractor originally proposed by René Thomas . [ 1 ] It has a simple form which is cyclically symmetric in the x, y, and z variables and can be viewed as the trajectory of a frictionally dampened particle moving in a ...
In applied mathematics and astrodynamics, in the theory of dynamical systems, a crisis is the sudden appearance or disappearance of a strange attractor as the parameters of a dynamical system are varied. [1] [2] This global bifurcation occurs when a chaotic attractor comes into contact with an unstable periodic orbit or its stable manifold. [3]
If a strange attractor C did exist in such a system, then it could be enclosed in a closed and bounded subset of the phase space. By making this subset small enough, any nearby stationary points could be excluded. But then the Poincaré–Bendixson theorem says that C is not a strange attractor at all—it is either a limit cycle or it ...
Turbulence, strange attractors and chaos. World Scientific. ISBN 978-9810223113. Ruelle, David (2008). Chaotic evolution and strange attractors. Vol. 1. Cambridge University Press. ISBN 978-0521368308. 1989 edition; Ruelle, David (2014). Elements of differentiable dynamics and bifurcation theory. Elsevier. ISBN 978-1483245881. 1989 1st edition [29]