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Bifurcation diagram for the Rössler attractor for varying Here, a {\displaystyle a} is fixed at 0.2, c {\displaystyle c} is fixed at 5.7 and b {\displaystyle b} changes. As shown in the accompanying diagram, as b {\displaystyle b} approaches 0 the attractor approaches infinity (note the upswing for very small values of b {\displaystyle b} ).
The attractor was first observed in simulations, then realized physically after Leon Chua invented the autonomous chaotic circuit which became known as Chua's circuit. [1] The double-scroll attractor from the Chua circuit was rigorously proven to be chaotic [2] through a number of Poincaré return maps of the attractor explicitly derived by way ...
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.
Chua's circuit (also known as a Chua circuit) is a simple electronic circuit that exhibits classic chaotic behavior. This means roughly that it is a "nonperiodic oscillator"; it produces an oscillating waveform that, unlike an ordinary electronic oscillator , never "repeats".
The Lorenz attractor is difficult to analyze, but the action of the differential equation on the attractor is described by a fairly simple geometric model. [24] Proving that this is indeed the case is the fourteenth problem on the list of Smale's problems. This problem was the first one to be resolved, by Warwick Tucker in 2002. [25]
The frequency response of this oscillator describes the amplitude of steady state response of the equation (i.e. ()) at a given frequency of excitation . For a linear oscillator with β = 0 , {\displaystyle \beta =0,} the frequency response is also linear.
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.
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.