Multiscroll attractor
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In the mathematics of
dynamical system In mathematics, a dynamical system is a system in which a Function (mathematics), function describes the time dependence of a Point (geometry), point in an ambient space, such as in a parametric curve. Examples include the mathematical models ...
s, the double-scroll attractor (sometimes known as Chua's attractor) is a
strange attractor In the mathematics, mathematical field of dynamical systems, an attractor is a set of states toward which a system tends to evolve, for a wide variety of starting conditions of the system. System values that get close enough to the attractor va ...
observed from a physical electronic
chaotic Chaotic was originally a Danish trading card game. It expanded to an online game in America which then became a television program based on the game. The program aired on 4Kids TV (Fox affiliates, nationwide), Jetix, The CW4Kids, Cartoon Netwo ...
circuit (generally, Chua's circuit) with a single
nonlinear In mathematics and science, a nonlinear system (or a non-linear system) is a system in which the change of the output is not proportional to the change of the input. Nonlinear problems are of interest to engineers, biologists, physicists, mathe ...
resistor (see Chua's diode). The double-scroll system is often described by a system of three nonlinear ordinary differential equations and a 3-segment piecewise-linear equation (see Chua's equations). This makes the system easily simulated numerically and easily manifested physically due to Chua's circuits' simple design. Using a Chua's circuit, this shape is viewed on an oscilloscope using the X, Y, and Z output signals of the circuit. This chaotic attractor is known as the double scroll because of its shape in three-dimensional space, which is similar to two saturn-like rings connected by swirling lines. The attractor was first observed in simulations, then realized physically after
Leon Chua Leon Ong Chua (; ; born June 28, 1936) is a Filipino-American electrical engineer and computer scientist. He is a professor in the electrical engineering and computer sciences department at the University of California, Berkeley, which he joined i ...
invented the
autonomous In developmental psychology and moral, political, and bioethical philosophy, autonomy is the capacity to make an informed, uncoerced decision. Autonomous organizations or institutions are independent or self-governing. Autonomy can also be defi ...
chaotic circuit which became known as Chua's circuit. The double-scroll attractor from the Chua circuit was rigorously proven to be chaotic through a number of Poincaré return maps of the attractor explicitly derived by way of compositions of the
eigenvector In linear algebra, an eigenvector ( ) or characteristic vector is a vector that has its direction unchanged (or reversed) by a given linear transformation. More precisely, an eigenvector \mathbf v of a linear transformation T is scaled by ...
s of the 3-dimensional state space. Numerical analysis of the double-scroll attractor has shown that its geometrical structure is made up of an infinite number of fractal-like layers. Each cross section appears to be a
fractal In mathematics, a fractal is a Shape, geometric shape containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension. Many fractals appear similar at various scale ...
at all scales. Recently, there has also been reported the discovery of hidden attractors within the double scroll. In 1999 Guanrong Chen (陈关荣) and Ueta proposed another double scroll chaotic attractor, called the Chen system or Chen attractor.


Chen attractor

The Chen system is defined as follows \frac=a(y(t)-x(t)) \frac=(c-a)x(t)-x(t)z(t)+cy(t) \frac=x(t)y(t)-bz(t) Plots of Chen attractor can be obtained with the Runge-Kutta method: parameters: a = 40, c = 28, b = 3 initial conditions: x(0) = -0.1, y(0) = 0.5, z(0) = -0.6


Other attractors

Multiscroll attractors also called ''n''-scroll attractor include the Lu Chen attractor, the modified Chen chaotic attractor, PWL Duffing attractor, Rabinovich Fabrikant attractor, modified Chua chaotic attractor, that is, multiple scrolls in a single attractor.


Lu Chen attractor

An extended Chen system with multiscroll was proposed by Jinhu Lu (吕金虎) and Guanrong Chen Lu Chen system equation \frac=a(y(t)-x(t)) \frac=x(t)-x(t)z(t)+cy(t)+u \frac=x(t)y(t)-bz(t) parameters:a = 36, c = 20, b = 3, u = -15.15 initial conditions:x(0) = .1, y(0) = .3, z(0) = -.6


Modified Lu Chen attractor

System equations: \frac=a(y(t)-x(t)), \frac=(c-a)x(t)-x(t)f+cy(t), \frac=x(t)y(t)-bz(t) In which f = d0z(t) + d1z(t - \tau ) - d2\sin(z(t - \tau )) params := a = 35, c = 28, b = 3, d0 = 1, d1 = 1, d2 = -20..20, tau = .2 initv := x(0) = 1, y(0) = 1, z(0) = 14


Modified Chua chaotic attractor

In 2001, Tang et al. proposed a modified Chua chaotic system \frac= \alpha (y(t)-h) \frac=x(t)-y(t)+z(t) \frac=-\beta y(t) In which h := -b \sin\left(\frac+d\right) params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0 initv := x(0) = 1, y(0) = 1, z(0) = 0


PWL Duffing chaotic attractor

Aziz Alaoui investigated PWL Duffing equation in 2000: PWL Duffing system: \frac=y(t) \frac=-m_1x(t)-(1/2(m_0-m_1))(, x(t)+1, -, x(t)-1, )-ey(t)+\gamma \cos(\omega t) params := e = .25, gamma = .14+(1/20)i, m0 = -0.845e-1, m1 = .66, omega = 1; c := (.14+(1/20)i),i=-25...25; initv := x(0) = 0, y(0) = 0;


Modified Lorenz chaotic system

Miranda & Stone proposed a modified Lorenz system:J.Liu and G Chen p834 \frac = 1/3*(-(a+1)x(t)+a-c+z(t)y(t))+((1-a)(x(t)^2-y(t)^2)+(2(a+c-z(t)))x(t)y(t))\frac \frac= 1/3((c-a-z(t))x(t)-(a+1)y(t))+((2(a-1))x(t)y(t)+(a+c-z(t))(x(t)^2-y(t)^2))\frac \frac = 1/2(3x(t)^2y(t)-y(t)^3)-bz(t) parameters: a = 10, b = 8/3, c = 137/5; initial conditions: x(0) = -8, y(0) = 4, z(0) = 10


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References


External links


The double-scroll attractor and Chua's circuit
* {{Chaos theory Chaos theory