Heterogeneous Earliest Finish Time

Double-scroll 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). 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 invented the autonomous 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 eigenvectors 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 at all scales. Recently, there has also been reported the discovery of hidden attractors within the double scroll.

Chen attractor

Chen attractor

a = 35, c = 27, b = 2.8,x(0) = -.1, y(0) = .3, z(0) = -.6

1999年Guanrong Chen(陈关荣) and Ueta proprosed another double scroll chaotic attractor。

Chen system:

$\frac{dx(t)}{dt}=a\ast (y(t)-x(t))$

$\frac{dy(t)}{dt}=(c-a)\ast x(t)-x(t)\ast z(t)+c\ast y(t)$

$\frac{dz(t)}{dt}=x(t)\ast y(t)-b\ast z(t)$

Plots of Chen attractor can be obtained with Runge-Kutta method：

parameters:a = 40, c = 28, b = 3

initial conditions:x(0) = -0.1, y(0) = 0.5, z(0) = -0.6

Multiscroll 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

Lu Chen attractor

An extended Chen system with muliscroll was proposed by Jinhu Lu(吕金虎）and Guanrong Chen

Lu Chen system equation $\frac{dx(t)}{dt}=a\ast (y(t)-x(t))$

$\frac{dy(t)}{dt}=x(t)-x(t)\ast z(t)+c\ast y(t)+u$

$\frac{dz(t)}{dt}=x(t)\ast y(t)-b\ast z(t)$

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

Maple plot of N scroll attractor based on Chen with sine and tau

System equations:.

$\frac{dx(t)}{dt}=a\ast (y(t)-x(t)),$

$\frac{dy(t)}{dt}=(c-a)\ast x(t)-x(t)\ast f+c\ast y(t),$

$\frac{dz(t)}{dt}=x(t)\ast y(t)-b\ast z(t)$

In which

$f=d0\ast z(t)+d1\ast z(t-\tau )-d2\ast \sin (z(t-\tau ))$

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

Modifed Chua chaotic attractor

9 scroll modified Chua chaotic attractor

xy plot of 9 scroll modified Chua chaotic attractor

In 2001Tang et al proposed a modified Chua chaotic system

$\frac{dx(t)}{dt}=\alpha \ast (y(t)-h)$

$\frac{dy(t)}{dt}=x(t)-y(t)+z(t)$

$\frac{dz(t)}{dt}=-\beta \ast y(t)$

In which

$h:=-b\ast sin(\frac{\pi \ast x(t)}{2\ast a}+d)$

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

PWL Duffing chaotic attractor xy plot

PWL Duffing chaotic attractor plot

Aziz Alaoui investigated PWL Duffing equation in 2000：。

PWL Duffing system:

$\frac{dx(t)}{dt}=y(t)$

$\frac{dy(t)}{dt}=-m1\ast x(t)-(1/2\ast (m0-m1))\ast (|x(t)+1|-|x(t)-1|)-e\ast y(t)+\gamma \ast cos(\omega \ast t)$

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

modified Lorenz attractor

Miranda & Stone proposed a modified Lorenz system：

$\frac{dx(t)}{dt}=1/3\ast (-(a+1)\ast x(t)+a-c+z(t)\ast y(t))+((1-a)\ast (x(t{)}^2-y(t{)}^2)+(2\ast (a+c-z(t)))\ast x(t)\ast y(t))$$\ast \frac{1}{3\ast \sqrt{x(t{)}^2+y(t{)}^2}}$

$\frac{dy(t)}{dt}=1/3\ast ((c-a-z(t))\ast x(t)-(a+1)\ast y(t))+((2\ast (a-1))\ast x(t)\ast y(t)+(a+c-z(t))\ast (x(t{)}^2-y(t{)}^2))$$\ast \frac{1}{3\ast \sqrt{x(t{)}^2+y(t{)}^2}}$

$\frac{dz(t}{dt}=1/2\ast (3\ast x(t{)}^2\ast y(t)-y(t{)}^3)-b\ast z(t)$

parameters： a = 10, b = 8/3, c = 137/5；

initial conditions： x(0) = -8, y(0) = 4, z(0) = 10

Rabinovich Fabrikant attractor xy plot

See also

• The double-scroll attractor and Chua's circuit