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.[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 of compositions of the eigenvectors of the 3-dimensional state space.[3]
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.[4] Recently, there has also been reported the discovery of hidden attractors within the double scroll.[5]
In 1999 Guanrong Chen (陈关荣) and Ueta proposed another double scroll chaotic attractor, called the Chen system or Chen attractor.[6] [7]
The Chen system is defined as follows
| dx(t) | |
| dt |
=a(y(t)-x(t))
| dy(t) | |
| dt |
=(c-a)x(t)-x(t)z(t)+cy(t)
| dz(t) | |
| dt |
=x(t)y(t)-bz(t)
Plots of Chen attractor can be obtained with the Runge-Kutta method:[8]
parameters: a = 40, c = 28, b = 3
initial conditions: x(0) = -0.1, y(0) = 0.5, z(0) = -0.6
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.[9]
An extended Chen system with multiscroll was proposed by Jinhu Lu (吕金虎) and Guanrong Chen
Lu Chen system equation
| dx(t) | |
| dt |
=a(y(t)-x(t))
| dy(t) | |
| dt |
=x(t)-x(t)z(t)+cy(t)+u
| dz(t) | |
| dt |
=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
System equations:
| dx(t) | |
| dt |
=a(y(t)-x(t)),
| dy(t) | |
| dt |
=(c-a)x(t)-x(t)f+cy(t),
| dz(t) | |
| dt |
=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
In 2001, Tang et al. proposed a modified Chua chaotic system[10]
| dx(t) | |
| dt |
=\alpha(y(t)-h)
| dy(t) | |
| dt |
=x(t)-y(t)+z(t)
| dz(t) | |
| dt |
=-\betay(t)
In which
h:=-b\sin\left(
| \pix(t) | |
| 2a |
+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
Aziz Alaoui investigated PWL Duffing equation in 2000:[11]
PWL Duffing system:
| dx(t) | |
| dt |
=y(t)
| dy(t) | |
| dt |
=-m1x(t)-(1/2(m0-m1))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma\cos(\omegat)
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;
Miranda & Stone proposed a modified Lorenz system:[12]
| dx(t) | |
| dt |
=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))
| 1 | |
| 3\sqrt{x(t)2+y(t)2 |
| dy(t) | |
| dt |
=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))
| 1 | |
| 3\sqrt{x(t)2+y(t)2 |
| dz(t) | |
| dt |
=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