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]

## Chen attractor

The Chen system is defined as follows[7]

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

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

${\displaystyle {\frac {dz(t)}{dt))=x(t)y(t)-bz(t)}$

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

## 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.[9]

### Lu Chen attractor

An extended Chen system with multiscroll was proposed by Jinhu Lu (吕金虎) and Guanrong Chen[9]

Lu Chen system equation

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

${\displaystyle {\frac {dy(t)}{dt))=x(t)-x(t)z(t)+cy(t)+u}$

${\displaystyle {\frac {dz(t)}{dt))=x(t)y(t)-bz(t)}$

parameters：a = 36, c = 20, b = 3, u = -15.15

initial conditions：x(0) = .1, y(0) = .3, z(0) = -.6

### Modified Lu Chen attractor

System equations:[9]

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

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

${\displaystyle {\frac {dz(t)}{dt))=x(t)y(t)-bz(t)}$

In which

${\displaystyle f=d0z(t)+d1z(t-\tau )-d2\sin(z(t-\tau ))}$

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[10]

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

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

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

In which

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

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

### PWL Duffing chaotic attractor

Aziz Alaoui investigated PWL Duffing equation in 2000:[11]

PWL Duffing system:

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

${\displaystyle {\frac {dy(t)}{dt))=-m_{1}x(t)-(1/2(m_{0}-m_{1}))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma \cos(\omega t)}$

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:[12]

${\displaystyle {\frac {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))}$${\displaystyle {\frac {1}{3{\sqrt {x(t)^{2}+y(t)^{2))))))$

${\displaystyle {\frac {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}))}$${\displaystyle {\frac {1}{3{\sqrt {x(t)^{2}+y(t)^{2))))))$

${\displaystyle {\frac {dz(t)}{dt))=1/2(3x(t)^{2}y(t)-y(t)^{3})-bz(t)}$

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

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

## References

1. ^ Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit" (PDF). IEEE Transactions on Circuits and Systems. IEEE. CAS-31 (12): 1055–1058. doi:10.1109/TCS.1984.1085459.
2. ^ Chua, Leon; Motomasa Komoru; Takashi Matsumoto (November 1986). "The Double-Scroll Family" (PDF). IEEE Transactions on Circuits and Systems. CAS-33 (11).
3. ^ Chua, Leon (2007). "Chua circuits". Scholarpedia. 2 (10): 1488. Bibcode:2007SchpJ...2.1488C. doi:10.4249/scholarpedia.1488.
4. ^ Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor". Scholarpedia. 2 (10): 1488. Bibcode:2007SchpJ...2.1488C. doi:10.4249/scholarpedia.1488.
5. ^ Leonov G.A.; Vagaitsev V.I.; Kuznetsov N.V. (2011). "Localization of hidden Chua's attractors" (PDF). Physics Letters A. 375 (23): 2230–2233. Bibcode:2011PhLA..375.2230L. doi:10.1016/j.physleta.2011.04.037.
6. ^ Chen G., Ueta T. Yet another chaotic attractor. Journal of Bifurcation and Chaos, 1999 9:1465.
7. ^ a b CHEN, GUANRONG; UETA, TETSUSHI (July 1999). "Yet Another Chaotic Attractor". International Journal of Bifurcation and Chaos. 09 (7): 1465–1466. Bibcode:1999IJBC....9.1465C. doi:10.1142/s0218127499001024. ISSN 0218-1274.
8. ^ 阎振亚著 《复杂非线性波的构造性理论及其应用》第17页 SCIENCEP 2007年
9. ^ a b c Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. Bibcode:2006IJBC...16..775L. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
10. ^ Chen, Guanrong; Jinhu Lu (2006). "Generating Multiscroll Chaotic Attractors: Theories, Methods and Applications" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 793–794. Bibcode:2006IJBC...16..775L. CiteSeerX 10.1.1.927.4478. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
11. ^ J. Lu, G. Chen p. 837
12. ^ J.Liu and G Chen p834