# Multiscroll attractor

*https://en.wikipedia.org/wiki/Multiscroll_attractor*

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]}

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

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]}

In which

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]}

In which

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:

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]}

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

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

## Gallery

## References

**^**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.**^**Chua, Leon; Motomasa Komoru; Takashi Matsumoto (November 1986). "The Double-Scroll Family" (PDF).*IEEE Transactions on Circuits and Systems*. CAS-33 (11).**^**Chua, Leon (2007). "Chua circuits".*Scholarpedia*.**2**(10): 1488. Bibcode: 2007SchpJ...2.1488C. doi: 10.4249/scholarpedia.1488.**^**Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor".*Scholarpedia*.**2**(10): 1488. Bibcode: 2007SchpJ...2.1488C. doi: 10.4249/scholarpedia.1488.**^**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.**^**Chen G., Ueta T. Yet another chaotic attractor. Journal of Bifurcation and Chaos, 1999 9:1465.- ^
^{a}^{b}CHEN, GUANRONG; UETA, TETSUSHI (July 1999). "Yet Another Chaotic Attractor".*International Journal of Bifurcation and Chaos*.**09**(7): 1465–1466. doi: 10.1142/s0218127499001024. ISSN 0218-1274. **^**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- ^
^{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. **^**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.**^**J. Lu, G. Chen p. 837**^**J.Liu and G Chen p834

## External links

- The double-scroll attractor and Chua's circuit
- Lozi, R.; Pchelintsev, A.N. (2015).
"A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case".
*International Journal of Bifurcation and Chaos*.**25**(13): 1550187. doi: 10.1142/S0218127415501874.