Multiscroll attractor

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.^[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 proprosed another double scroll chaotic attractor.^[6]

Chen system:

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

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

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

Plots of Chen attractor can be obtained with Runge-Kutta method：^[7]

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.^[8]

Lu Chen attractor

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

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

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

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

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

System equations:.^[9]

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

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

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

In which

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

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]

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

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

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

In which

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

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

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

PWL Duffing system:

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

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

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]

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

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

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

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

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

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.
↑ Chua, Leon; Motomasa Komoru; Takashi Matsumoto (November 1986). "The Double-Scroll Family" (PDF). IEEE Transactions Circuits and Systems. CAS-33 (11).
↑ Chua, Leon (2007). "Chua circuits". Scholarpedia. doi:10.4249/scholarpedia.1488.
↑ Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor". Scholarpedia. 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. doi:10.1016/j.physleta.2011.04.037.
↑ Chen G., Ueta T. Yet another chaotic attractor. Journal of Bifurcation and Chaos, 1999 9:1465.
↑ 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
↑ Chen, Guanrong; Jinhu Lu (2006). "GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
1 2 Jinhu Lu
↑ Chen, Guanrong; Jinhu Lu (2006). "GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 793–794. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
↑ J.Lu et al p837
↑ J.Liu and G Chen p834

Chaos theory

Chaos theory	Anosov diffeomorphism Bifurcation theory Butterfly effect Chaos theory in organizational development Complexity Control of chaos Dynamical system Edge of chaos Fractal Predictability Quantum chaos Santa Fe Institute Synchronization of chaos Unintended consequences

Chaotic maps (list)	Arnold tongue Arnold's cat map Baker's map Complex quadratic map Complex squaring map Coupled map lattice Double pendulum Double scroll attractor Duffing equation Duffing map Dyadic transformation Dynamical billiards outer Exponential map Gauss map Gingerbreadman map Hénon map Horseshoe map Ikeda map Interval exchange map Kaplan–Yorke map Logistic map Lorenz system Multiscroll attractor Rabinovich–Fabrikant equations Rössler attractor Standard map Swinging Atwood's machine Tent map Tinkerbell map Van der Pol oscillator Zaslavskii map

Chaos systems	Bouncing ball dynamics Chua's circuit Economic bubble Tilt-A-Whirl

Chaos theorists	Michael Berry Leon O. Chua Mitchell Feigenbaum Martin Gutzwiller Brosl Hasslacher Michel Hénon Edward Norton Lorenz Aleksandr Lyapunov Benoît Mandelbrot Henri Poincaré Otto Rössler David Ruelle Oleksandr Mykolaiovych Sharkovsky Floris Takens James A. Yorke

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Multiscroll attractor

Multiscroll attractors

Lu Chen attractor

Modified Lu Chen attractor

Modified Chua chaotic attractor

PWL Duffing chaotic attractor

Modified Lorenz chaotic system

See also

References