Multiscroll attractor

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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]

Chen attractor[edit]

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

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[edit]

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[edit]

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[edit]

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

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

Modifed Chua chaotic attractor[edit]

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[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[edit]

PWL Duffing chaotic attractor xy plot
PWL Duffing chaotic attractor plot

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[edit]

modified Lorenz attractor

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

Rabinovich Fabrikant attractor xy plot

See also[edit]

References[edit]

  1. ^ Matsumoto, Takashi (December 1984). "A Chaotic Attractor from Chua's Circuit". IEEE Transactions on Circuits and Systems (IEEE). CAS-31 (12): 1055–1058. 
  2. ^ Chua, Leon; Motomasa Komoru, Takashi Matsumoto, (November 1986). "The Double-Scroll Family". IEEE Transactions Circuits and Systems. CAS-33 (11). 
  3. ^ Chua, Leon (2007). "Chua circuits". Scholarpedia. doi:10.4249/scholarpedia.1488. 
  4. ^ Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor". Scholarpedia. doi:10.4249/scholarpedia.1488. 
  5. ^ Leonov G.A., Vagaitsev V.I., Kuznetsov N.V. (2011). "Localization of hidden Chua's attractors". Physics Letters A 375 (23): 2230–2233. 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. ^ 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
  8. ^ Chen, Guanrong; Jinhu Lu (2006). "GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS". International Journal of Bifurcation and Chaos 16 (4): 775–858. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. 
  9. ^ a b Jinhu Lu
  10. ^ Chen, Guanrong; Jinhu Lu (2006). "GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS". International Journal of Bifurcation and Chaos 16 (4): 793–794. doi:10.1142/s0218127406015179. Retrieved 2012-02-16. 
  11. ^ J.Lu et al p837
  12. ^ J.Liu and G Chen p834