To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Multiscroll attractor

From Wikipedia, the free encyclopedia

Double-scroll attractor from a simulation
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 proposed another double scroll chaotic attractor.[6]

Chen system:

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

LuChenAttractor3D.svg

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

LuChenAttractorModified3D.svg

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

ChuaAttractorModified.svg

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

PWLDuffingAttractor.svg

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

LorenzModified3D.svg

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

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.
  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. Bibcode:2007SchpJ...2.1488C. doi:10.4249/scholarpedia.1488.
  4. ^ Chua, Leon (2007). "Fractal Geometry of the Double-Scroll Attractor". Scholarpedia. 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. ^ 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" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 775–858. Bibcode:2006IJBC...16..775L. 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" (PDF). International Journal of Bifurcation and Chaos. 16 (4): 793–794. Bibcode:2006IJBC...16..775L. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
  11. ^ J.Lu et al p837
  12. ^ J.Liu and G Chen p834

External links

This page was last edited on 10 May 2019, at 04:32
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.