陈氏吸引子

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.

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.

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]

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. The double-scroll attractor from the Chua circuit was rigorously proven to be chaotic 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.

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.

In 1999 Guanrong Chen (陈关荣) and Ueta proposed another double scroll chaotic attractor, called the Chen system or Chen attractor.

In 1999 Guanrong Chen (陈关荣) and Ueta proposed another double scroll chaotic attractor, called the Chen system or Chen attractor.[6][7]

The Chen system is defined as follows

Chen 系统的定义如下

Chen attractor

The Chen system is defined as follows[7]

parameters: a = 40, c = 28, b = 3

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

initial conditions: x(0) = -0.1, y(0) = 0.5, z(0) = -0.6

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

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.

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

Plots of Chen attractor can be obtained with the Runge-Kutta method:[8]

Lu Chen Attractor

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

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

parameters: a = 40, c = 28, b = 3

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

= alpha (y (t)-h) </math >

initial conditions: x(0) = -0.1, y(0) = 0.5, z(0) = -0.6

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

= x (t)-y (t) + z (t) </math

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]

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

=-beta y (t) </math >

Lu Chen attractor

In which

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

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

[ math > h: =-b sin left (frac { pi x (t)}{2a } + d right) </math >

Lu Chen system equation

params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0

10.82，beta = 14.286，a = 1.3，b = . 11，c = 7，d = 0

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

initv := x(0) = 1, y(0) = 1, z(0) = 0

Initv: = x (0) = 1，y (0) = 1，z (0) = 0

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

PWL Duffing Attractor

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

Aziz Alaoui investigated PWL Duffing equation in 2000:

2000年，Aziz Alaoui 调查了 PWL 杜芬振子:

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

PWL Duffing system:

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

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

[ math > frac { dx (t)}{ dt } = y (t) </math ]

Modified Lu Chen attractor

$\displaystyle{ \frac{dy(t)}{dt}=-m_1x(t)-(1/2(m_0-m_1))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma \cos(\omega t) }$

< math > frac { dy (t)}{ dt } =-m _ 1x (t)-(1/2(m _ 0-m _ 1)(| x (t) + 1 |-| x (t)-1 |)-ey (t) + gamma cos (omega t) </math >

Lu Chen Attractor modified

System equations:[9]

params := e = .25, gamma = .14+(1/20)i, m0 = -0.845e-1, m1 = .66, omega = 1; c := (.14+(1/20)i)，i=-25...25;

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

initv := x(0) = 0, y(0) = 0;

Initv: = x (0) = 0，y (0) = 0;

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

Lorenz systemmodified

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

Miranda & Stone proposed a modified Lorenz system:

In which

$\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}} }$

< math > 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)) </math > frac {1}{ x (t) ^ 2 + y (t)2} </math >

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

$\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}} }$

(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))(x (t) ^ 2-y (t) ^ 2)) </math > frac {1}{ x (t) ^ 2 + y (t)2} </math > frac {1}{ x (t) ^ 2 + y (t)2} </math >

params := a = 35, c = 28, b = 3, d0 = 1, d1 = 1, d2 = -20..20, tau = .2

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

1/2(3x (t) ^ 2y (t)-y (t) ^ 3)-bz (t) </math >

initv := x(0) = 1, y(0) = 1, z(0) = 14

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

Modified Chua chaotic attractor

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

In 2001, Tang et al. proposed a modified Chua chaotic system[10]

{{Gallery

{画廊

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

|title=

2012年10月11日

|width=160 | height=170

160 | height = 170

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

|align=center

| align = center

|footer=

2012年10月22日 | footer =

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

|File:Maple plot Chen Attractor.jpg

|alt1=Chen attractor


1 = Chen attractor

In which

|Chen attractor


|File:Chen chaos attractor plot.png

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

|alt2=a = 35, c = 27, b = 2.8,x(0) = -.1, y(0) = .3, z(0) = -.6


| alt2 = a = 35，c = 27，b = 2.8，x (0) =-. 1，y (0) = . 3，z (0) =-. 6

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


| parameters < nowiki > a = 35，c = 27，b = 2.8，x (0) =-. 1，y (0) = . 3，z (0) =-. 6

params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0

|File:Lu Chen chaotic attractor.gif

|alt3=Lu Chen attractor


3 = Lu Chen attractor

initv := x(0) = 1, y(0) = 1, z(0) = 0

|Lu Chen attractor


| Lu Chen attractor

|File:N scroll generalized Chen attractor 41 frames.gif

PWL Duffing chaotic attractor

|alt4=Maple plot of N scroll attractor based on Chen with sine and tau


| alt4 = 基于 Chen 和正弦和 τ 的 n 个涡卷吸引子的枫树图

PWL Duffing Attractor
|Maple plot of N scroll attractor based on Chen with sine and tau


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

|File:9 scroll modified Chua attractor.png

9 scroll modified Chua attractor.png

|alt5 = 9 scroll modified Chua chaotic attractor


| alt5 = 9涡卷修正蔡氏混沌吸引子

PWL Duffing system:

|9 scroll modified Chua chaotic attractor


| 9涡卷修正蔡氏混沌吸引子

|File:9 scroll modified Chua attractor xt plot.png

9 scroll modified Chua attractor xt plot.png

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

|alt6 = xy plot of 9 scroll modified Chua chaotic attractor


| alt6 = 9涡卷修正蔡氏混沌吸引子的 xy 图

|xy plot of 9 scroll modified Chua chaotic attractor


9涡卷修正蔡氏混沌吸引子 | xy 图

$\displaystyle{ \frac{dy(t)}{dt}=-m_1x(t)-(1/2(m_0-m_1))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma \cos(\omega t) }$

|File:PWL Duffing chaotic attractor xy plot.gif

|alt7 = PWL Duffing chaotic attractor xy plot


7 = PWL Duffing 混沌吸引子 xy plot

params := e = .25, gamma = .14+(1/20)i, m0 = -0.845e-1, m1 = .66, omega = 1; c := (.14+(1/20)i)，i=-25...25;

|PWL Duffing chaotic attractor xy plot


|File:PWL Duffing chaotic attractor plot.gif

initv := x(0) = 0, y(0) = 0;

|alt8 = PWL Duffing chaotic attractor plot


8 = PWL Duffing 混沌吸引子图

|PWL Duffing chaotic attractor plot


Modified Lorenz chaotic system

|File:Trillium attractor.png

Lorenz systemmodified
|alt9 = modified Lorenz attractor


9 = 修正的洛伦兹吸引子

Miranda & Stone proposed a modified Lorenz system:[12]

|modified Lorenz attractor


| 修正的 Lorenz 吸引子

|File:Rabinovich Fabricant xy plot 0.15.png

$\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}} }$

|alt10 = Rabinovich Fabrikant attractor xy plot


10 = Rabinovich Fabrikant attractor xy plot

|Rabinovich Fabrikant attractor xy plot


| Rabinovich Fabrikant 吸引子 xy 阴谋

$\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}} }$

}}

}}

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

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

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

Gallery

| last1=Lozi | first1=R.

1 = Lozi | first1 = r.

}}

|alt1=Chen attractor

|Chen attractor


|File:Chen chaos attractor plot.png

|alt2=a = 35, c = 27, b = 2.8,x(0) = -.1, y(0) = .3, z(0) = -.6


Category:Chaos theory

This page was moved from wikipedia:en:Multiscroll attractor. Its edit history can be viewed at 陈氏吸引子/edithistory

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". Physics Letters A. 375 (23): 2230–2233. Bibcode:2011PhLA..375.2230L. doi:10.1016/j.physleta.2011.04.037. Unknown parameter |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. Recently, there has also been reported the discovery of hidden attractors within the double scroll. 双涡卷吸引子的数值分析表明，它的几何结构是由无穷多个分形层构成的。每一个横截面在所有尺度上都是不规则的。最近，也有报道发现隐藏吸引子内的双涡卷。 url= ignored (help)
6. Chen G., Ueta T. Yet another chaotic attractor. Journal of Bifurcation and Chaos, 1999 9:1465.
7. 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.
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. 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. doi:10.1142/s0218127406015179. Retrieved 2012-02-16.
11. J. Lu, G. Chen p. 837
12. J.Liu and G Chen p834