陈氏吸引子

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文件:DoubleScrollAttractor3D.svg
Double-scroll attractor from a simulation

Double-scroll attractor from a simulation

一个模拟的双涡卷吸引子


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.

在动力系统的数学中,双涡卷吸引子(有时称为蔡氏吸引子)是一个奇怪的吸引子,观察到一个物理电子混沌电路(一般称为蔡氏电路)和一个非线性电阻(见蔡氏二极管)。双涡卷系统通常由三个非线性常微分方程组和一个三段分段线性方程组组成。由于蔡氏电路设计简单,使得该系统易于进行数值模拟和物理实现。


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.

使用蔡氏电路,这种形状是在示波器上观察使用的 x,y,和 z 输出信号的电路。这个混沌吸引子被称为双涡卷,因为它的形状是三维空间,类似于由旋转线连接的两个类似土星的环。


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.

该吸引子首先在模拟中被观察到,然后在蔡明亮发明了自治混沌电路后物理实现,这就是蔡明亮电路。利用三维状态空间的特征向量的组合明确地导出了 Chua 电路的双涡卷吸引子的庞加莱回归映射,严格地证明了该吸引子是混沌的。


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

参数 a = 40,c = 28,b = 3


[math]\displaystyle{ \frac{dx(t)}{dt}=a(y(t)-x(t)) }[/math]

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

初始条件: x (0) =-0.1,y (0) = 0.5,z (0) =-0.6


[math]\displaystyle{ \frac{dy(t)}{dt}=(c-a)x(t)-x(t)z(t)+cy(t) }[/math]


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.

多涡卷吸引子又称 n 涡卷吸引子,包括 Lu Chen 吸引子、修正 Chen 混沌吸引子、 PWL Duffing 吸引子、 Rabinovich Fabrikant 吸引子、修正 Chua 混沌吸引子,即在单个吸引子中多个涡卷。

[math]\displaystyle{ \frac{dz(t)}{dt}=x(t)y(t)-bz(t) }[/math]


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

Lu Chen Attractor

陆晨吸引子


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


[math]\displaystyle{ \frac{dx(t)}{dt}= \alpha (y(t)-h) }[/math]

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

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


[math]\displaystyle{ \frac{dy(t)}{dt}=x(t)-y(t)+z(t) }[/math]

= 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]

[math]\displaystyle{ \frac{dz(t)}{dt}=-\beta y(t) }[/math]

=-beta y (t) </math >


Lu Chen attractor

In which

在其中

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

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

[ 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


[math]\displaystyle{ \frac{dx(t)}{dt}=a(y(t)-x(t)) }[/math]

initv := x(0) = 1, y(0) = 1, z(0) = 0

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


[math]\displaystyle{ \frac{dy(t)}{dt}=x(t)-x(t)z(t)+cy(t)+u }[/math]


PWL Duffing Attractor

杜芬吸引子

[math]\displaystyle{ \frac{dz(t)}{dt}=x(t)y(t)-bz(t) }[/math]

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:

工务试验所杜芬系统:


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

[math]\displaystyle{ \frac{dx(t)}{dt}=y(t) }[/math]

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


Modified Lu Chen attractor

[math]\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]

< 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 >

文件:LuChenAttractorModified3D.svg
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;

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


[math]\displaystyle{ \frac{dx(t)}{dt}=a(y(t)-x(t)), }[/math]

initv := x(0) = 0, y(0) = 0;

Initv: = x (0) = 0,y (0) = 0;


[math]\displaystyle{ \frac{dy(t)}{dt}=(c-a)x(t)-x(t)f+cy(t), }[/math]


Lorenz systemmodified

对 Lorenz 系统进行了改进

[math]\displaystyle{ \frac{dz(t)}{dt}=x(t)y(t)-bz(t) }[/math]

Miranda & Stone proposed a modified Lorenz system:

米兰达和斯通提出了一个改进的洛伦兹系统:


In which

[math]\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)) }[/math][math]\displaystyle{ \frac{1}{3\sqrt{x(t)^2+y(t)^2}} }[/math]

< 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 >


[math]\displaystyle{ f = d0z(t) + d1z(t - \tau ) - d2\sin(z(t - \tau )) }[/math]

[math]\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)) }[/math][math]\displaystyle{ \frac{1}{3\sqrt{x(t)^2+y(t)^2}} }[/math]

(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

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

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;

参数 a = 10,b = 8/3,c = 137/5;


Modified Chua chaotic attractor

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

初始条件: x (0) =-8,y (0) = 4,z (0) = 10

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


{{Gallery

{画廊

[math]\displaystyle{ \frac{dx(t)}{dt}= \alpha (y(t)-h) }[/math]

|title=

2012年10月11日


|width=160 | height=170

160 | height = 170

[math]\displaystyle{ \frac{dy(t)}{dt}=x(t)-y(t)+z(t) }[/math]

|align=center

| align = center


|footer=

2012年10月22日 | footer =

[math]\displaystyle{ \frac{dz(t)}{dt}=-\beta y(t) }[/math]

|File:Maple plot Chen Attractor.jpg

文件: Maple plot Chen Attractor.jpg


|alt1=Chen attractor

1 = Chen attractor

In which

|Chen attractor

陈吸引子


|File:Chen chaos attractor plot.png

文件: Chen chaos attractor plot.png

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

|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

文件: Lu Chen chaotic attraction. 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

文件: 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

文件:PWLDuffingAttractor.svg
PWL Duffing Attractor
|Maple plot of N scroll attractor based on Chen with sine and tau

基于正弦和 τ 的 Chen 的 n 个涡卷吸引子的枫树图

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

[math]\displaystyle{ \frac{dx(t)}{dt}=y(t) }[/math]

|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

[math]\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]

|File:PWL Duffing chaotic attractor xy plot.gif

文件 | 文件: PWL Duffing 混沌吸引子 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

杜芬混沌吸引子 xy 图


|File:PWL Duffing chaotic attractor plot.gif

文件: 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

杜芬混沌吸引子图

Modified Lorenz chaotic system

|File:Trillium attractor.png

文件: Trillium attractor.png

文件:LorenzModified3D.svg
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

文件: Rabinovich Fabricant xy plot 0.15. png

[math]\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)) }[/math][math]\displaystyle{ \frac{1}{3\sqrt{x(t)^2+y(t)^2}} }[/math]

|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

[math]\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)) }[/math][math]\displaystyle{ \frac{1}{3\sqrt{x(t)^2+y(t)^2}} }[/math]

}}

}}


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


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.

模板:Gallery

}}

|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

范畴: 混沌理论


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. 7.0 7.1 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. 9.0 9.1 9.2 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