查看“︁多卷波混沌吸引子”︁的源代码

'''多卷波混沌吸引子'''（N scroll chaotic attractor）也称'''N卷波吸引子'''，是實際[[混沌理论|混沌]]電路（一般而言，是[[蔡氏電路]]）加上一個[[非線性]]電阻（例如{{le|蔡氏二極體|Chua's Diode}}）而產生的[[奇異吸引子]]。多卷波混沌吸引子可以用三個非線性[[常微分方程]]以及三段的片段連續線性方程來描述。這可以簡化系統的數值模擬，也因為蔡氏電路的設計簡單，也很容易實作。

多卷波混沌吸引子在保密数码通讯，同步预测等方面有重要应用。

==超混沌陈氏吸引子==


陈氏系统：
<math>\frac{\mathrm{d}x(t)}{\mathrm{d}t}=a*(y(t)-x(t)), </math>

<math>\frac{\mathrm{d}y(t)}{\mathrm{d}t}=(c-a)*x(t)-x(t)*f+c*y(t), </math>

<math>\frac{\mathrm{d}z(t)}{\mathrm{d}t}=x(t)*y(t)-b*z(t)</math>

其中 <math>f</math> 为调控函数：<ref>XINZHI LIU   MULTI-SCROLL CHAOTIC AND HYPERCHAOTIC ATTRACTORS GENERATED FROM CHEN SYSTEM, International Journal of Bifurcation and Chaos, Vol. 22, No. 2 (2012) 1250033-2</ref>
===正弦调控函数===
[[File:51 frame N scroll modified Chen attractor.gif]]
[[File:51 frame N scroll modified Chen attractor x axe vs time.gif|thumb|500px|51 frame N scroll modified Chen attractor x axe vs t]]

<math>f=g*z(t)-h*\sin(z(t))</math>

参数：<syntaxhighlight>:= a = 35, c = 28, b = 3, g = 1, h = -25..25;</syntaxhighlight>

初始条件：<syntaxhighlight>initv := x(0) = 1, y(0) = 1, z(0) = 14;</syntaxhighlight>

利用[[Maple]]中{{le|龙格－库塔-菲尔伯格法|Runge–Kutta–Fehlberg method}}（Runge–Kutta–Fehlberg法，简称 RKF45）可得数字解并做图。

{| class="wikitable"
|-
! h !! 卷波数
|-
| 5 || 4
|-
| 8 || 6
|-
| 22 || 14
|-
|}

===延时正弦函数===
[[File:N scroll generalized Chen attractor 41 frames.gif|400px|thumb|right|N scroll attractor based on Chen with sine and tau]]

<math> f = d0*z(t) + d1*z(t -  \tau ) - d2*\sin(z(t - \tau ))</math>

参数：<syntaxhighlight>params := a = 35, c = 28, b = 3, d0 = 1, d1 = 1, d2 = -20..20, tau = .2;</syntaxhighlight>

初始条件：<syntaxhighlight>initv := x(0) = 1, y(0) = 1, z(0) = 14;</syntaxhighlight>

利用[[Maple]]中[[龙格－库塔-菲尔伯格法]]（Runge–Kutta–Fehlberg法，简称 RKF45）可得数字解并做图。

==超混沌蔡氏吸引子==
2001年Tang等提出改进的蔡氏吸引子系统：.<ref>{{cite journal|last=Chen|first=Guanrong|coauthors=Jinhu Lu|title=GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS|journal=International Journal of Bifurcation and Chaos|year=2006|volume=16|issue=4|pages=793-794|url=http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf|accessdate=2012-02-16|archive-date=2012-01-06|archive-url=https://web.archive.org/web/20120106122526/http://www.ee.cityu.edu.hk/~gchen/pdf/LC-IJBC06-survey.pdf|dead-url=no}}</ref>


<math>\frac{\mathrm{d}x(t)}{\mathrm{d}t}= \alpha*(y(t)-h)</math>

<math>\frac{\mathrm{d}y(t)}{\mathrm{d}t}=x(t)-y(t)+z(t)</math>

<math>\frac{\mathrm{d}z(t)}{\mathrm{d}t}=-\beta*y(t)</math>

其中

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

参数：<syntaxhighlight>params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0;</syntaxhighlight>

初始条件：<syntaxhighlight>initv := x(0) = 1, y(0) = 1, z(0) = 0;</syntaxhighlight>

利用[[Maple]]中[[龙格－库塔-菲尔伯格法]]（Runge–Kutta–Fehlberg法，简称 RKF45）可得数字解并做图：

[[File:9 scroll modified Chua attractor.png|thumb|center|500px|alt=9  scroll|9 卷波 超混沌蔡氏吸引子]]
[[File:9 scroll modified Chua attractor xt plot.png|thumb|center|500px|9 卷波 超混沌蔡氏吸引子]]

==延龄草型混沌吸引子==
[[File:Trillium attractor.png|thumb|right|300px|延龄草型混沌吸引子]]

1993年 Miranda & Stone 提出下列方程组：<ref>J.Liu and G Chen p834</ref>

<math>\frac{\mathrm{d}x(t)}{\mathrm{d}t} = 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>*\frac{1}{3*\sqrt{x(t)^2+y(t)^2}}</math>

<math>\frac{\mathrm{d}y(t)}{\mathrm{d}t}= 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>*\frac{1}{3*\sqrt{x(t)^2+y(t)^2}}</math>

<math>\frac{\mathrm{d}z(t)}{\mathrm{d}t} = 1/2*(3*x(t)^2*y(t)-y(t)^3)-b*z(t)</math>

参数：<math> a = 10,\quad b = \frac{8}{3}, \quad c = \frac{137}{5}</math>

初始条件：<math>x(0) = -8, \quad y(0) = 4, \quad z(0) = 10</math>

利用[[Maple]]中[[龙格－库塔-菲尔伯格法]]（Runge–Kutta–Fehlberg法，简称 RKF45）可得数字解并做图：

==PWL 杜芬混沌吸引子==
2000年Aziz Alaoui 提出 PWL Duffing 方程：<ref>J.Lu et al p837</ref>。

PWL 杜芬方程：

<math>\frac{\mathrm{d}x(t)}{\mathrm{d}t}=y(t)</math>

<math>\frac{\mathrm{d}y(t)}{\mathrm{d}t}=-m1*x(t)-(1/2*(m0-m1))*(|x(t)+1|-|x(t)-1|)-e*y(t)+\gamma*cos(\omega*t)</math>

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

初始条件：<syntaxhighlight>initv := x(0) = 0, y(0) = 0;</syntaxhighlight>

利用[[Maple]]中[[龙格－库塔-菲尔伯格法]]（Runge–Kutta–Fehlberg法，简称 RKF45）可得数字解并做图：

[[File:PWL Duffing chaotic attractor xy plot.gif|thumb|center|300px|PWL Duffing chaotic attractor xy plot]]
[[File:PWL Duffing chaotic attractor plot.gif|thumb|300px|center|PWL Duffing chaotic attractor plot]]

==参考文献==
{{reflist}}

==外部連結==
*[http://www.chuacircuits.com/howtobuild4.php The double-scroll attractor and Chua's circuit] {{Wayback|url=http://www.chuacircuits.com/howtobuild4.php |date=20200217183104 }}
* {{cite journal
 |last1        = Lozi
 |first1       = R.
 |last2        = Pchelintsev
 |first2       = A.N.
 |title        = A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case
 |journal      = International Journal of Bifurcation and Chaos
 |year         = 2015
 |volume       = 25
 |issue        = 13
 |pages        = 1550187
 |doi          = 10.1142/S0218127415501874
 |url          = https://hal.archives-ouvertes.fr/hal-01323625/document
 |ref          = harv
 |access-date  = 2020-10-08
 |archive-date = 2019-05-02
 |archive-url  = https://web.archive.org/web/20190502151350/https://hal.archives-ouvertes.fr/hal-01323625/document
 |dead-url     = no
}}

{{吸引子}}
{{混沌理论}}

[[category:非线性常微分方程]]
[[category:混沌理论]]
[[Category:吸子]]