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 電路(一般而言,是蔡氏電路 )加上一個非線性 電阻(例如蔡氏二極體 )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。
陈氏系统:
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{\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=a*(y(t)-x(t)),}
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{\displaystyle {\frac {\mathrm {d} y(t)}{\mathrm {d} t}}=(c-a)*x(t)-x(t)*f c*y(t),}
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{\displaystyle {\frac {\mathrm {d} z(t)}{\mathrm {d} t}}=x(t)*y(t)-b*z(t)}
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{\displaystyle f}
为调控函数:[ 1]
51 frame N scroll modified Chen attractor x axe vs t
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{\displaystyle f=g*z(t)-h*\sin(z(t))}
参数:
:= a = 35, c = 28, b = 3, g = 1, h = -25..25;
初始条件:
initv := x(0) = 1, y(0) = 1, z(0) = 14;
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。
N scroll attractor based on Chen with sine and tau
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{\displaystyle f=d0*z(t) d1*z(t-\tau )-d2*\sin(z(t-\tau ))}
参数:
params := a = 35, c = 28, b = 3, d0 = 1, d1 = 1, d2 = -20..20, tau = .2;
初始条件:
initv := x(0) = 1, y(0) = 1, z(0) = 14;
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。
2001年Tang等提出改进的蔡氏吸引子系统:.[ 2]
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{\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=\alpha *(y(t)-h)}
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{\displaystyle {\frac {\mathrm {d} y(t)}{\mathrm {d} t}}=x(t)-y(t) z(t)}
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{\displaystyle {\frac {\mathrm {d} z(t)}{\mathrm {d} t}}=-\beta *y(t)}
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{\displaystyle h:=-b*sin({\frac {\pi *x(t)}{2*a}} d)}
参数:
params := alpha = 10.82, beta = 14.286, a = 1.3, b = .11, c = 7, d = 0;
初始条件:
initv := x(0) = 1, y(0) = 1, z(0) = 0;
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:
9 卷波 超混沌蔡氏吸引子
9 卷波 超混沌蔡氏吸引子
延龄草型混沌吸引子
1993年 Miranda & Stone 提出下列方程组:[ 3]
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{\displaystyle {\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))}
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{\displaystyle *{\frac {1}{3*{\sqrt {x(t)^{2} y(t)^{2}}}}}}
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{\displaystyle {\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}))}
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{\displaystyle *{\frac {1}{3*{\sqrt {x(t)^{2} y(t)^{2}}}}}}
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{\displaystyle {\frac {\mathrm {d} z(t)}{\mathrm {d} t}}=1/2*(3*x(t)^{2}*y(t)-y(t)^{3})-b*z(t)}
参数:
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{\displaystyle a=10,\quad b={\frac {8}{3}},\quad c={\frac {137}{5}}}
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{\displaystyle x(0)=-8,\quad y(0)=4,\quad z(0)=10}
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:
2000年Aziz Alaoui 提出 PWL Duffing 方程:[ 4] 。
PWL 杜芬方程:
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{\displaystyle {\frac {\mathrm {d} x(t)}{\mathrm {d} t}}=y(t)}
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{\displaystyle {\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)}
参数:
params := e = .25, gamma = .14 (1/20)*i, m0 = -0.845e-1, m1 = .66, omega = 1; c := (.14 (1/20)*i),i=-25..25;
初始条件:
initv := x(0) = 0, y(0) = 0;
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 Duffing chaotic attractor xy plot
PWL Duffing chaotic attractor plot
^ 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
^ Chen, Guanrong; Jinhu Lu. GENERATING MULTISCROLL CHAOTIC ATTRACTORS: THEORIES, METHODS AND APPLICATIONS (PDF) . International Journal of Bifurcation and Chaos. 2006, 16 (4): 793–794 [2012-02-16 ] . (原始内容存档 (PDF) 于2012-01-06).
^ J.Liu and G Chen p834
^ J.Lu et al p837