Multiscroll attractor

Lu Chen Attractor

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

Lu Chen system equation

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

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

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

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

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

Lu Chen Attractor modified

System equations:[9]

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

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

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

In which

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

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

Chua Attractor

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

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

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

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

In which

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

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 Attractor

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

PWL Duffing system:

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

${\displaystyle {\frac {dy(t)}{dt}}=-m_{1}x(t)-(1/2(m_{0}-m_{1}))(|x(t)+1|-|x(t)-1|)-ey(t)+\gamma \cos(\omega t)}$

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;

Lorenz systemmodified

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

${\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}}}}}}$

${\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}}}}}}$

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

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

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