Multiscroll attractor

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.

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]

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.^{[6] [7]}

The Chen system is defined as follows^[7]

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

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

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

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

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

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

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]

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 {dx(t)}{dt}}=a(y(t)-x(t))}

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

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

System equations:^[9]

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

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

In which

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

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

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

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

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

In which

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

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

PWL Duffing system:

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

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

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 {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 {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 {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 {1}{3{\sqrt {x(t)^{2}+y(t)^{2}}}}}}

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

• The double-scroll attractor and Chua's circuit
• Lozi, R.; Pchelintsev, A.N. (2015). "A new reliable numerical method for computing chaotic solutions of dynamical systems: the Chen attractor case". International Journal of Bifurcation and Chaos. 25 (13): 1550187–1550412. Bibcode:2015IJBC...2550187L. doi:10.1142/S0218127415501874. S2CID 12339358.

• Chaos theory

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