Flows with Strange Attractors

Select Attractor

Lorenz Strange Attractor

dx/dt = σ(y - x)
dy/dt = x(ρ - z) - y
dz/dt = xy - βz

σ = 10
ρ = 28
β = 8/3

Rössler Attractor

dx/dt = - y - z
dy/dt = x + ay
dz/dt = b + z(x - c)

a = 0.1
b = 0.1
c = 14

Hénon Map

x_n+1 = y_n + 1 - ax_n²
y_n+1 = bx_n
(This is a 2-D attractor, so there is no z coordinate, technically)

a = 1.4
b = 0.3

Chua's Attractor (Double Scroll Attractor)

dx/dt = α(y - x - f(x))
dy/dt = x - y + z
dz/dt = - βy
f(x) = m₁x + (m₀-m₁)/2 * |x+1| - |x-1|)

The equation for f(x) was obtained from matlab code provided here

α = 15.6
β = 28
m₀ = -1.143
m₁ = -0.714

Liénard-Van der Pol Oscillator

dx/dt = y
dy/dt = μ(1 - x²)y - x

This is a 2-D oscillator. I showed flow for various random values of μ. It also looks nice if you set z = μ

Rabinovich-Fabrikant System

dx/dt = y(z - 1 + x²) + γx
dy/dt = x(3z + 1 - x²) + γy
dz/dt = - 2z(α + xy)

α = 0.98
γ = 0.1

Tinkerbell Map

x_n+1 = x_n² - y_n² + ax_n + by_n
y_n+1 = 2x_ny_n + cx_n + dy_n
(This is a 2-D dynamical system, so there is no z coordinate, technically)

a = 0.9
b = -0.6013
c = 2
d = 0.5

Ikeda Map

x_n+1 = 1 + u(x_ncos(t_n) - y_nsin(t_n))
y_n+1 = u(x_nsin(t_n) + y_ncos(t_n))
t_n = 0.4 - 6/(1 + x_n² + y_n²)

(This is a 2-D dynamical system, so there is no z coordinate, technically)

u = 0.85

Rainey Dynamical System

dx/dt = yz
dy/dt = xz
dz/dt = xy

Chris Rainey is a former REU/illiMATH student who made up his own set of ODEs for a dynamical system

Fu Attractor

dx/dt = α(y + x)/z
dy/dt = x(β - x - y - z)
dz/dt = xyz/γ

α = 18
β = 42
γ = 15

My own creation (a reward for hard work)