Strange Attractors An attractor is a set of points described by a dynamical system. Some attractors exhibit chaotic behavior -- these strange attractors are often visually striking and intricate despite being described by simple mathematical expressions. Here are some colored visualizations, with brighter values indicating that the system spent more time in that area. Source code can be found here. This work is inspired partly by work done by Paul Bourke. back to home
 Peter de Jong Attractor $$\begin{eqnarray} x_{t + 1} = \sin{(a y_t)} - \cos{(b x_t)} \\ y_{t + 1} = \sin{(c x_t)} - \cos{(d y_t)} \end{eqnarray}$$ $$a=-0.709, b=1.638, c=0.452, d=1.740$$
 Thomas' Cyclically Symmetric Attractor $$\begin{eqnarray} \frac{dx}{dt} &=& \sin{(y)} - bx \\ \frac{dy}{dt} &=& \sin{(z)} - by \\ \frac{dz}{dt} &=& \sin{(x)} - bz \\ \end{eqnarray}$$ $$b = 0.208186$$
 Generalized Chua's Circuit $$\begin{eqnarray} \frac{dx}{dt} &=& \alpha (y - h(x)) \\ \frac{dy}{dt} &=& x - y + z \\ \frac{dz}{dt} &=& -\beta y - \gamma z \end{eqnarray}$$ $$\displaystyle h(x) = m_{2n - 1}x + \frac{1}{2}\sum_{k=1}^{2n-1}(m_{k-1} - m_k)(|x + b_k| - |x - b_k|)$$ For the 3-Double scroll attractor pictured: $$\alpha=9, \beta=14.286, \gamma=0$$ $$m_0 = \frac{-1}{7}, m_1 = m_3 = m_5 = \frac{2}{7}, m_2 = m_4 = \frac{-4}{7}$$ $$b_1 = 1, b_2 = 2.15, b_3 = 3.6, b_4 = 8.2, b_5 = 13$$
 Aizawa Attractor $$\begin{eqnarray} \frac{dx}{dt} &=& (z - \beta) x - \delta y \\ \frac{dy}{dt} &=& \delta x - (z - \beta) y \\ \frac{dz}{dt} &=& \gamma + \alpha z - \frac{z^3}{3} - (x^2+y^2)(1 + \epsilon z) + \zeta z x^3 \end{eqnarray}$$ $$\alpha=0.95, \beta=0.7, \gamma=0.65, \delta=3.5, \epsilon=0.25, \zeta=0.1$$
 Clifford Attractor $$\begin{eqnarray} x_{t + 1} &=& \sin{(a y_t)} + c \cos{(a x_t)} \\ y_{t + 1} &=& \sin{(b x_t)} + d \cos{(b y_t)} \\ \end{eqnarray}$$ $$a=-1.7, b=1.8, c=-1.9, d=-0.4$$

Pseudocode for Visualization

            set initial conditions of system
initialize 2D histogram for axes of choice
until desired:
propagate system dynamics
increment system state in histogram
for bin in 2D histogram:
create pixel according to bin value
display image