Fractal Home
Mira- and Martin Attractors
The machine on this page presents two different types of strange attractors, both sorts of orbit-fractals with a two-dimensional iteration systems: The Mira attractor and the Martin attractor, also known as Hopalong. Hopalong orbits were discovered by Barry Martin from the Aston University, Birmingham, England. The name Hopalong is derived from the fact, that such an image is built of points hopping along on an elliptical path starting from one point in the center.
The Mira attractor, also known as the Gumowski–Mira attractor, originates from a nonlinear two-dimensional recurrence relation introduced by I. Gumowski and C. Mira in the 1970s. In contrast to Hopalong orbits, its dynamics are dominated by algebraic nonlinearities, leading to highly structured, filament-like patterns with pronounced symmetry.
Both system are classic examples of controlled deterministic chaos.
Mira & Martin Combo Attractor Machine
Mira Attractor Formulas
\[ \begin{align*} x_{n+1} &= b \, y_n + \Bigl[ a \, x_n - (1-a)\cdot \frac{2 x_n^2}{1 + x_n^2} \Bigr] \\[1mm] y_{n+1} &= -x_n + \Bigl[ a \, x_{n+1} - (1-a)\cdot \frac{2 x_{n+1}^2}{1 + x_{n+1}^2} \Bigr] \end{align*} \]
Martin Attractor (Hopalong) Formulas
\[ \begin{align*} x_{n+1} &= \begin{cases} y_n - \sqrt{|b x_n - c|}, & x_n > 0 \\[1mm] y_n + \sqrt{|b x_n - c|}, & x_n < 0 \\[1mm] y_n, & x_n = 0 \end{cases} \\[1mm] y_{n+1} &= a - x_n \end{align*} \]
Use the Combo machine by toggling the type or changing the parameter slides at the bottom of the attractor canvas. The new figure is drawn immediately. The figures depend sensitively on the numerals behind the decimal point.
The color is changed every 10000 dots, until the 50 predefined colors are used, then the colors are repeated.