This explorable illustrates the structure and beauty of strange attractors of two-dimensional discrete maps. These maps generate sequences of pairs of number $(x_n,y_n)$ where the index $n=0,1,2,…$ denotes the step of the iteration process that starts at the point $(x_0,y_0)$. The map is defined by two functions $f(x,y)$ and $g(x,y)$ that determine the point $(x_{n+1},y_{n+1})$ given $(x_{n},y_{n})$:

$$ x_{n+1}=f(x_{n},y_{n}) $$

$$y_{n+1}=g(x_{n},y_{n})$$

The orbit of such a map is the sequence of points $(x_n,y_n)$ in the plane.

Attractors

One is usually interested in the asymptotics of the map, so the behavior as $n\rightarrow\infty$. The orbits can, for example, approach a fixpoint (a single point in the plane), they can approach a limit cycle, a set of points that repeat after some number of iterations, they can also be quasi-periodic or they can approach something more complex, a strange attractor.

This is what is observed in the above examples. For each of the systems, the explorable iterates the map 200000 times and the attractor is shown in the display panel. Each of the example maps has 2-4 parameters that you can vary.

You can use the sliders to vary the parameters for each system and preset the reset button for their default values.

Clifford Attractor

The Clifford attractor is generated by the map

$$ x_{n+1}=\sin (\alpha , y_n)+ \gamma \cos(\alpha, x_n) $$

$$ y_{n+1}=\sin (\beta , x_n)+ \delta \cos(\beta, y_n) $$

with four parameters, discovered by and named after Clifford A. Pickover.

Bedhead Attractor

This one is generated by the map

$$ x_{n+1}=y_n \sin (x_n y_n/\beta)+ \cos(\alpha x_n-y_n) $$

$$ y_{n+1}=x_n + \sin (y_n)/\beta $$

and has only two parameters.

Fractal Dream Attractor

This attractor is generated by a map similar to the Clifford attractor.

$$ x_{n+1}= \sin (\beta, y_n)+ \gamma, \sin(\beta , x_n) $$

$$ y_{n+1}=\sin (\alpha, x_n)+ \delta , \sin(\alpha , y_n) $$

It has a bit more symmetry and four parameters.

Gumowski-Mira Attractor

The map for this one looks a bit more complicated (see below) but has only two parameters. However, it can generate a very rich set of attractor shapes that are reminiscent of shapes and geometries observed in natural systems. See for yourself. The underlying map is defined by:

$$ x_{n+1}= \beta y_n+f(x_n) $$

$$ y_{n+1}= f(x_{n+1})-x_n $$

with

$$ f(x)=\alpha x+\frac{2(1-\alpha)x^2}{(1+x^2)^2} $$

Inspiration

This explorable was inspired by Mike Bostock’s wonderful Observable on the Clifford attractor that generates super high-quality renderings of the Clifford attractor in real time.

Other sources of inspiration were Jason Rampe’s blog on 2D-attractors and Paul Bourke’s Site with really beautiful images of strange attractors.