A second technique, often called "hopalong", is normally used to represent the strange attractor of a chaotic system, for example, the well known Julia set. In this case each coordinate generated by the series is drawn as a small point, ie: we hop-along from one point to the next. For an image on a plane the series is either an equation of a complex variable or else there are two interrelated equations, one for the x and one for the y coordinate. As an example consider the following function:
This series of x,y coordinates is specified by an initial point xo,yo and three constants a,b, and c. The following is an example where a=0.4, b=1, and c=0.
Interestingly for strange attractors the initial point does not matter (except for a few special cases), ie: all initial coordinates xo,yo result in the same image. In other words, the image shows the x,y pairs that can be generated by the series, any initial point will generate the same set of points although they will be generated in a different order. Another example attributed to Peter de Jong uses the two equations
This gives swirling tendrils that appear three dimensional, an example is shown below where a = -2.24, b = -0.65, c = 0.43, d = -2.43.