Dynamical systems

All the pictures in the gallery are illustrations of some sort of (mathematical) dynamical system. In abstract terms a dynamical system (X, f) is defined as a set X, together with a function f : X  → X which maps points of X to other points of X. As a convenient notation we use f n(x) to mean the function f applied n times to the variable x, so f 3(x) = f(f(f(x))). We can think of a function f as moving the points of X around, so if we take a point x0 as a starting point and iterate (apply) f repeatedly, putting x1 = f(x0), x2 = f(x1) = f 2(x0), . . . , xn = f(xn−1) = f n(x0), . . . then the sequence of iterates {x0, f(x0), f 2(x0), . . . , f n(x0), . . . } = {x0, x1, x2, . . . , xn, . . . } is known as the orbit of x0 under f. We put f 0(x0) = x0 and use {f n(x0)} as a convenient shorthand for the orbit of x0 under f. Orbits are the paths or trajectories mapped out (in X) by repeated iterations of f.

If an orbit returns to its starting point x0 after a finite number of iterations then x0 is known as a periodic point. Either a point is periodic or it isn't. If we have some way of representing points of X as pixels in a digital image, then by testing whether a point is periodic or not, and colouring the corresponding pixel black or white respectively, provides us with a simple method for creating a black and white image using the dynamical system.

As another example, suppose we iterate some random orbits and count the number of times, n(x) say, each point x is landed on by an orbit. Then we can use the value of n(x) to assign a colour to the pixel corresponding to x.

The point of these examples is to show that even for a general dynamical system (X, f), where no extra structure like position or distance is assumed, the idea of producing a picture of it occurs naturally as soon as we start considering the mathematical nature of its dynamics.

In fact all the pictures in the gallery are snapshots of specific dynamical systems, where the colour of the pixels is directly (or indirectly) related to the way the corresponding points are moving and this is the reason we call the pictures “Dynamical Designs”. The dynamical systems used are all 2-dimensional, lying in the real or complex plane (or a close relative of them) and each of them is from one of the following three general categories.

Trigonometric functions (T)

Complex dynamics (CD)

Directed graph iterated function systems (DGIFSs)

This list is by no means exhaustive as there are many other types of dynamical system that can be used to make interesting pictures. Also these categories are not mutually exclusive either. For example it is possible for some Julia sets, which can be constructed as repellers in the CD category, to also be constructed as standard IFS attractors in the DGIFS category. However we have labelled each picture T, CD or DGIFS, in order to show the actual dynamical system that was modelled by our computer program in the creation of the picture. The pictures labelled CD and DGIFS are all images of fractals. The numbers in the labels have no special significance and lower-case letters are added only to group pictures together whenever they are variations on a theme.

All the dynamical systems we use are examples of discrete dynamical systems where points make discrete jumps from one point to the next along an orbit. This behaviour is particularly apparent in the CD category where points can jump about by large distances on the iteration of a complex function. It is not so obvious in the T category where the jumps are so small that the behaviour approximates a continuous dynamical system, with points appearing to flow smoothly along (almost) continuous paths. For the DGIFS category the underlying space X is of a different nature to the categories T and CD, although it remains closely related to either the real or complex plane. Even so, all the pictures labelled DGIFS are still pictures of discrete dynamical systems with points moving discrete distances along their orbits.