Some information on chaos
Mathematical chaos has been a hot topic in recent years. If you have seen Jurassic Park, you probably know a little about it already. Chaos is a property of some deterministic systems. A deterministic system is one in which future states depend strictly on the current conditions. They can be modelled by dynamical systems. Historically the idea has been that all processes ocurring in the universe are deterministic, and that if we knew enough of the rules governing the behavior of the universe and had measurements about its current state we could predict what would happen in the future. These ideas have been applied with a great deal of success to falling objects, tide prediction, and many other systems. However, there have always been things which we have never been able to predict, things such as the weather. This was written off for a long time as being due to our incomplete knowledge of the system. With the development of chaos however, a new idea has emerged.
The essence of chaos is something called sensitive dependence on initial conditions. For the systems in which our ability to make predictions has been good, only a reasonable approximation of the inital state is neccessary for prediction. The paths, or orbits, of inital conditions which are nearby to one another stay close together. A reasonable approximation of the current state yeilds a reasonable approximation of the future state. With sensitive dependence, this is not the case. In a system which exhibits sensitive dependence on initial conditions, reasonable approximations of the initial state do not provide reasonable approximations of the future state. The orbits of nearby initial conditions diverge until one can no longer discern any indication that they were once similar to one another. In order to make useful long term predictions in such a system, one needs measurements of initial conditions with infinite accuracy, which are impossible to obtain. This means that systems with a deterministic underpinning can generate behavior which seems random and unpredictable. A good example of such a system is the logistic map
The idea of deterministic chaos did not come to full fruition until after the advent of the computer, which could be used to model systems which were previously unapproachable by traditional methods. One of the most common methods of modelling a system is by way of a phase space diagram. A phase space diagram is a diagram created by plotting the various dependent variables of a system against one another. For example, one could create a phase space diagram by plotting a moving object's position versus its velocity in two dimensions, or even by plotting its position vs. velocity vs. acceleration in three dimensions. The end result is something that looks like a single point moving along a path through space (this is the orbit that I spoke of earlier). This orbit can do several things: it can settle down to one point and stop, it can travel around in a circle, it can exhibit behavior which is predictable but not quite repetitive. It can also behave chaotically by travelling unpredictably on a path within a certain region of space, known as a chaotic attractor. Chaotic attractors can take may forms, but they all fall into a category of objects known as fractals.