Chaos Theory - Background

Chaos Theory, as understood in mathematics and physics, deals with the behavior of certain nonlinear dynamic systems that, under certain conditions, exhibit the phenomenon known as chaos. Systems that exhibit mathematical chaos are considered to be deterministic (i.e. not determined randomly) and therefore to some extent orderly.

Examples of such chaotic systems include the atmosphere, the solar system, plate tectonics, turbulent fluids, economies, and population growth. However Chaos Theory is perhaps most famously characterised by sensitivity to initial conditions, which implies that two such systems with however small a difference in their initial state eventually will end up with a finite difference between their states, and that two deterministic systems with identical initial conditions will remain identical.

An example of such sensitivity is the so-called butterfly effect, whereby the flapping of a butterfly's wings is imagined to create tiny changes in the atmosphere which over the course of time cause it to diverge from what it would have been and potentially cause something as dramatic as a hurricane or tornado to occur.

It should be noted that the technical use of the word chaos is at odds with common usage (which suggests complete disorder). It should also be distinguished from the related field of physics known as Quantum Chaos Theory which studies non-deterministic systems following the laws of Quantum Mechanics.

Chaos Theory History

The origins of chaos theory date back to around 1900 and the studies of Henri Poincaré on the problem of the motion of three objects in mutual gravitational attraction. Poincaré found that there can be orbits which are nonperiodic and yet not eternally increasing nor approaching a fixed point.

From the mid 20th century, developments in Chaos Theory progressed more rapidly when it first became evident for some scientists that conventional linear theory could not adequately explain the observed behavior of certain experiments, such as those found found in logistic mapping. However it was with the advancement of computer technology that Chaos Theory was able to develop further, since much of its mathematics involves the repeated iteration of simple mathematical formulas, impractical to perform manually.

An early pioneer of Chaos Theory was Edward Lorenz whose interest in chaos came about accidentally through his work on weather prediction in 1961. Lorenz was using a computer to run his weather simulation and wanted to see a sequence of data again, so to save time he started the simulation in the mid-course, taking the input data from a printout of the data from the initial analysis. To his surprise, the weather that the machine began to predict was completely different from the weather calculated on the first run. Lorenz tracked this apparent anomoly down to the computer printout, which had rounded variables off to 3 significant figures, whilst the computer worked internally with a precision of 5 significant figures. This difference is tiny and the consensus at the time would have been that it should have had practically no effect. However Lorenz had discovered that small changes in initial conditions produced large changes in the long-term outcome.

The term chaos as used in mathematics and physics was coined by the applied mathematician James A Yorke in a paper published in 1975.

Strange Attractors

One way of visualizing motion of any type is to make a phase diagram of the motion. In such a diagram, time is implicit and each axis represents one dimension of the state. For example, a system at rest will be plotted as a point and a system in periodic motion will be plotted as a simple closed curve.

A phase diagram for a given system will depend upon a set of parameters including its initial state, and such diagrams frequently reveal that the system exhibits the same motion for all initial states in a region around the motion, almost as though the system is attracted to the motion. Such attractive motion is appropriately termed an attractor for the system, and is found very frequently in what are known as forced dissipative systems.

However, whilst many types of motion give rise to very simple attractors, such as points and circle-like curves, termed limit cycles, chaotic motion gives rise to what are termed strange attractors - attractors possessing great detail and complexity. For instance, a simple three-dimensional model of the Lorenz weather system gives rise to what has become known as the Lorenz attractor - one of the best-known chaotic system diagrams, resembling the wings of a butterfly.

Strange attractors are considered to have a fractal structure. The term fractal was coined in 1975 by Benoît Mandelbrot, from the Latin word fractus (meaning "broken"), and refers to a geometric object which is rough or irregular on all scales of length, and so appears to be completely broken up. However, a characteristic of many fractals is that they can be divided into parts, each of which is similar to the original object. Fractals are said to possess infinite detail, and they may actually have a self-similar structure that occurs at different levels of magnification. In many cases, a fractal can be generated by a repeating pattern, in a typically recursive or iterative process.

The above definition of fractal geometry leads us on to consider Chaos Theory and fractal geometry as a way to model the movements of the financial markets.

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