Chaos theory |
(a) Introduction The traditional approach to science has been to find the simplest laws that enable sense to be made of the world around us. Newton's laws of motion are a good example. However they have been derived for an idealized situation in which practical complications such as friction and air resistance are largely ignored. In mathematical terms, they deal with linear systems in which the equations expressing their behaviour involve directly proportional relationships between quantities. Such equations are readily solved and so enable predictions about future behaviour to be made. This is the Newtonian 'mechanistic' or 'deterministic' view in which the universe is regarded as a piece of clockwork (see p. 167) whose future is determined by the past. It held sway for 200 years before it was challenged early in the twentieth century by two developments. The first was the theory of relativity, which shattered the idea of absolute space and time. The second was quantum mechanics, with its 'uncertainty principle' which stated that it was impossible to measure both the position and the velocity of a particle simultaneously. For most of the last 100 years the exploration of the building blocks of matter and the forces of nature has been spear-headed by particle physicists. They have attempted to reduce our understanding of the physical world to explanations in terms of its constituent entities such as quarks and gluons. While this 'reductionist' approach has had its successes, it has been accompanied by narrow specialization. Some scientists have felt that broader questions relating to everyday problems and systems have been by-passed and left unanswered. Many of these are undoubtedly complex and apparently lacking in any kind of order. They involve non-linear equations that are notoriously difficult to solve. For these problems it would appear that a quite different approach is needed. (b) The chaologist's approach It is against the above background that chaos theory emerged in the 1960s and 70s. It developed for three reasons: (i) the availability of computers for repeating thousands of calculations again and again (and so taking a system through a great many steps); (ii) the existence of a new type of geometry called topology (based on work done by the French mathematician Poincaré at the end of the 19th century): and (iii) an increase of scientific interest in non-linear systems where disorder and unpredictability, i.e. chaos (in a scientific sense but not 'mess' as the everyday meaning of the word suggests), have been shown to be common characteristics. The chaologist's approach involves creating a mathematical model of the system under study (e.g. the atmosphere) in the form of equations (usually non-linear), containing the variable quantities that affect its behaviour (e.g. temperature, pressure, humidity, wind speed and direction, etc.). The equations and data are then fed into a computer which is programmed to perform a large number of similar operations, i.e. to iterate, resulting in an output that represents the final predicted state of the system (e.g. a weather forecast). How close the prediction is to reality depends on how good a likeness the model is to the actual system. The output is commonly in the form of computer graphics that are colourful shapes which repeat themselves on a smaller and smaller scale- an effect known as 'self-similarity'. In 1975 Mandelbrot, an American mathematician and physicist, who has played a major role in developing chaos theory, proposed the word fractal (from the Latin for broken for such shapes. One adorns the front cover of this book,as they do on many T-shirts, posters and videos. A tree is a natural fractal. The formation of a fractal from a triangle is shown in Fig. 25.32 This process can be repeated indefinitely. For those versed in the new geometry, conclusions can be drawn from the shape and repeating pattern of the computer graphics, enabling limited predictions to be made about the behaviour of the system. A feature of chaotic systems revealed by this approach is that it shows how many changing systems are greatly affected by their initial state. Even if the states are very similar to start with, they rapidly diverge. This is called the 'butterfly effect'. In the context of weather forecasting, the often-quoted statement about the flapping of a butterfly's wings in the Amazonian jungle causing a tornado in Texas, derives from this 'sensitivity to initial conditions' of chaotic systems.
(c) Applications of chaos theory Chaos theory, though still in its infancy, now permeates almost every branch of science and has highlighted the interdisciplinary aspect of modern research. It has brought together the computer, one of today's chief research tools, and abstract mathematics. There now exists a way of probing complex dynamical problems of the real world and finding patterns where disorder and unpredictability, i.e. chaos, formerly appeared to reign. It attempts to bridge the gap between knowing what one item (e.g. atom, cell) will do and what millions do. Initially chaos theory emerged from the work on meteorology in the early 1960s by Lorenz at the Massachusetts Institute of Technology, where he discovered the 'butterfly effect'. Today the study of the atmosphere's movements, which determines our weather, is an important application. The theory is used by biologists dealing with changing bird and insect populations and with the action of cells. Medical researchers use it to help estimate the spread of diseases such as AIDS. Chemists use it to study certain catalytic reactions that can behave oddly while physicists employ it to gain insight into the motion of electrons in atoms and the loss of particles in accelerators. Not all electrical circuits behave predictably and electronic engineers find the theory helps in understanding their unusual behaviour. The way nuclear power stations rock during earthquakes, the reaction of offshore oil platforms to large waves and the sudden disappearance of ships in high seas are all investigated by engineers using chaos theory. Such events usually involve non-linear dynamical systems. The complex fluctuations of financial and commodity markets are studied by economists using the theory. Astronomers and cosmologists try, on the one hand, to model the early stages of the birth of the universe using chaos theory and, on the other, to predict how it will end. Their conclusions so far seem to suggest that, since it is a chaotic system, prediction of its ultimate fate can only be very limited. It is very much an open system whose future does not necessarily depend on its earlier states, as did the 'deterministic' Newtonian universe. (d) Pendulums, order and chaos Pendulums have long been regarded as providing classical examples of simple, orderly motion that can be used to control clocks because of their regularity. However, a closer look reveals that this is not always so. (i) Simple pendulum. The period of a simple pendulum of a certain length is constant for all angles of swing so long as they are small (p.179). If the angle gets too large, the period depends on it and varies, due to the gravitational force pulling the bob to the vertical position no longer being directly proportional to the angle of swing. The equation describing its motion is then non-linear and its period becomes unpredictable, i.e. it exhibits the chaotic behaviour characteristic of non-linear systems. (ii) Driven conical pendulum. One arrangement is shown in principle in Fig. 25.33a, p. 524. The point of suspension of the string oscillates horizontally in a straight line when the motor (whose speed has to be precisely controlled) sets the eccentric into rotation. The amplitude of the oscillation is small compared with the length of the pendulum. Two very different types of motion of the bob can be observed. The first type occurs if the driving frequency is just greater than the pendulum's natural frequency, when it swings freely; it is non-chaotic. Initially the bob moves parallel to the drive, gaining in amplitude since near-resonance occurs due to the closeness of the driving and natural frequencies. Subsequently the swings acquire a component perpendicular to the drive, causing the bob to settle into an almost circular path. It makes one complete orbit during one period of the drive. This motion, which is maintained for as long as the system is driven, is orderly and predictable except that it is impossible to say beforehand whether the circulation will be clockwise or anti-clockwise.
The second type of motion occurs when the driving frequency is just less than the natural frequency; it is chaotic. Fig. 25.33b shows some of the many possible paths described by the bob; in general, each one is an ellipse but the length, width, orientation and direction of rotation vary with time. There is no observable order to the changes. Furthermore, if the experiment is repeated, the sequence of orbits followed is quite different and unpredictable. If we wished to study the unstable chaotic behaviour shown in the second case using chaos theory, we would have to make detailed observations of the bob's motion over say a period and feed these into a computer programmed with the system's equations of motion. The predictions about future motion would agree with the actual motion initially but eventually it would fail. The more exact the observations fed to the computer, the longer would agreement last but in the end the observed motion would differ from that predicted. The discrepancy escalates very rapidly as time passes and militates against reliable prediction. Unfortunately we can never know precisely how the bob is moving at a certain time since a slight puff of air or an external vibration could alter it. And as the 'butterfly effect' tells us, the smallest change in a chaotic system affects its behaviour profoundly. The 'sensitivity to initial conditions' is the feature which distinguishes such systems from well ordered ones.
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[From "Advanced Physics" by Tom Duncan]
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