Introduction to Chaos

(By Dr. Tariq Abdullah, Life Member KSS)

The word chaos is used in everyday life to describe behaviour that is unpredictable, disordered complicated and irregular. We use this word to describe the traffic on our roads for example. In science we try to study phenomena that have predictable behaviour and are relatively simple. We look for regular patterns and try to explain and understand these patterns. If we come across complicated behaviour in studying natural phenomena we think it is because many bodies, particles, units or elements are interacting together and producing this complicated or patternless behaviour. If we do study the properties of a system composed of many particles like a gas we reach an understanding by ignoring the motion of a single particle but consider the average properties of many particles. This gives rise to those branches of science based on statistical or probabilistic approaches. Random behaviour is then seen to emerge when studying a system composed of many particles whereas predictable behaviour is expected in systems with a small number of elements or particles. Our weather system is then an example of a random system within which there are many elements and the weather frequently behaves in an unpredictable fashion. Our planetary system is then an example of a predictable system with regular, ordered behaviour and this system just has a few elements, the handful of planets with the sun in the centre.

The discovery that gave birth to the science of chaos is that a  system with a small number of elements can give rise to a behavior that is complicated and practically unpredictable. The randomness is inherent in the system rather than introduced from outside disturbances.

The birth of the Science of Chaos came with the work  of Edward Lorenz who made a computer model for use in metrology. He discovered the Butterfly effect. This effect means that small variations in initial conditions can result in completely different behaviour of a system. It is almost like saying the flapping of the wings of a butterfly can determine whether a storm will or will not occur. The model of the weather system that Lorenz developed appears inherently unpredictable. Lorenz work shows that a model with a few interacting elements can have chaotic or unpredictable behaviour.

The weather is generally considered to be an unpredictable system. Can we have chaotic behaviour in a system that we generally think of as predictable and ordered. The planetary system is an example of such a system. The motion of the planets appears to predictable as clockwork and we can send spaceships to the nearby planets because we are able to predict their positions years in advance. The prediction of the appearance of Halley's comet is just such an example. The science of  chaos is now raising the question does and can unpredictability be found to exist within the apparent regularity of planetary motion.

What are the necessary conditions for chaotic behaviour? One of these is non-linearity. We can understand non-linearity in terms of feedback.

If we change the input into a system then the output will change. If some part of the output is fed back as input into the system we have a feedback process. Stable processes involve negative feedback. Such processes are involved in for example in maintaining our body temperature in different temperaure environments. Any disturbance in input will result only in a temporary change of output which because of negative feedback will return to its stable value. We can describe this process by saying the disturbance in input results in transient behaviour which eventually disappears and the system returns to s stable state which is also called the attractor. Unstable processes involve positive feedback. An example of a positive feedback
process concerns the longterm changes in the average earth's atmospheric temperature. An increase in temperature will result in the polar ice caps shrinking and as a result less sunlight will be reflected from the earth back into space and so the temperature of the atmosphere may further increase. This may result in an unstable process. We can thus imagine that complex feedback loops can result in complex behaviour.

Feedback processes can easily be modeled in mathematical or computer models. Computers are very good at doing iterations. Take a number, square it and add to it a constant and call this the new number. Repeat the process. This can be described by the equation

X -> C + sqrt(X)

Such an equation is called a difference equation. Scientists have carried out computer experiments with such difference equations and have found a surprising variety of behaviours. For some value of the constant the system settles down after some time to a steady state or equilibrium value. This single value is called a point attractor. For another value of the constant the system settles down to an oscillation between two values. This is called a periodic attractor. Sometimes even stranger behavior is observed. The sequence of points never repeat themselves and we have aperiodic or chaotic behaviour. This is called a strange attractor. 

Changing the constant in our difference equation is like changing the external conditions of our system and as a result we see the attractor may change. The equilibrium state or the oscillatory state are simple and ordered motions. We thus observe the pathway from order to chaos involves a change from a simple attractor to a strange attractor. An example of this change can even be seen in the study of a dripping tap as we slowly open the tap.

Scientists have studied the changes in attractor with changes in external conditions in physical systems and mathematical models. A very significant discovery was made by Mitchell Feigenbaum who showed that different systems may approach chaos in similar ways. This discovery is called Universality. It implies that simple models may be used to understand
complex behaviour.

Another example of this phenomena that simple rules can generate complex behaviour is seen in the study of fractals.The most famous fractal is the Mandelbrot set. What are fractals? These are geometrical objects with fractional dimension. A point, a line, a sheet, a ball are all objects with integral dimension. They are generally smooth objects. Most natural objects, a tree or a stone are irregular objects. A fractal is an object that has the same irregularity or crinkliness at different magnifications. Natural objects such as a snowflake or coastline are fractals. With the advancement of computers and computer graphics in particular we can generate fractal objects on a computer screen which have the same degree of complexity under magnification. The strange attractor that Edward Lorenz discovered almost thirty years ago is a fractal and since then many other strange attractors have been seen to give rise to chaotic behaviour in physical
systems and mathematical models.

The science of chaos is enabling us to understand the transition from order to chaos. The transition from regular smooth flow to turbulent flow is important in fields as diverse as the mixing of industrial gases in a chemical plant to the flow of air over the wings of a supersonic
aeroplane. With the advent of the science of chaos we expect to find islands of chaos within order and islands of order within chaos. Our planetary system is generally an ordered system. The motion of the planets appear ordered and regular. In the planetary system islands of chaos have been discovered in the motion of asteroids in the asteroid belt between Mars and Jupiter and in the motion of the moon of Jupiter called Hyperion.

The atmosphere of the planet Jupiter is highly turbulent but within this turbulent flow we have the red spot of Jupiter. It has been seen for centuries  and is an island of order within disorder. Before the advent of the tools  of chaos theory irregular behaviour was considered too complicated to study and due to perhaps external disturbances or noise within the
system. Now we see that apparently random behavior may be due to the internal dynamics of a system rather than due to external fluctuations. It is this first type of motion that is called chaotic motion. The science of chaos also teaches us to be very skeptical about long term predictions of complex systems such as the environment because of the unpredictability of chaotic systems.