Linear and nonlinear systems

Whatever the shortcomings of conventional modelling, a wide range of physical systems can, in fact, be satisfactorily approximated as regular and continuous. This can be often traced to a crucial property known as linearity.

A linear system is one in which cause and effect are related in a proportionate fashion. As a simple example consider stretching a string of elastic.

Figure 1. The length of an elastic string, y, is said to be 'linearly' related to the stretching force, x, when the graph of y against x is a straight line. For a real string non-linear behaviour sets in when the stretching becomes large.

If the elastic stretches by a certain length for a certain pull, it stretches by twice that length for twice the pull. This is called a linear relationship because if a graph is plotted showing the length of the string against the pulling force it will be a straight line (Figure 1). The line can be described by the equation y=ax+b, where y is the length of the string, x is the force, and a and b are constants.

If the string is stretched a great deal, its elasticity will start to fail and the proportionality between force and stretch will be lost. The graph deviates from a straight line as the string stiffens; the system is now non-linear. Eventually the string snaps, a highly non-linear response to the applied force.

A great many physical systems are described by quantities that are linearly related. An important example is wave motion. A particular shape of wave is described by the solution of some equation (mathematically this would be a so-called differential equation, which is typical of nearly all dynamical systems). The equation will possess other solutions too; these will correspond to waves of different shapes. The property of linearity concerns what happens when we superimpose two or more waves. In a linear system one simply adds together the amplitudes of the individual waves.

Most waves encountered in physics are linear to a good approximation, at least as long as their amplitudes remain small. In the case of sound waves, musical instruments depend for their harmonious quality on the linearity of vibrations in air, on strings, etc. Electromagnetic waves such as light and radio waves are also linear, a fact of great importance in telecommunications. Oscillating currents in electric circuits are often linear too, and most electronic equipment is designed to operate linearly. Non-linearities that sometimes occur in faulty equipment can cause distortions in the output.

A major discovery about linear systems was made by the French mathematician and physicist Jean Fourier. He proved that any periodic mathematical function can be represented by a (generally infinite) series of pure sine waves, whose frequencies are exact multiples of each other. This means that any periodic signal, however complicated, can be analysed into a sequence of simple sine waves. In essence, linearity means that wave motion, or any periodic activity, can be taken to bits and put together again without distortion.

Linearity is not a property of waves alone; it is also possessed by electric and magnetic fields, weak gravitational fields, stresses and strains in many materials, heat flow, diffusion of gases and liquids and much more. The greater part of modern science and technology stems directly from the for tunate fact that so much of what is of interest and importance in modern society involves linear systems. Roughly speaking, a linear system is one in which the whole is simply the sum of its parts. Thus, however complex a linear system may be it can always be understood as merely the conjunction or superposition or peaceful coexistence of many simple elements that are present together but do not 'get in each other's way'. Such systems can therefore be decomposed or analysed or reduced to their independent component parts. It is not surprising that the major burden of scientific research so far has been towards the development of techniques for studying and controlling linear systems. By contrast, nonlinear systems have been largely neglected. In a non-linear system the whole is much more than the sum of its parts, and it cannot be reduced or analysed in terms of simple subunits acting together. The resulting properties can often be unexpected, complicated and mathematically intractable.

In recent years, though, more and more effort has been devoted to studying non-linear systems. An important result to come out of these investigations is that even very simple non-linear systems can display a remarkably rich and subtle diversity of behaviour. It might be supposed that complex behaviour requires a complex system, with many degrees of freedom, but this is not so. We shall look at an extremely simple non-linear system and find that its behaviour is actually infinitely complex.

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