One group of approaches involves replacement of the nonlinear system by (in some sense) a linear approximation. Analysis of the linear system then gives information about the original nonlinear system. The describing function method and Lyapunov's first method are in this group. A second approach attempts to find useful information about a system without solving the system equations. Lyapunov's second or direct method has such an approach to determine approximate regions of stability. A third approach uses what I have called 'envelope techniques'. Here, the nonlinearity is enveloped graphically in a linear segment and a 'worst-case' stability analysis is then possible. Numerical solution of nonlinear equations is, of course, always possible but is of limited usefulness in obtaining an overall insight into system behaviour. A useful graphical technique for the display of nonlinear system behaviour is the phase-plane. Despite the fact that the display is virtually restricted to second-order systems, it allows most of the important concepts of nonlinear analysis to be illustrated.

The describing function method can be considered as an extension to nonlinear systems of the usual Nyquist stability criterion. The origins of the method can be traced back to nonlinear mechanics (Krylov and Bogoliuboff, 1932) but the first workers to apply the approach specifically to control systems were Tustin (1947) and Kochenburger (1950).

The replacement of a nonlinear function by a linear approximation is known as linearisation. The motivation for linearisation is to allow analysis of a nonlinear problem by linear techniques. The results of such analysis need to be interpreted with care to ensure that the linearising approximation does not cause unacceptably large errors.

When stability is guaranteed by the second method of Lyapunov, the system response may still be unsatisfactory in that it is highly oscillatory and/or takes an inordinately long time to settle down. For instance, a numerical iterative loop whose convergence was guaranteed by the second method, converged so slowly as to be quite useless in practice. This aspect is particularly important when the Lyapunov method is used as a design tool. The designer must ensure that a satisfactory rate of convergence is obtained.

Oscillations were first studied as part of nonlinear mechanics and a long standing and comprehensive literature exists. In particular, there are fundamental theories concerning the existence of stable periodic solutions. Some knowledge of this mathematical background is helpful in interpreting the results of nonlinear system analyses. Conversely, if it is required to construct a mathematical model of a stable periodic system, such as a biological oscillator, it will be essential to postulate a mechanism that has the necessary properties.

Gradient systems have some particularly simple properties. They cannot exhibit oscillatory behaviour and their stability can be investigated using a natural Lyapunov function that can always be found by integration. An arbitrary dynamic system can be represented as the sum of a gradient system and a conservative system. An analysis of the system in its decomposed form is shown to lead to a Lyapunov-like algorithm. The algorithm can be applied to yield directly a graphical indication of the qualitative behaviour of nonlinear second-order systems.