Nonlinear Dynamics

Abstract

The term ‘nonlinear dynamics’ covers a multitude of different phenomena. Most real-life systems are nonlinear to a greater or lesser degree, but they do not receive the space they really deserve owing principally to the lack of methods available for their complete mathematical analysis. Linear systems are much more readily solved. It is true that there are systems that are naturally linear. For example, motion under gravity without resistance or with resistance proportional to speed is linear, as is orbital motion under the inverse square law. Vibrations and other oscillatory motions are approximated by linear differential equations, and of course these approximations are well worth doing and lead to very useful models. The danger is in overdoing the amount that can be deduced, thinking that linearity is reality. It isn’t. In this chapter we introduce methods that can be used with nonlinear differential equations, in particular those that arise from real problems. The systematic study of nonlinear differential equations is well beyond the scope of this text (see, for example, the excellent text by D. W. Jordan and P. Smith, Nonlinear Ordinary Differential Equations, Oxford University Press, 2nd edn, 1989, which happens to be on my shelves for an exposition on the subject). Here we shall give only a few practical examples and define enough new methods to be able to cope with them. In recent times there has been much public attention on nonlinear mathematics, not least because we have just acquired the means to draw elegant pictures (for example, the Mandelbrot set ‘gingerbread man’) and make executive toys (for example, two magnets between which is an iron pendulum bob suspended with a steel wire which when set swinging dodges about in a seemingly random fashion). The principles upon which these rest lie squarely in the realm of nonlinear mathematics and are given a brief airing at the end of this chapter. To start however, we introduce the phase plane which may be familiar to some readers. The example used to introduce it will be familiar to all.