View from the Pennines: Analysis on Time Scales

To be truly accepted in the village you need to be able to trace back your local roots for several generations. Everyone else is referred to as a ‘comer-in’. The recent redevelopment of an abandoned mill has created an influx of comers-in, with some good and some bad effects. The moors and valley also have their share of botanical comers-in: rhodedendrons have occupied some moorland areas, and Himalayan balsam spreads speedily along the canal paths. Both these plants originate in the mountain ranges of China and the high moorland hereabout suits them. They are beautiful plants in their different ways but the fact that they are not local causes some concern. For the past week I have been wrestling with what I am beginning to regard as a mathematical comer-in: analysis on time scales. Recently it has been the subject of a cover story in New Scientist [9], special issues of two international journals (Journal of Computational and Applied Mathematics [1] and Dynamic Systems and Applications, see [6]) and two books [3, 4]. Our library does not have the more recent book [4] and the introductory book [3] has been permanently on loan, but the other sources have in turn intrigued and infuriated me. The basic idea seems reasonable and interesting. There are many similarities betwen results of differential calculus and discrete calculus. For example, the continuous time differential equation dx dt = ax has solution ex0, whilst the analogous discrete time equation x(t + 1) − x(t) = ax(t) with t ∈ Z has solution (1 + a)x0. Provided 1 + a > 0 this can be written as ex0 where b = ln(1+a), so is there a formalism (or theory) in which the different cases can be treated uniformly? In the late 1980s, Stefan Hilger, then a graduate student at Augsburg in Germany, developed just such a formalism [7]. Using Hilger’s ideas, applied mathematicians have started to develop and apply the theory. Many of the applications to date are from the biological sciences, since the techniques seem particularly well adapted to periods of latency or dormancy in systems, so that the dynamics is a mixture of continuous time development and discrete time development [9].