This is a lively and interesting survey of a lively and interesting area of current research, namely the application of non-standard analysis to singular perturbation problems for ordinary differential equations. In particular, the authors discuss certain singular solutions (called ”ducks”) of nonlinear second-order equations depending on a small parameter that behave like the classical relaxation oscillations of the van der Pol equation.
The paper opens with a brief introduction to the basic ideas of non- standard analysis which are then illustrated by non-standard formulations of some important results from the theory of ordinary differential equations, including the existence and uniqueness theorem and the Poincaré-Bendixson theorem. This is followed by a discussion of the van der Pol equation using both standard and non-standard techniques.
The remainder of the paper is devoted to a study of duck solutions. A duck is a solution of a singularly perturbed equation that behaves like the cycle of the van der Pol equation in the Liénard phase plane, that is, it is a trajectory of a fast-slow vector field \((\dot x=f(x,y,\epsilon)\), \(\dot y=\epsilon^{-1}g(x,y,\epsilon)\) as \(\epsilon\to 0)\) which at first moves along the attracting part of the slow curve and then rapidly along the repelling part of the fast curve. Both periodic and non-periodic duck solutions are studied, and the paper contains a number of interesting and useful examples.
Finally there is a discussion of the original contributions of the Strasbourg school that pioneered the non-standard study of these and other asymptotic phenomena, and the bibliography of ninety references is complete and up-to-date. All in all, this paper is an important contribution to the singular perturbation literature that conveys the utility and the beauty of the non-standard approach.