A Consistency Theory of Singular Perturbations of Differential Equations

Source abstract

Many physical systems are described by singularly perturbed differential equations. It is known that good approximations of the solutions can be obtained by neglecting the perturbations [14]. This procedure can be justified using the results of A. N. Tihonov ([15] to [17]) and subsequent generalizations ([1] to [4], [7], [8] and [10], [11]). In this paper we introduce concepts of consistency. These are extensions of total stability and were set up in order to give a formal description of the fact that the perturbations can be neglected. These concepts have been investigated using auxiliary functions of Lyapunov type. We obtain theorems that generalize several known results [1], [7], [11], [17], and are expected to handle many practical situations.