A case of impulsive singularity

Abstract

The paper considers an impulsive system with singularities. Different types of problems with singular perturbations have been discussed in many books. In Bainov and Kovachev’s book [4]several articles cited therein consider impulse systems with small parameter involving only differential equations. The parameter is not in the impulsive equation of the systems. In our present the small parameter is inserted into the impulse equation. This is the principal novelty of our study. Furthermore, for the impulsive function, we found a condition that prevents the impulsive function to blow up as the parameter tends to zero. So we have significantly extended the singularity concept for discontinuous dynamics.

The singularity of the impulsive part of the system can be treated in the manner of perturbation theory methods. This article is a continuation of [1] work. In our present research, we apply the method of the paper [1]. Our goal is to construct an approximation with higher accuracy and to obtain the complete asymptotic expansion. We construct a uniform asymptotic approximation of the solution that is valid in the entire close interval by using the method of boundary functions [22]. An illustrative example using numerical simulations is given to support the theoretical results.