Academic Journal of Applied Mathematical Sciences ISSN(e): 2415-2188, ISSN(p): 2415-5225 Vol. 4, Issue. 8, pp: 77-89, 2018 URL: http://arpgweb.com/?ic=journal&journal=17&info=aims Academic Research Publishing Group *Corresponding Author 77 Original Research Open Access Equivalent Construction of Ordinary Differential Equations from Impulsive System I. M. Esuabana Department of Mathematics, University of Calabar, Calabar, Cross River State, Nigeria U. A. Abasiekwere * Department of Mathematics and Statistics, University of Uyo, Uyo, Akwa Ibom State, Nigeria J. A. Ugboh Department of Mathematics, University of Calabar, Calabar, Cross River State, Nigeria Z. Lipcsey Department of Mathematics, University of Calabar, Calabar, Cross River State, Nigeria Abstract We construct an ordinary differential equation representation of an impulsive system by a bijective transformation that structurally maps the solutions of the initial value problem of the impulsive differential equations to the solutions of the initial value problems of the ordinary differential equations. Established in this work is the relationship between impulsive differential equations and ordinary differential equations which play a fundamental role in qualitative analysis of the former. It is also established that an dimensional impulsive differential equation can be represented in terms of a dimensional ordinary differential equation. Figures are used to demonstrate the practicability of the methodology developed. Keywords: Stability; Impulsive; Ordinary differential equations; Bijective transformation. CC BY: Creative Commons Attribution License 4.0 1. Introduction Various evolutionary processes from fields such as population dynamics, aeronautics and engineering are characterized by the fact that they undergo abrupt changes of state at certain moment of times between intervals of continuous evaluation. Since the duration of this changes are often very small compared to the total duration of the process, such changes can be reasonably well approximated as being instantaneous changes of state, or impulses. These processes tend to be more suitably modelled by impulsive differential equations, which allow for discontinuities in evaluation of the state. Impulsive differential equations are usually defined by a pair of equations, an ordinary differential equation to be satisfied during the continuous portion of evolution and a difference equation defining the discrete impulsive actions. Impulsive differential equations seem to have received very little attention not until in 1980s when interest in the area began to gather momentum. Among the earliest articles on impulsive differential equations was a seminar paper by Milman and Myshkis [1] where they considered differential equations with impulses occurring when certain spatio-temporal relations were satisfied [2-4]. Research into impulsive differential equations had culminated in the publishing of several monographs and articles [5-8]. These authors consider an impulsive differential equation to be an ordinary differential equation coupled with a difference equation to be satisfied at certain fixed or variable impulse times. The resulting solutions are thereby piecewise continuous with discontinuities occurring at these impulse times. This approach enabled them to apply many well established results for ordinary differential equations to these systems in order to develop the qualitative theory of impulsive differential equations which is still at its infancy. A few recent results in this new area can be found in [9-14]. Due to the nature of impulsive processes which are momentarily exposed to harsh impacts, their qualitative analysis is more complicated than that of ordinary differential equations. This work seeks to fill this gap by formulating a differential system that is equivalent to the impulsive system, thereby simplifying the analysis of the later. To help in our investigation, we will define some important concepts.